Integration of In Vitro and In Vivo Models to Predict Cellular and Tissue Dosimetry of Nanomaterials Using Physiologically Based Pharmacokinetic Modeling

Abstract

Nanomaterials (NMs) have been increasingly used in a number of areas, including consumer products and nanomedicine. Target tissue dosimetry is important in the evaluation of safety, efficacy, and potential toxicity of NMs. Current evaluation of NM efficacy and safety involves the time-consuming collection of pharmacokinetic and toxicity data in animals and is usually completed one material at a time. This traditional approach no longer meets the demand of the explosive growth of NM-based products. There is an emerging need to develop methods that can help design safe and effective NMs in an efficient manner. In this review article, we critically evaluate existing studies on in vivo pharmacokinetic properties, in vitro cellular uptake and release and kinetic modeling, and whole-body physiologically based pharmacokinetic (PBPK) modeling studies of different NMs. Methods on how to simulate in vitro cellular uptake and release kinetics and how to extrapolate cellular and tissue dosimetry of NMs from in vitro to in vivo via PBPK modeling are discussed. We also share our perspectives on the current challenges and future directions of in vivo pharmacokinetic studies, in vitro cellular uptake and kinetic modeling, and whole-body PBPK modeling studies for NMs. Finally, we propose a nanomaterial in vitro to in vivo extrapolation via physiologically based pharmacokinetic modeling (Nano−IVIVE−PBPK) framework for high-throughput screening of target cellular and tissue dosimetry as well as potential toxicity of different NMs in order to meet the demand of efficient evaluation of the safety, efficacy, and potential toxicity of a rapidly increasing number of NM-based products.

Recent rapid advancements in nanotechnology have enabled efficient synthesis of nanomaterials (NMs) or nanoparticles (NPs) with a variety of physicochemical properties for several different applications, such as consumer products, industrial, and medical applications.¹⁻³ As the usage of NMs increases, human exposure to these materials inevitably intensifies. The effect of NM exposure on human health could be diagnostic, therapeutic, or detrimental depending on the NM itself (usually related to its physicochemical properties), the drug that the NM carries, the external exposure dose, frequency and duration, the internal dose at the target organ and cells, as well as the sensitivity of the species, organs, and cells.⁴⁻⁶ Many animal studies have shown that overexposure to NMs can cause adverse effects on different organ systems, such as the lungs, liver, and blood.⁶⁻⁸ A few human studies have also associated accidental or occupational exposure to NMs with adverse effects.⁸⁻¹⁰ The increasingly widespread use of NMs and the reported toxicity data have caused concern on the potential risk of NMs and raised a need to develop strategies to properly assess the risk of NMs.

Multiple studies have attempted to develop methods for risk assessment of different NMs, including titanium dioxide, gold, and silver NPs.^4,11−13 However, existing methods/models are limited to a certain type of NMs, and some of the models have not been adequately validated due to a lack of in vitro or in vivo data. High variations in the type and physicochemical properties of NMs, the type of target cells, and the number of cells in the body, as well as the route of exposure, create an impossible task to perform risk assessment of different NMs using conventional animal toxicity data-based approaches. To address this challenge and to keep up with the rapid expansion of NMs, mechanistic computational models that can efficiently predict NM in vitro and in vivo dosimetry based on its physicochemical properties as well as extrapolate in vitro results to in vivo are needed.

One approach that can help address the challenge of NM risk assessment is to build in vitro kinetic models to accurately quantify or predict target cellular dose based on physicochemistry, and then extrapolate in vitro kinetics and toxicity to in vivo via a whole-body physiologically based pharmacokinetic (PBPK) model.¹⁴⁻¹⁷ Multiple in vitro kinetic or whole-body PBPK modeling studies have been performed for different NMs based on either in vitro or in vivo dosimetry, with very few models that can integrate in vitro with in vivo.^13,18,19 In this review article, we discuss: (1) the current progress in in vivo pharmacokinetics (PK) or toxicokinetics (TK), in vitro kinetic modeling, and whole-body PBPK modeling as well as in vitro to in vivo extrapolation (IVIVE) approaches; (2) a NM in vitro to in vivo extrapolation via physiologically based pharmacokinetic (Nano-IVIVE-PBPK) framework for high-throughput screening of the target cellular and tissue dosimetry as well as potential toxicity of different NMs solely based on their physicochemical properties; and (3) the challenges and our future perspectives in this field.

In line with the objectives of this review article, the manuscript is categorized according to different sections: (1) in vivo PK/TK, (2) in vitro kinetic modeling, (3) in vivo PBPK modeling, (4) IVIVE, and (5) challenges and future perspectives. Within each section, we will review the state of the art in each area. In the in vivo PK/TK section, we will introduce the PK of NMs that have been widely researched in vivo, including metal-based nanoparticulate systems, polymer-based NPs, and lipid-based NPs. More specifically, we will focus on gold nanoparticles (AuNPs), poly(lactic-co-glycolic) acid (PLGA) NPs, and liposomes as representative NPs for different nanocarrier systems owing to their demand in clinical use as approved biomaterials or biomaterials at various stages of clinical development,²⁰⁻²⁴ thus human health risk assessment of these NPs is relevant. In the sections of in vitro kinetic modeling, in vivo PBPK modeling, and IVIVE, we will comprehensively review existing studies on almost all types of NPs as these sections are the focus of this review. In the last section, besides summarizing the challenges and sharing our future perspectives in the field, we will also highlight how this review article furthers the knowledge and understanding of the field.

Pharmacokinetics/Toxicokinetics of Nanoparticles In Vivo

PK or TK characteristics are important in the evaluation of the safety, efficacy, and potential toxicity of nanoparticulate systems, whether through intended (e.g., drug nanocarrier) or accidental exposure. PK or TK models are also important tools in human health risk assessment as these models can be used to predict the target organ dosimetry of the modeled substance, which is directly related to its toxicity risk. PK or TK parameters that describe absorption, distribution, metabolism, and excretion (ADME) properties of a xenobiotic include absorption rate constant (K_a), half-life (t_1/2), clearance (Cl), volume of distribution (V_d), mean residence time (MRT), and area under the concentration curve (AUC).

Gold Nanoparticles

In metal-based NP systems, plasmonic AuNPs have been widely studied because of their variety of diagnostic and therapeutic functionalities for biomedical applications.^25,26 Moreover, gold is highly biocompatible, which promises successful biomedical applications of AuNPs profiting from plasmonic or other effects.²⁷⁻²⁹ Dubaj et al. studied the PK, biodistribution, and potential toxic effects of polyethylene glycol (PEG)-coated AuNPs after a single intravenous injection. Rats were injected with a single dose of 0.7 mg/kg body weight, and the concentrations of PEG-AuNPs were examined at 1 h, 4 h, 24 h, 7 days, and 28 days postexposure.¹⁹ They analyzed the kinetics of gold clearance using a two-compartment PK model, which revealed a rapid drop in the initial amount with a biodistribution half-life of 1.56 h. In contrast, the elimination phase showed a considerably longer half-life of 57.0 h. More than 80% of the PEG-AuNPs injected dose was eliminated from circulation within 24 h. Accumulation of AuNPs in blood, lungs, and kidneys were found to be decreased, whereas accumulation in the liver (1.546 μg/g in 1 h to 2.153 μg/g in 28 days) and spleen (2.693 μg/g in 1 h to 4.410 μg/g in 28 days) were found to be increased. In another study by Cho et al., AuNPs with similar chemistry but smaller in size (13 nm) were studied using 6-week-old male BALB/c mice.³⁰ They found a significant increase in liver inflammation with apoptosis of liver hepatocytes, which is an indication of NP-induced toxicity. The percentages of retained NPs in the liver and spleen at 7 days post-treatment ranged from 39% to 45% and from 12% to 20% for doses of 0.85 and 4.26 mg/kg, respectively. These AuNPs were found trapped in liver Kupffer cells and spleen macrophages. A clear indication of necrosis was observed in hepatocytes.³⁰ Mean maximum plasma NP levels were concentration dependent, but the volumes of distribution were not affected by dosages.³⁰

A systemic study was done by Bailly et al. to evaluate the PK, biodistribution, and safety of laser-ablated dextran-coated AuNPs.³¹ These dextran-coated AuNPs were found to be rapidly eliminated from the blood circulation and accumulated preferentially in the liver and spleen, without inducing liver or kidney toxicity (Figure 1A,B). Nearly 50% of the injected dose was accumulated in the liver after 14 days postinjection (Figure 1B).³¹ The authors applied a bicompartmental model to determine mean PK parameters, which shows an elimination half-life (T_1/2β) of 5.12 h (Figure 1C).

Figure 1.

Distribution and pharmacokinetics of dextran-coated AuNPs. The concentration of AuNPs in (A) liver and (B) spleen was determined after a single injection (1 mg/kg) at 24 h and 7 and 14 days (n = 3, data are mean ± SEM). (C) Pharmacokinetic profile of dextran-coated AuNPs (1 mg/kg) at 5, 15, 30, 45, 60 min and 4 h in the plasma after a single intravenous administration (n = 5, data are mean ± SEM). A point for 0 min corresponds to theoretical administered concentration (10,000 ng/mL). Reprinted with permission under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/) from ref (31). Copyright 2019 The Authors.

A detailed kinetic study was performed by De Jong et al. where the authors focused on the influence of PEGylated AuNPs sizes in in vivo tissue distribution.³² In this study, a rat model was intravenously injected via tail vein with AuNPs of 10, 50, 100, and 250 nm sizes, respectively. Organ analysis was done after 24 h postinjection. A difference was observed between the distribution of the 10 nm particles and the larger particles, with most of the particles in the liver and spleen.³² This study clearly indicated a size-dependent response in the distribution with larger particles accumulated more in the liver and spleen.³² An interesting observation that can be drawn from this study is the absence of larger particles in the brain, whereas detectable amounts of smaller 10 nm particles were found in the brain. The brain is one of the challenging organs to deliver NPs for various reasons, with the blood-brain barrier being significantly difficult to cross. This study indicated that smaller particles are more suspectable to cross such barriers (0.3% injected dose in the brain), therefore, could hold promise in drug delivery to the brain.

A similar size-dependent study was done by Sonavane et al. with PEGylated AuNPs of average sizes of 15, 50, 100, and 200 nm.³³ Considerable accumulation of particles of all sizes was identified in the liver. The major difference in the studies between De Jong et al.³² and Sonavane et al. is the blood level of AuNPs between rats and mice. For rats, the blood content of AuNPs at 24 h was nearly comparable with the content in the main accumulating organs;³³ whereas, for mice,³² the blood level was an order of magnitude lower than the contents in the accumulating organs. The reason behind these differences was not discussed in the literature; however, we assume the differences in blood concentration between rats and mice are presumably due to the differences in their blood circulation times, excretion, as well as anatomical and immunological differences.^34,35 Most importantly, both studies show the blood-brain barrier penetration potential of AuNPs of a size smaller than 20 nm.

Polymer-Based Nanoparticles

The second NM that has been widely used is polymer-based. Among various polymer-based nanoparticulate systems, the PLGA^36,37 polymeric NP system has been extensively used both in laboratory research and preclinical setting owing to its great degree of biocompatibility as its degradation produces lactic and glycolic acids, which are biogenic and are byproducts of various metabolic pathways in the body under normal physiological conditions.³⁸ Acute toxicological analysis of PLGA NPs was studied by Chen et al. using healthy mice.³⁹ These PLGA NPs were surface coated with chitosan and are biotinylated to maximize cellular delivery, 218.4 ± 21.0 nm in size with a ζ potential of 24.15 ± 1.41 mV. Biochemical analysis and hematological indexes for both male and female mice between all tested groups did not show a significant difference (intravenous injected 300 mg/kg PLGA NPs). The histopathological exam also confirmed that the tissues were healthy, and the particles were not toxic.³⁹ After one single intravenous injection of PLGA NPs, the elimination half-life was 16.624 h and the mean residence time was 19.067 h.³⁹

Pioneer work in cell membrane coating of PLGA was carried out in Prof. Liangfang Zhang’s laboratory at the University of California, San Diego.⁴⁰ A study on red blood cell membrane-coated PLGA NPs highlighted the elimination half-life of 39.6 h in contrast to 15.8 h for PEG-coated PLGA NPs.⁴⁰ Considering studies by Chen et al.³⁹ and Hu et al.,⁴⁰ RBC membrane coating is superior in retarding in vivo clearance compared to the conventional PEG or chitosan stealth coating. Nguyen et al. prepared magnetically active PLGA NPs made up of phospholipid conjugated gadolinium and coated with RBC membrane.⁴¹ In this case, Nguyen et al. took advantage of gadolinium to assay the NP kinetics and found an elimination half-life of 11.6 h. This value is significantly shorter than that reported by Hu et al.,⁴⁰ which could be possibly due to the presence of surface gadolinium and the analytical technique used in the measurement process. These studies highlight the role of surface chemistry in PK or TK of NPs.

Wang et al. studied the PK of anticancer drug gemcitabine-loaded PLGA NPs with poly(vinyl alcohol) (PVA) coatings of size 241 ± 36.2 nm using a male Sprague–Dawley rat (180–200 g) model and calculated various PK parameters, such as t_1/2, AUC, and MRT, which were 5.01 h, 10,986.17 h·μg/L, and 4.96 h, respectively.⁴² Similar PLGA NPs stabilized with PVA with an average size of ∼200 nm show an elimination half-life of 8.25 ± 3.19 h in male albino rabbits.⁴³ In another report, a PLGA end-functionalized with glycine was micellized with methotrexate (MTX), forming 110 ± 5 nm particles. This MTX-loaded particle enhances the biological residence of MTX by 2-fold with a drug elimination half-life of 4.30 h (elimination half-life of free MTX was 2.47 h).⁴⁴ Pitchaimani et al., reported the circulation half-life and PK of biomimetic NPs made up of PLGA.⁴⁵ This NP was coated with the membrane of natural killer cells, and PK studies were conducted in the MCF-7 tumor-bearing immunodeficient mouse model.⁴⁵ The t_1/2, AUC, V_d, Cl, and MRT from a two-compartmental model were found to be 9.51 ± 6 h, 1069 ± 507 %ID·h·mL^–1, 3.85 ± 1.16 mL, 0.28 ± 0.14 mL·h^–1, and 13.61 ± 1.38 h, respectively. Rafiei and Haddadi conducted a comparative PK study of bare PLGA and PEG-PLGA NPs with an aim to enhance the blood residence time of docetaxel.⁴⁶ Docetaxel-loaded PEG-PLGA NPs with a hydrodynamic size of 186.7 ± 2.9 nm showed an elimination half-life of 15.87 ± 1.66 h, whereas bare PLGA NPs exhibited 6.05 ± 0.78 h, indicating the surface passivation with PEG is the key to enhance its residence time, confirming the role of PEG in helping NPs evade from clearance mechanisms present in the systemic circulation and the body.⁴⁶

Owing to their ease of modification in various functionalities and engineering to subcellular size, PLGA NP-based delivery systems have a distinct advantage in drug delivery and implantable medical devices, such as but not limited to reducing drug dosage while achieving pharmaceutical effects, minimizing side effects, protecting the drug from degradation, and enhancing drug stability. These NPs can penetrate deep into the tissue through fine capillaries or can take advantage of leaky vasculature to improve deep tissue drug delivery and are generally taken up efficiently by the cells. Studies highlight the importance of NP chemistry and size to alter its in vivo kinetics, which in turn modifies the kinetics of its cargo. Similarities presented in these cited studies are the use of a murine model. In contrast, the differences in the size and the surface chemistry of PLGA make it challenging to predict a conclusive direction. Therefore, more studies with a focus on unified parameters of sizes and surface properties are warranted to picture a realistic scenario to better understand the PK of PLGA NPs. Table 1 summarizes the elimination half-life of gold and PLGA polymeric nanoparticulate systems.

Table 1. Characterization and Pharmacokinetics of Gold, PLGA, and Liposomal Nanoparticles^a.

Nanoparticles	Surface coatings and candidate drug dose (mg/kg)	Hydrodynamic size (nm)	Animal model	t1/2β (elimination half-life, h)	MRT (whole body, h)	Refs
AuNPs	PEG (MW: 5 kDa)	13.1 ± 3.02	Rat (M)	57.0	74.6	Dubaj et al.19
AuNPs	PEG (MW: 5 kDa)	27.6 ± 6.6	Mouse (M)	32.65 ± 11.64	47.11 ± 16.80	Cho et al.30
AuNPs	Dextran	46	Mouse (F)	5.12	NC	Bailly et al.31
PLGA	Chitosan/biotinylated	218.4 ± 21.0	Mouse (M/F)	16.64	19.06	Chen et al.39
PLGA	Red blood cell membrane	80	Mouse (M)	39.6	NC	Hu et al.40
PLGA	PEG (MW: 2 kDa)	80	Mouse (M)	15.8	NC	Hu et al.40
PLGA	PVA	241 ± 36.2	Rat (M)	5.01 ± 1.73	4.96 ± 0.71	Wang et al.42
PLGA	PVA	∼200	Rabbit (M)	8.25 ± 3.19	13.75 ± 3.48	Haggag et al.43
PLGA	Natural killer cell membrane	134 ± 4.4	Mouse (F)	9.51 ± 5.74	13.61 ± 1.38	Pitchaimani et al.45
PLGA	No surface coatings	140 ± 9	Mouse (F)	6.05 ± 0.78	9.29 ± 0.45	Rafiei and Haddadi46
PLGA	PEG	186.7 ± 2.9	Mouse (F)	15.87 ± 1.66	18.46 ± 2.82	Rafiei and Haddadi46
Liposome	Lipid	121.3 ± 48.7	Mouse (F)	20.49 ± 2.71	NC	Deng et al.51
	Drug: 16-Dehydropregnenolone
Liposome	PEG	∼100	Rat	23.6	NC	Working and Dayan53
	Drug: Doxorubicin (DOXIL); 1 mg/kg
Liposome	PEG	∼100	Rat	35.0	NC	Siegal et al.55
	Drug: Doxorubicin (DOXIL); 6 mg/kg
Liposome	PEG	∼100	Dog	27.0	NC	Gabizon et al.54
	Drug: Doxorubicin (DOXIL); 6 mg/kg
Liposome	PEG, Estrone	137.93 ± 1.22	Mouse (F)	20.98 ± 8.31	NC	Han et al.57
	Drug: Paclitaxel; 166.39 mg/kg
Liposome	Lipid, cardiolipin	150 ± 20	Mouse (M/F)	6.38	NC	Pal et al.66
	Drug: 7-ethyl-10-hydroxycamptothecin (SN-38); 10 mg/kg
Liposome	Lipid, cardiolipin	150 ± 20	Dog (M/F)	1.38–6.42	NC	Pal et al.66
	Drug: 7-ethyl-10-hydroxycamptothecin (SN-38); 1.2 mg/kg
Liposome	PEG	120 ± 6.0	Mouse (F)	11.0	NC	Immordino et al.67
	Drug: Docetaxel; 3 mg/kg
Liposome	Natural killer cell membrane	88  ±  1	Mouse (F)	18.0	NC	Pitchaimani et al.186
	Drug: Doxorubicin; 5 mg/kg

^aMW, molecular weight; NC, Not calculated; PEG, polyethylene glycol; PLGA, poly(lactic-co-glycolic) acid; PVA, poly(vinyl alcohol).

Liposomal Nanoparticles

The third nanoparticulate system that we discuss here is a liposomal NP. Unlike metal and core–shell polymeric NPs, liposomes have an aqueous core shield with a phospholipid bilayer. Liposomes were discovered by Alec D. Bangham in 1965⁴⁷ and were approved as therapeutic NPs for cancer treatment. They represent a large proportion of clinical-stage nanotherapeutics due to their biodegradable, biocompatible, nontoxic, and nonimmunogenic composition.^20,48−50 Deng et al. discussed the importance of the liposomal formulation of 16-dehydropregnenolone (16-DHP), a potent antitumor compound.⁵¹ In this report, the authors evaluated PK by measuring 16-DHP concentrations in plasma after a single dose intravenous injection. A liposomal formulation of 16-DHP was compared with free 16-DHP administration and found an order of magnitude enhancement in the plasma residence time of liposomal 16-DHP. The calculated t_1/2 for liposomal 16-DHP was 20.49 ± 2.71 h, whereas free administrated 16-DHP was 5.99 ± 2.01 h. The first clinically approved liposomal anticancer drug is DOXIL,⁵² which is a doxorubicin (DOX) encapsulated liposome with a PEG surface. The hydrodynamic size of DOXIL is ∼100 nm. A comparative study was carried out to understand the PK parameters of free DOX and liposomal DOX (DOXIL) in rats and dogs. Free DOX displays biphasic nature with a rapid decline in concentrations in plasma, and the half-life of the first phase is 5–10 min. In contrast, DOXIL has an elimination half-life within the range of 20–35 h.⁵³ The DOXIL exhibits plasma concentrations of DOX several hundred-fold greater several hours after injection in liposome-treated animals than in animals treated with free drug.⁵⁴⁻⁵⁶ It has been identified that the drug found in plasma remains encapsulated in the liposome, which is not yet bioavailable. Therefore, to enhance its bioavailability, development of fast drug releasing liposomes or revisiting DOXIL formulation is warranted, which can be achieved by developing NPs sensitive toward external or internal stimuli and by enhancing vascular penetration of NPs.

Recently, Han et al. studied the PK of paclitaxel-loaded liposomes made up of soy lecithin phospholipid, cholesterol, and PEG₂₀₀₀.⁵⁷ These liposomes were functionalized with estrone to target positive estrogen receptors (ER) in breast cancer. These ER-targeted liposomes showed an elimination half-life of 20.98 ± 8.31 h and enhanced drug accumulation in tumor sites. Highly studied liposomal hydrophobic drugs include paclitaxel,⁵⁸⁻⁶¹ 9-nitrocamp-tothecin,⁶²⁻⁶⁵ SN-38 (an active metabolite of irinotecan),^24,66 docetaxel,⁶⁷cis-Bis-neodecanoato-trans-R,R-1,2-diaminocyclohexane platinum(II),⁶⁸⁻⁷⁰ and 7-t-butyldi-methylsilyl-10-hydroxycamptothecin.⁷¹ These liposomal formulations are based on the principle of stealth coating with PEG, which is a commercially available hydrophilic polymer. This prevents NPs from the reticuloendothelial system (RES), hence enhancing the plasma residence time as shown in Table 1, which highlights the importance of the surface properties of NPs for improving tissue distribution and drug PK.

The aforementioned liposome formulations mostly use naturally occurring phospholipids. With naturally occurring phospholipids being a major component in these formulations, toxicities of liposomes were found to arise mainly due to the use of other additives to alter its stability or kinetics. The main targets for potential toxic effects of liposomes are the RES system and formed elements of the blood upon parenteral administration.⁷² Platelet aggregation, erythrocyte hemolysis, immunological hypersensitivity reaction due to repeated administration, etc. are the major safety concern with liposomes, which are primarily caused due to the use of nonlipid constituents. In addition, the impact on RES cells such as macrophage saturation by liposomes leads to immunosuppression and increases the risk of infections. For example, administration of DOXIL in mice showed a dose-dependent clearance saturation effect due to partial blockade of the RES cells in the liver, which was not observed in the case of free DOX.⁷³ Although the liposomal drug delivery systems have greatly impacted many medical fields, the use of nonlipid additives needs to be carefully analyzed while considering risk versus benefit.

Uptake and Release Kinetics of Nanoparticles In Vitro

Effects of particle size, surface charge, and cell type have received abundant attention with respect to their effects on NP uptake and removal kinetics by cells in vitro. Herein, we review studies on the kinetics of in vitro cellular uptake, organized by particle type. A brief overview of in silico research that contributed to the understanding of NP-cell interaction kinetics is included as well. Table 2 provides an overview of the work discussed.

Table 2. A Summary of In Vitro Cellular Uptake and Release Kinetic Studies of Different Nanoparticles.

Refs	Particle type	Particle shape	Particle size (nm)	Particle surface charge	Cell type	Methodology
Gold Nanoparticles (AuNP)
Bancos and Tyner170		Spherical	10	Negative	RAW 264.7	in vitro
Chithrani et al.74		Spherical	14/50/74/100	Negative	HeLa	in vitro
		Rod-shaped	40 × 14/75 × 14	Negative	HeLa
Chithrani and Chan75		Spherical	14/30/50/74/100	N/A	HeLa, SNB19, STO	in vitro
		Rod-shaped	20 × 30/14 × 50/7 × 42	N/A
Bartczak et al.78		Spherical	15	Negative	HUVEC	in vitro
Au et al.77		Cage	35/50/90	N/A	U87MGwtEGFR	in vitro
Rischitor et al.171		Spherical	20–95	Negative	A549	in vitro
Cho et al.100		Spherical	15/54/100	Negative	SK-BR-3	in vitro
		Cage	62/118	Negative
		Rod-shaped	16 × 40	Positive
Lunnoo et al.115		Spherical	2/4/6/8/10	Negative/Neutral/Positive	Lipd bilayer models	in silico
		Rod-shaped	2	Neutral
		Cage	2	Neutral
		Plate	2	Neutral
		Hexapod	2	Neutral
Moore et al.178	Poly(vinylpyrrolidone) coated	Spherical	116	N/A	J774A.1	in vitro & in silico
Petrovic et al.79		Spherical	1.4	N/A	A549	in vitro
Dubaj et al.19	PEG coated	Spherical	10	Negative	TH1, A549, Hep G2, 16HBE	in vitro & in silico
Gupta and Rai116		Spherical	2/4/5	Neutral	N/A	in silico
		Spherical	3	Negative/Neutral/Positive
Nangia and Sureshkumar117		Spherical	4	Negative/Neutral/Positive	N/A	in silico
		Pyramid	2 × 2 × 4	Negative/Neutral/Positive
		Prolate	4 × 0.25	Negative/Neutral/Positive
		Cone	2 × 4	Negative/Neutral/Positive
		Rod-shaped	4 × 1	Negative/Neutral/Positive
		Cube	4 (diagonal)	Negative/Neutral/Positive
Li et al.123		Spherical	5.0	Positive	HeLa	in vitro
		Spherical	3.23/4.5/7.1	Positive	N/A	in silico
Polystyrene Nanoparticles (PS)
Kim et al.167	Carboxylated	Spherical	40	Negative	A549	in vitro
		Spherical	100	Negative
Doiron et al.80		Spherical	20/100/200/500	Negative	HUVEC	in vitro and in silico
Lesniak et al.168	Carboxylated	Spherical	40/100	Negative	A549	in vitro and in silico
Johnston et al.82	Carboxylated	Spherical	20/200	N/A	C3A, HepG2, IRGC	in vitro
Khanbeigi et al.83		Spherical	50/100/200/700/1000	Negative	J774A.1	in vitro and in silico
Yaehne et al.111	Carboxylated	Spherical	20/50/100/250	Negative	HUVEC	in vitro
	Aminated	Spherical	100	Negative
Poly(lactic-co-glycolic acid) Nanoparticles (PLGA)
Derakhshande et al.96		Spherical	110–950	Negative	Caco-2	in vitro
Cartiera et al.97		Spherical	95	N/A	OK, Caco-2, HBE	in vitro
Panyam and Labhasetwar95		Spherical	97	Negative	VSMC	in vitro
Reix et al.187		Spherical	180	Negative	Caco-2	in vitro
Dou et al.98	Chitosan coated	Spherical	531/532	Positive	Caco-2	in vitro
		Spherical	358/472	Neutral/Negative
Other Polymeric Nanoparticles
Bitounis et al.172	Cellulose nanocrystal	Spherical	267 × 25	N/A	culture medium only	in vitro
Dombu et al.91	Maltodextrin	Spherical	60	Positive	16HBE14o-	in vitro
Fernando et al.90	PFBT	Spherical	18	N/A	J774A.1	in vitro
Je et al.87	Chitosan	Spherical	200–1000	Positive	Caco-2	in vitro
Sulheim et al.92	PBCA	Spherical	177	Negative	PC3/RBE4	in vitro
	POCA	Spherical	151	Negative
Zaki et al.86	Chitosan HA-coated	Spherical	276	Negative	J774.2/L929	in vitro
	Chitosan uncoated	Spherical	220	Positive
Journey et al.88	PEG	Rod-shaped	800 × 100 × 100; 400 × 100 × 100	Negative	HUVEC	in vitro
		Disc-shaped	325 × 100/220 × 100	Negative
Kulikov et al.93	N-vinyl-2-pyrrolidone oligomer NP - Amphiphilic Polymeric	Spherical	105/216	N/A	CRL 2429, U87	in vitro
Pontes-Quero et al.94	α-Tocopheryl methacrylate/1-vinyl-2-pyrrolidone/N-vinylcaprolactam - Amphiphilic polymeric	Spherical	130	Negative	HC-a, RAW 264.7	in vitro
Vasey et al.89	Poly(lactide)-poly(carbonate) PEGylated	Spherical	67	Negative	U87, GIN8, GIN28	in vitro
Liposomes
Martina et al.99	Liposome- containing Fe2O3 (8 nm)	Spherical	200	N/A	J774	in vitro and in silico
Slabu et al.102	magnetoliposomes (ML)	Spherical	11	Negative	MiaPaCa-2, BxPC-3, L929	in vitro and in silico
Yaehne et al.111	Liposome DOTAP	Spherical	55	Negative	HUVEC	in vitro
	Liposome DOPC	Spherical	50/82	Negative
Yang et al.169		Spherical	100–200	Negative	HeLa	in vitro
Zhang et al.101		Spherical	120	Negative	RAW 264.7	in vitro
Oxide Nanoparticles
Kim et al.167	Silica SiO2	Spherical	50	Negative	A549	in vitro
Blechinger et al.106	Silica SiO2	Spherical	310	Negative	HUVEC, HeLa	in vitro
Cohen et al.177	Silica SiO2	Spherical	12/14/55	N/A	N/A	in vitro and in silico
	Ag/SiO2	Spherical	5–10	N/A	N/A
	Titania TiO2	Spherical	21	N/A	N/A
	Ceria CeO2	Spherical	13/30	N/A	N/A
Lesniak et al.168	Silica SiO2	Spherical	50	Negative	A549	in vitro and in silico
Torrano and Bräuchle107	Silica SiO2	Spherical	310	Negative	HUVEC/HeLa	in vitro
	Ceria CeO2	Spherical	8/30	Negative	HMEC-1
Vranic et al.108	Silica SiO2	Spherical	50/100	Negative	NCI-H292	in vitro
Superparamagnetic Iron Oxide Nanoparticles (SPION)
Cohen et al.177		Spherical	5	N/A	N/A	in vitro & in silico
Lunov et al.103	Carboxydextran-coated	Spherical	20/60	Negative	Macrophage	in vitro & in silico
Wilhelm et al.81		Spherical	9	Negative	HeLa/RAW264.7	in vitro & in silico
Zhang et al.105		Spherical	7/15/30	Positive	MCF-7	in vitro
Quantum Dots (QD)
Jiang et al.110	CdSe/ZnS QD	Spherical	4	Negative	HeLa	in vitro
Ohta et al.113	Silicon QD	Spherical	3	Positive	HUVEC	in vitro and in silico
Xiao et al.109	CdSe/ZnS QD -COOH	Spherical	20–30	Negative	MCF-7/MCF-10A	in vitro
	CdSe/ZnS QD -PEG	Spherical	20–30	Neutral
	CdSe/ZnS QD -PEG-NH2	Spherical	20–30	Positive
Yaehne et al.111	CdSe/ZnS QD	Spherical	12–14	Negative	HUVEC	in vitro and in silico
Price and Gesquiere18	CdSe/ZnS QD	Spherical	14	Negative	J774A.1, C166, AML12, SV40 MES13	in vitro and in silico
Other Nanoparticles
Li and Xing76	Generic NP with ligand	Spherical	N/A	N/A	N/A	in silico
Jin et al.114	SWCNT single walled carbon nanotube	Rod-shaped	130/320/430/660	N/A	NIH-3T3	in vitro and in silico
Gao et al.120	Generic NP	Spherical	6.25/12.5/25/37.5	N/A	N/A	in silico
Ginzburg and Balijepalli124	Generic NP	Spherical	1.6/2.0/2.4/2.8/3.2	Neutral/Charged	N/A	in silico
Yu and Zhang121 and Chen et al.122	Generic NP	Spherical	2.5/3.3/3.8/4/4.5/6/9	N/A	N/A	in silico
Xiong et al.118	Generic NP	Spherical	AR = 1	N/A	N/A	in silico
		Oblate	AR = 0.5
		Prolate	AR = 3
Shen et al.119	Generic NP	Spherical	25/27.5/30/50/75/100	N/A	N/A	in silico
		Oblate	AR = 0.3
		Prolate	AR = 3

Gold Nanoparticles

One of the leading NPs researched is AuNPs due to their potential biomedical and biosensing applications. Size and shape effects on AuNP-cell interaction were illustrated by differences in kinetics and saturation of cellular dose. For spherical particles with citric acid coating, uptake was maximum for 50 nm AuNPs, while 14 and 100 nm showed the least uptake in HeLa (cervix carcinoma) cells.⁷⁴ Uptake for spherical AuNPs saturated around 6 h, and uptake half-life was 2.10, 1.90, and 2.24 h for 14, 50, and 74 nm AuNPs, respectively. This corresponds to rates of 1.72 × 10^–5, 1.63 × 10^–3, and 1.71 × 10^–3 ng/h based on the unit conversion from the number of NPs per hour to nanograms per hour reported by Chithrani et al.⁷⁴ When citric acid was replaced with transferrin on the AuNP surface, uptake kinetics showed comparable profiles, but the number of transferrin-coated AuNPs internalized was three times less compared to the citrate-coated AuNPs. Comparing these results with AuNP rods of similar dimensions (74 × 14 nm, 1:5 aspect ratio), cells internalized spherical 74 and 14 nm AuNPs were 500% and 375% more than the rod-shaped particle. Although surface chemistry or protein corona formation could play a role, it was noted that 1:3 aspect ratio rods internalize more easily. This investigation was extended by the investigators to quantify exocytosis for transferrin-coated AuNPs while considering a broader group of cell lines in vitro.⁷⁵ Smaller AuNPs were exocytosed at a faster rate (14 nm two times faster than 74 nm AuNPs) and at a larger fraction than larger NPs (fraction of 14 nm AuNPs five times higher than for 100 nm NPs). Exocytosis kinetics were found to plateau by 1 h for the smallest AuNPs, while for larger ones, it took about 5 h. The differences were proposed to relate to the thermodynamics of wrapping of the NPs and receptor diffusion kinetics, a mechanism that was later modeled by Li et al. for spherical particles, see below.⁷⁶ Overall, endocytosis and exocytosis were found to depend on NP size, but kinetics were different between the two processes. In another study, gold nanocages coated with antibodies and PEG were evaluated with respect to U87MGwtEGFR glioblastoma cells.⁷⁷ About 75% NPs were internalized with the rest of the NPs interacting with cells remaining surface-bound. Uptake approached saturation quickly by 1 h incubation time, after which the uptake rate slowed considerably. An increase in the number of antibodies on the Au nanocage surface increased uptake. The uptake of Au nanocages was also size-dependent, with the smaller 35 nm NP internalizing abundantly and 90 nm NPs internalizing about 60% less. The exocytosis of AuNPs coated with two different peptides (KATWLPPR and KPRQPSLP) was investigated in human umbilical vein endothelial cells (HUVECs).⁷⁸ Both types of particles show that over half of the particles remain in the cells after 6 h. Furthermore, the KATWLPPR AuNPs showed a progressive increase in the exocytosis rate until it appears to approach saturation at 6 h, while the KPRQPSLP AuNPs show a complex kinetic profile that suggests reuptake of NPs after 4 h.

Recently, AuNPs have also been studied with in vitro 3D cell culture.⁷⁹ An A549 human lung carcinoma 3D model provided the biologically relevant modular structure and presence of extracellular matrix for NP localization, specifically with respect to these altering the kinetics of NP uptake. Slow distribution of AuNPs from the extracellular matrix to intracellular space was observed, in contrast to 2D in vitro culture where rapid uptake kinetics are observed (see above). The kinetics in 3D culture showed two phases, an earlier phase with a relatively higher uptake rate and a second phase with a slower uptake rate. Up to 28 days, no saturable uptake was observed. An example of correlating in vitro kinetics of spherical PEG-coated AuNPs to in vivo observations was reported recently as well.¹⁹ Lung epithelial, lung bronchial, and liver cells exhibit comparable kinetics of interaction with the AuNPs, while the renal proximal tubular epithelial cells showed an order of magnitude faster kinetics. The authors modeled an equilibrium between culture media and cell space (membrane + internalized) and extracted rates of entry and exit from the cell space. The ratio of the rates, basically the equilibrium constant for first-order mass transfer between media and cell space, was used as the PBPK partition coefficient. Notable deviations between prediction and in vivo observations were attributed to the nonmechanistic approach used in modeling.

Polymeric Nanoparticles

Polymeric NPs have been considered extensively as well due to their biocompatibility and potential biodegradability. The kinetics of polystyrene NPs were investigated in quantitative detail for membrane adsorption, detachment, and cell internalization.⁸⁰ Particle size was found to considerably affect NP-cell interactions and was attributed to a reduced number of binding sites for NPs per cell with increasing particle size. The kinetic model applied to determine the corresponding rate constants is based on the Wilhelm et al. mass transfer scheme that is based on Langmuir adsorption.⁸¹ In this model the number of particles adsorbed on the cell membrane is proportional to the concentration of NPs in the culture media near the cell membrane, noting that NP sedimentation can affect this concentration (discussed in the next section). The change in the number of particles bound to the cell membrane over time can then be described as

1

while the number of NPs internalized follows from

2

In these equations N_mem is the number of NPs adsorbed on the cell membrane, N_sites is the number of available binding sites (varies with time depending on number of NPs bound), N_i is the number of particles internalized, C is the concentration of NPs in the culture media near the cell membrane, and k_a, k_d, and k_i are the rate constants for NP adsorption, desorption, and internalization, respectively. The rate constant k_a has units of inverse concentration times inverse time, and k_d and k_i have units of inverse time, where the authors expressed time in seconds and the rate constant is in reference to a change in the number of particles. The solution to this model yields the time evolution of the number of particles bound to the cell membrane and the number of NPs internalized:

3

with B the binding capacity of the cell and

4

Sample results of applying this model are shown in Figure 2. Uptake of carboxylated polystyrene NPs by hepatocytes was found to depend on time, size, and in vitro environment.⁸² For 200 nm NPs, uptake was sparse, with most NPs located on the cell surface. The 20 nm NPs were internalized, both earlier and more abundantly. Findings were consistent for the different hepatocytes studied, although the C3A cell line showed earlier compartmentalization of NPs. In addition, after 30 min, 20 nm NPs started to accumulate within bile canaliculi leading the authors to conclude that NP excretion within bile is the mechanism of elimination. It is noteworthy that when cells were incubated with 20 nm NPs in presence of serum that uptake increased over time, but in absence of serum, the NPs agglomerated which led to reduced NP uptake. For the 200 nm NPs, no effect was found on agglomeration and uptake remained low. Interaction of polystyrene NPs with murine macrophages was evaluated while taking into account the differences between the administered dose and the delivered dose via the in vitro sedimentation, diffusion, and dosimetry (ISDD) model.⁸³ The authors found that only a small fraction of the administered dose interacts with cells in vitro regardless of particle size. It was speculated that low-density NPs and short incubation times exacerbate this effect. Internalized dose correlated well with the delivered dose in all cases. 50 and 1000 nm particles showed more internalization by macrophages. Macrophages are known to prefer the uptake of microparticles,⁸⁴ while sub-100 nm particles allow for more cellular interaction through a potentially wider scope of internalization pathways.⁸⁵

Figure 2.

Sample results from the (A) Doiron et al.⁸⁰ studies for polystyrene NPs (PS) and (B) Price and Gesquiere¹⁸ studies for quantum dots (QDs). Modeling was performed using a single cell model of particle-cell interaction based on Langmuir adsorption and chemical kinetics, respectively. Corresponding rate constants are summarized in the table. Panel A reproduced with permission from ref (80). Copyright 2011 John Wiley and Sons. Panel B adapted with permission under a Creative Commons Attribution CC BY-NC license (https://creativecommons.org/licenses/by-nc/4.0/) from ref (18). Copyright 2020 The Authors, some rights reserved; exclusive license AAAS.

Uptake of uncoated chitosan particles by mouse macrophages and fibroblasts was reported to be very rapid, within a few minutes of incubation.⁸⁶ In contrast, hyaluronic acid (HA)-coated chitosan particles required tens of minutes of incubation time to observe any appreciable uptake. Uptake kinetics was saturable with clear dose dependence and a Michaelis–Menten-like profile presaturation. In addition, macrophages exhibited a greater uptake capacity for uncoated NPs than fibroblasts, while the uptake capacity for HA-coated NPs was 2 orders of magnitude lower. The authors of this study reported that HA-coated chitosan particles were stable in size and ζ potential. Conversely, the uncoated chitosan NPs showed an increase in size with reversal of ζ potential in culture media. This was attributed to protein adsorption, which was hypothesized to be largely absent for the HA-coated NPs. These results illustrate how protein adsorption can drastically change NP properties and experimental outcomes in vitro. The size and surface charge dependence of chitosan NP uptake by colorectal adenocarcinoma (Caco-2) cells were studied.⁸⁷ It was found that increasing particle size led to increased positive ζ potential. Uptake of the chitosan NPs by Caco-2 cells increased in the 200–600 nm size range, while larger particles with slightly higher ζ potential showed less uptake. These results suggest complex interactions that affect NP uptake other than chitosan NP size.

Jurney et al. compared the in vitro kinetics of nonspherical PEG hydrogel NPs between static culture and dynamic cell culture in microfluidic devices.⁸⁸ Notable outcomes are that in the presence of shear flow NP-cell interaction is reversed, likely due to increased motion of larger nonspherical NPs that pushes them to the cell wall allowing for increased cell interaction. PEGylated poly(lactide)-poly(carbonate)-DOX NPs showed first-order kinetics with a faster uptake rate in the first hour followed by slower uptake rates in the following five hours.⁸⁹ Comparing three glioblastoma cell lines, uptake was higher for GIN8 than GIN28, with U87 showing the lesser uptake.

Conjugated polymer NPs fabricated from the polymer poly(fluorene-alt-benzothiadiazole) (PFBT) were tested for their in vitro kinetics with J774A.1 macrophages.⁹⁰ A linear correlation between the applied dose and the internalized dose was observed. Similarly, a linear relationship was found between amounts of NPs internalized and incubation time. For cationic maltodextrin NPs interacting with human bronchial epithelial cells, the endocytosis rate was found to exceed NP binding.⁹¹ Further research suggests the NP binding to anionic sites on the cell membrane is required before their endocytosis. In addition, uptake was saturable but trended downward after media replacement, which was attributed to the exocytosis of the NPs. The rate of exocytosis increased substantially after 1 h. The uptake of a series of poly(alkyl cyanoacrylate) NPs was found to depend on polymer composition. Different cell types also interacted differently with the NPs, resulting in different internalization pathways. The results of another study⁹² showed that rat brain endothelial cell line RBE4 internalized significantly more poly(butyl cyanoacrylate) (PBCA) and poly(octyl cyanoacrylate) (POCA) NPs than prostate cancer cell line PC3. Uptake saturated at 3 h for RBE4 and the uptake rate for PC3 slowed down around the same time without showing saturation. N-Vinyl-2-pyrrolidone oligomer amphiphilic NPs showed linear kinetics for uptake by fibroblasts and glioblastoma cells, although the uptake was tracked for only 1 h.⁹³ The authors of the study speculated that due to the particle size and inert surface, macropinocytosis and receptor-mediated endocytosis were suppressed. Another amphiphilic polymeric NP made from three polymers including N-vinyl-2-pyrrolidone polymer was cultured in vitro with HC-a and RAW 264.7 cells.⁹⁴ Uptake kinetics were different between the two cell types. A linear zero-order kinetic profile was observed for the RAW 264.7 macrophages for the duration of the study (24 h). For HC-a, a first-order kinetic profile with saturation at 8 h was noticeable. The difference was attributed to the phagocytic properties of macrophages.

Poly(lactic-co-glycolic acid) Nanoparticles

A study on 97 nm PLGA NPs with vascular smooth muscle cells (VSMC) revealed that the NP uptake was dose- and time-dependent, with a fairly linear correlation between applied dose and internalization while uptake saturated at 4 h.⁹⁵ With the removal of the NPs from the cell surroundings in vitro exocytosis of NPs took place rapidly. Roughly 65% of the internalized fraction was exocytosed in 30 min and proceeded more slowly afterward. The work reported by the authors suggests that the exocytosis of NPs proceeds through an energy-dependent process. The effect of protein corona formation was illustrated when exocytosis was not observed in serum-free media, indicating that NPs carry media proteins that potentially interact with the exocytotic pathway. PLGA NPs of varying sizes (110–950 nm) with negative ζ potential showed more transepithelial transport for the smaller sizes, with a limited effect of the applied dose, while the kinetics profile appeared nonsaturable.⁹⁶ A similar observation was made for 95 nm PLGA NPs and human bronchial epithelial (HBE) cells, although a comparison of the same PLGA NPs with opossum kidney (OK) and Caco-2 cells shows that uptake was saturable and nearly indistinguishable from control until 24 h, respectively.⁹⁷ NP uptake by HBE cells was found to be slower compared to OK cells, while both cell types exhibit exocytosis of PLGA NPs. These observations illustrate the role of cell type in cell-NP interactions. In another report, 180 nm PLGA NPs were found to internalize with a linear kinetic profile that did not saturate after 6 h. Exocytosis took place after replacing NPs containing media with NP-free media. 80% of NPs were removed from cells within 2 h to the basolateral medium. Coating PLGA NPs with chitosan increased the rate of uptake and transcellular transport for Caco-2 cells, with higher doses showing saturable uptake at 2 h.⁹⁸

Liposomes

The interaction of liposomes loaded with 8 nm iron oxide NPs with macrophages was studied, and the effect of a PEG coating on the liposomes was evaluated.⁹⁹ The liposomes bound with J774 macrophages were taken up by endocytosis. The PEG coating limited the association of liposomes with the cells, which limited internalization. This effect of the PEG coating was also reported for AuNPs.¹⁰⁰ However, the kinetics of the liposome-macrophage interactions were not affected by the polymer coating. Both liposome types showed first-order kinetics for endocytosis with identical rate constants. Interestingly, the maximum number of liposomes that could be endocytosed per cell as determined from the first order kinetic model was lower for the PEG-coated liposomes by a factor of 1.5. A recent report by Zhang et al. using RAW 264.7 macrophages on the other hand indicates zero-order uptake kinetics of liposomes by these macrophages, suggesting effects of liposome composition, coating, and cell line.¹⁰¹ Magentoliposome (ML) in vitro kinetics were studied and framed within a mathematical model by Slabu et al.¹⁰² Pancreatic tumor cell uptake kinetics were compared with healthy fibroblast cells. MiaPaCa-2 pancreatic tumor cells internalized three times more ML than BxPC-3 (another pancreatic tumor cell line) and L929 murine fibroblast cells. Notably, these observations were made while ML adsorption rate constants were nearly the same. These results were modeled as a three-step process including adsorption, internalization (endocytosis), and exocytosis. The authors started with the model described above and extended this further with the inclusion of exocytosis. They introduced the assumptions that (i) only a finite fraction (F₀) of the cell membrane is available for NP internalization, (ii) the same fraction of the cell membrane is available for NP exocytosis, and (iii) exocytosis follows after endocytosis. The latter is simulated by an activation function Θ(t) that varies the delay and magnitude of exocytosis. The membrane subfraction participating in endocytosis is F_endo and for exocytosis is F_exo. With these conditions eq 2 is then reworked as follows to represent endocytosis and exocytosis, respectively:

5

6

with N_e the number of NPs removed from the cell interior and k_e the rate constant for exocytosis. The rate constant k_e has units of inverse time with time reported in minutes in reference to a change in the number of particles, and the fraction F and activation function Θ are unitless.

Superparamagnetic Iron Oxide Nanoparticles and Other Oxide Nanoparticles

Wilhelm et al. conducted a kinetic analysis of membrane adsorption and internalization for 9 nm SPION to compare tumor cell and macrophage properties.⁸¹ Neither HeLa cells nor RAW macrophages showed a saturable kinetic profile after 12 h incubation with SPION, although uptake rates had significantly slowed down. The authors also reported a kinetic model to further analyze the results. Their reported kinetic model consists of a two-step process involving Langmuir adsorption followed by internalization and has formed the basis for recent advancements in the modeling of NP-cell interactions as discussed above. Binding parameters were consistent between cell types, but internalization capacity was 1 order of magnitude greater for RAW macrophages. Carboxydextran-coated SPION were incubated with blood-derived macrophages to test the effect of particle size.¹⁰³ The results of the study show that 60 nm NPs are internalized 10 times more effectively than 20 nm NPs of the same composition, demonstrating NP size dependence of cell-NP interaction. In addition, the internalization was observed to saturate in less than 3 h in both cases. In a mathematical model developed by the investigators, the uptake mechanism is described as contributions from NP capture and NP wrapping by the macrophage cell membrane involving multiple receptors, based on the Gao et al. model.¹⁰⁴ The size effect of SPION uptake kinetics by MCF-7 human mammary adenocarcinoma epithelial cells was reported recently.¹⁰⁵ Different NP sizes all followed first-order kinetics but with saturation at a higher internalized dose for 15 nm versus 7 and 30 nm.

Larger (310 nm) silica NPs were found to be internalized twice as much by HeLa cells than HUVEC cells after 24 h. Interestingly, within the first 4 h of NP-cell interaction, uptake was much higher for HUVEC cells, after which uptake by HeLa cells accelerated 20-fold.^106,107 These results illustrate differing uptake rates for the same NPs by different cell types, with nonlinear kinetic profiles at different incubation times. For another type of oxide NPs, ceria, the kinetic profile was vastly different.¹⁰⁷ A strong increase in the number of internalized NPs was observed between 3 and 24 h, with no obvious size dependence. Even though the particles were smaller, they aggregated into approximately 300 nm clusters before internalization. When smaller 50 and 100 nm silica NPs were incubated with NCI-H292 lymphocytes, a saturable time- and dose-dependent internalization kinetics was found, although the 100 nm particles showed high adsorption on the cell membrane.¹⁰⁸

Quantum Dots and Carbon Nanotubes

A series of CdSe/ZnS QD coated with carboxylate, PEG, and PEG-amine were studied for the kinetics of their interaction with malignant and nontumorigenic human mammary epithelial cells.¹⁰⁹ Carboxylated QD were internalized with a linear kinetic profile up to 40 h incubation time, while the PEG coating blocked internalization for both those QD types over 12 h of study. Zwitterionic CdSe/ZnS QDs were primarily endocytosed via the clathrin-mediated mechanism by HeLa cells with accumulation observed on the cell membrane in conjunction with uptake saturation.¹¹⁰ The investigators suggest that clusters of particles rather than individual particles need to be wrapped for successful completion of internalization, as corroborated by computational modeling discussed herein. Exocytosis proceeded partially by active transport of QD to the periphery of the cell, and the fraction of QD exocytosis saturated at 2 h. The effects of NP size and surface charge were studied for a series of QD, liposomes, and polystyrene NPs by Yaehne et al.¹¹¹ Deposition rate constants were found to depend on surface charge when the researchers removed effects from particle size. In addition, NP kinetics were found to be controllable by size rather than chemistry. We recently reported kinetics studies on the interaction of negatively charged CdSe/ZnS core–shell QD with a number of cell types using a ratiometric assay.^18,112 Adsorption and internalization were found to dominate the kinetics, with adsorption being about 2 orders of magnitude slower. In addition, we observed significant NP degradation that needed to be taken into account for correct modeling of the data. To extract rate constants from the data, we built a chemical kinetics model that includes degradation of the NPs for correct modeling. The model consists of three compartments representing the culture media, cell membrane, and internal cell space. The NP dose evolution for the media compartment is represented by

7

with [Med] and [Mem], the concentration of NPs in the culture media and NPs adhered to the cell membrane, respectively, reported in nanomolar (nM). The cell membrane compartment separates the culture medium from the internal space of the cell. Once adsorbed to the cell membrane, NPs can go back to media or enter the cell as expressed by

8

Once NPs enter the cell space compartment, there is a chance they become degraded. This is expressed with a term that contains a first-order rate constant for the degradation process (k_deg). If in vitro empirical work shows that the NP does not degrade, this k_deg can simply be set to zero so that the degradation term is not included in the modeling of the data. The NP dose evolution for the cell space compartment is then described as

9

with [Cell] the concentration of NPs (nM) in the cell interior at time t. All rate constants were reported as per hour in reference to a change in the molar concentration of particles. Sample results of applying this model are shown in Figure 2. The results of this in vitro work were applied to in vivo predictions of QD predisposition.¹⁸ The mechanistic approach in the modeling resulted in a satisfactory agreement between predicted and observed in vivo data for multiple species.

Interaction of silicon quantum dots (Si-QDs) with HUVEC cells resulted in near saturation of uptake by 6 h of exposure.¹¹³ The dose-dependent response changed the number of particles per cell under those conditions but the kinetic profiles remained similar. Removing Si-QDs from the in vitro media was followed by exocytosis of the NP from cells with a plateau reached by 2.5 h. A kinetic model based on the mass balance of the Si-QDs allowed the authors to extract the dissociation constant between receptors in endosomes and Si-QD aggregates and was found to be the determinant in NPs remaining in cells instead of being removed.

DNA-Coated Single-Walled Carbon Nanotubes

DNA-coated single-walled carbon nanotubes (SWCNT) with sizes ranging from 130 to 660 nm showed maximum uptake by fibroblasts for intermediate sizes.¹¹⁴ The investigators reported a model based on the Gao et al. model,¹⁰⁴ where the SWCNT were reversibly bound to the cell membrane into clusters of receptor-bound complexes, which when of sufficient size lowers the elastic energy for wrapping enough to be overcome by the free energy reduction resulting from the receptor-particle association. From the model, the endocytosis rate constant and exocytosis rate constant were obtained for the SWCNT and compared to modeling based on literature data for AuNPs^74,75 and PLGA NPs.⁹⁵ The endocytosis rate constant of SWCNT was estimated to be about 1000 times that of AuNPs, while the exocytosis rate constants were similar for SWCNT, PLGA, and AuNPs, noting that each NP was studied with different cell lines.

In Silico Evaluation of Nanoparticle-Cell Interactions

Computational modeling has also made significant contributions to the understanding and quantification of in vitro NP-cell interactions. More commonly used approaches include coarse-grained molecular dynamics (CGMD) and dissipative particle dynamics (DPD) as more manageable versions of all-atom molecular dynamics (AAMD). In addition, other approaches such as thermodynamic models have been reported to study NP-cell interactions.

Recently, Lunnoo et al. studied the interactions of AuNPs and AuNP aggregates with cell membranes using CGMD simulations while considering size, shape, and surface charge.¹¹⁵ The investigators calculated rate constants for internalization and translocation half-life and found that aggregated AuNPs had less ability to transport through the lipid bilayer than the corresponding individual NP. Nanohexapods exhibited the highest cellular uptake, while spherical AuNPs showed distinct surface charge and size dependence of their transport through the plasma membrane. In an earlier study by Gupta et al. using CGMD simulations to consider AuNP-membrane interactions of uncharged smaller particles, these were found to permeate the membrane more easily although all sizes considered entered the lipid bilayer with relative ease.¹¹⁶ Entry was facilitated by disruptions of the membrane caused by the NP. However, the charge effect on this observation was found to be significant, with small cationic and anionic particles showing mainly adhesion and minimal permeation. The interplay between NP shape and charge was evaluated previously while attempting to maintain NP size consistent.¹¹⁷ Internalization rates increased over 30 orders of magnitude with increasing cationic charge density for prolate NPs while the effect increased to 60 orders of magnitude for spherical NPs. Negatively charged NPs were repelled from the cell membrane. Nonspherical NPs reoriented near the cell membrane to enhance NP-membrane contact, with the potential to disrupt the membrane structure. Shape and size dependence on the cooperative wrapping of NPs by lipid membranes was further studied with CGMD by Xiong et al.¹¹⁸ The authors reported complex dependencies that resulted in various wrapping structures. Interestingly, Shen et al. recently reported that deformable spherical NPs require interaction with more receptors to induce membrane wrapping, more so with increasing NP size.¹¹⁹ Oblate and prolate NPs on the other hand showed more complex behaviors that depend on the orientation of the NP on the membrane.

The effect of particle size and interaction strength with membrane on endocytosis has also been evaluated with DPD simulations.¹²⁰ The investigators found two wrapping modes depending on the extent of attraction between the NP and cell membrane. Weak interaction resulted in slow and limited wrapping, with larger particles accelerating the dynamics of membrane wrapping. For strong NP-cell membrane interactions wrapping dynamics accelerated with decreasing particle size, however. In another DPD investigation, a cooperative effect in NP endocytosis was considered.¹²¹ The researchers reported that smaller particles (2.5 nm) cluster to induce membrane deformation and wrapping, while intermediate-size particles (4.0 nm) aggregate into pearl-chain arrangements, both internalizing in cooperative fashion. Larger particles (6.0 nm) were found to internalize individually. The investigators followed this study up with further DPD simulations to understand these size effects on the NP internalization pathways.¹²² The way the NPs aggregate on the membrane surface was found to affect the mechanism of internalization. Li et al. compared in vitro experimental results with DPD simulations to investigate the endocytosis of charged AuNPs in the context of cooperativity.¹²³ Both experiment and theory demonstrate that positively charged AuNPs enter the cell together through cooperative endocytosis. Even though the like-charged NPs repel each other, the investigators conclude that membrane deformation counteracts this effect.

A thermodynamic model to determine the optimal size for receptor-mediated endocytosis of spherical NPs was reported by Li et al.⁷⁶ The model describes the cell membrane during receptor-mediated endocytosis with two regions: bound and free membrane regions. Receptors on the cell surface behave accordingly, with a change of receptor state leading to cell membrane deformation. Thermodynamic contributions are then described by receptor binding free energy changes, loss of entropy of the receptors, and free energy changes due to cell membrane deformation. From this model, it is deduced that receptor density and release of energy upon ligand-receptor binding are the main factors that control the optimal NP size for receptor-mediated endocytosis. Size and interaction-dependent thermodynamics of NP adhesion effects on cell membrane were discussed by Ginzberg et al.¹²⁴ The researchers found that uncharged or hydrophobic NPs cause membrane swelling by incorporating into the membrane. Conversely, charged or hydrophilic particles can remove phospholipids from the membrane to form micelles, potentially rupturing the membrane and leaving nanosized holes. These effects were found to correlate with the charge density of the particle surface.

Physiologically Based Pharmacokinetic Modeling of Nanoparticles

Currently, there are approximately 50 PBPK modeling studies for different types of NPs in the literature. These PBPK models were developed for different applications, such as nanomedicine development and NM toxicity and risk assessment. The major features and applications for each of these existing PBPK modeling studies are summarized in Table 3 for inorganic NPs and Table 4 for organic NPs and hybrid NPs. Some of these studies only simulate the PK of a particular NP, but other studies simulate the PK of both the NP itself and the active pharmaceutical ingredient that is carried by and released from the NPs. Schematics of representative PBPK model structures for different types of NPs are provided in Figure 3. In the overview below, we will discuss the progress of PBPK modeling for different NP types, including inorganic NPs, such as QD, AuNPs, titanium dioxide, silver, and other metal NPs as well as organic NPs, such as polymeric NPs, liposomes, and other organic NPs.

Table 3. Summary of Selected Physiologically Based Pharmacokinetic Modeling Studies for Inorganic Nanoparticles^a.

Refs	Physicochemical properties of administered NPs	Routes	Modeling species and sex	Main features and applications for the developed models
Quantum dots (QDs)
Lin et al.125	QD705: 18.5 nm CdTe/ZnS with PEG coating	IV	Mice (M)	(1) Perfusion-limited PBPK model; (2) time-dependent Hill equation used for the distribution coefficients and metabolic rate constant; and (3) health and potential beneficial implications are suggested
Lee et al.127	QD705: IV, 13 nm (HD) CdTe/ZnS with PEG coating	IV and intradermal	Mice (M/F/NM) and rats (M)	(1) Perfusion-limited PBPK model; (2) validated across different QDs, species, and administration routes; (3) perfusion-limited model is not adequate to explain some complicated PK exhibited by different QDs; and (4) self-agglomeration of various QDs and redistribution to various tissues through the lymphatic system need to be considered
	QD621: intradermal, 37 nm (HD) CdSe/CdS with amphiphilic polymer coating
	QD525: IV, 12 nm (HD), CdSe/ZnS with PEG coating
	QD800: IV, 21 nm (HD), CdSe/ZnS with PEG coating
	QD-LM: IV, 7–25 nm (HD) CdSe/ZnS with LM coating
	QD-BSA: IV, 80 nm (HD) CdSe/ZnS with BSA coating
Liang et al.128	Experimental data: IV and SC: 4.2 nm (HD) and −37 mV CdTe/CdS QDs	IV and SC	Mice (M/F) and rats (F)	(1) Permeability-limited PBPK model; (2) Hill function-based RES uptake from tissue interstitium (lungs, spleen, kidneys, and rest of body) and vascular space (liver); (3) LSA: 1% increase in parameters; (4) prediction capability across routes and species; and (5) long-circulating biodistribution due to RES uptake and release
	Independent data: IV: 3.5 nm (HD) MPA-coated CdTe QDs; 18.5 nm CdTe/ZnS QDs with PEG coating; 21.2 nm (HD) and −4.4 mV CdSe/ZnS QDs with PEG coating
Metallic NPs
MacCalman et al.149	Silver NPs: 17.1 nm (MMD)	Inhalation	Rats (M/F)	(1) Route- and particle-specific and permeability-limited PBPK model; and (2) optimal parameter estimates were found by minimizing the model mean squared error
	Iridium NPs: 15–20 nm (CMD)
Lankveld et al.188	20, 80, and 110 nm AgNPs with mean surface area per particle of 1.3 × 103, 2.0 × 104, and 4.0 × 104 nm2, respectively	IV	Rats (M)	(1) Permeability-limited and size-specific PBPK model with elimination pathways incorporated into blood compartment; (2) aggregation and/or agglomeration in larger AgNPs (80 and 110 nm); and (3) accumulations in the lung, liver, and spleen indicate potential toxicity and risk
Bachler et al.11	Silver NPs: IV: 20, 80, and 110 nm NPs with mean surface area per particle of 1.3 × 103, 2.0 × 104, and 4.0 × 104 nm2, respectively	IV, oral, dermal, and inhalation	Rats (M/F) and humans (M)	(1) Size-dependent and permeability-limited PBPK models incorporating dissolution kinetics for 1–150 nm AgNPs; (2) RES uptake included in the lung, liver, and spleen; (3) LSA with 1% change in parameters; (4) no significant dissolution from AgNPs to Ag ions; (5) similar storage processes for ionic Ag and AgNPs; and (6) human risk assessment based on different exposure scenarios
	Oral: 14 and 60 nm NPs
	Inhalation: 17.1 nm (MMD) and 14.77 nm (GM) NPs
Demin et al.189	Silver NPs with size of 35 ± 15 nm (TEM) radiolabeled with 110mAg	Oral	Rats (M)	(1) Perfusion-limited PBPK model; (2) predictability of dose-dependent NP disposition under acute and subacute exposure scenarios; and (3) employed in vitro data to estimate cytotoxicity
Bachler et al.12	TiO2 NPs: IV: 20 nm (TEM) rutile TiO2 with surface charge of −12.5 mV and 20–30 nm uncoated TiO2 with surface area of 48.6 m2/g	IV, oral, and dermal	Mice (M/F/NM), rats (M), and humans (M/F)	(1) Permeability-limited PBPK model; (2) RES uptake considered in the lung, liver, and spleen; (3) LSA with 1% change in parameters; (4) size and crystalline structure of TiO2 NPs had minor effects on the disposition fate; (5) particle agglomerate in vivo at high internal exposure and uptake by macrophages in RES; and (6) ingestion risk of TiO2 NPs in the German population is small
	Oral: 25 and 80 nm TiO2 NPs
	Dermal: 21 nm anatase/rutile TiO2 and 25 and 60 nm rutile TiO2 with surface areas of 50, 80, and 40 m2/g, respectively
Laomettachit and Liangruksa190	TiO2 NPs: 20–30 nm uncoated NPs with surface area of 48.6 m2/g	IV	Rats (M)	(1) Perfusion-limited PBPK model; and (2) cytotoxicity assessment based upon tissue dosimetry of TiO2 NPs
Laomettachit et al.191	TiO2 NPs: 20 nm (TEM) rutile with surface charge of −12.5 mV and 20–30 nm uncoated TiO2 NPs with surface area of 48.6 m2/g	IV	Mice (NM) and rats (M)	(1) Permeability-limited PBPK model linked with cell-response model to predict cell death in the liver; (2) RES uptake was not considered; (3) LSA with ±10% change; and (4) negligible tissue damage from low-dose exposure but larger cell fractions needed to join in the cell cycle to recover the original cell mass
Bachler et al.166	Gold NPs: 1.4–80 nm (TEM) and 2.9–85 nm (HD) S-TPP (SO3–)-modified AuNPs with ∼ (−20) mV; 2.8 nm (TEM) carboxyl (positive charge) and amino (negative charge) groups-modified AuNPs; 22 nm (CMD) AuNPs with surface area of 1.5 × 103 nm2	Inhalation	Mice (F) and rats (F)	(1) Permeability-limited PBPK model; (2) RES uptake included in the lung, liver, and spleen; (3) LSA: 1% change in parameters; (4) in vitro assays to distinguish between mucociliary- and biliary-cleared NPs in GI and feces; and (5) combined in vitro and in silico methods to potentially replace short-term animal studies for assessing NP pulmonary fate and biodistribution
Lin et al.126	Gold NPs: 13, 16, 20, 80, and 100 nm PEG-coated AuNPs where 20 and 80 nm AuNPs were 111In labeled and measured from tumor-bearing mice	IV	Mice (M/F)	(1) Permeability-limited PBPK model has superior prediction power over perfusion-limited PBPK model; (2) RES uptake mechanisms: uptake from the tissue interstitium (13 nm) or capillary blood (100 nm); (3) Hill function was implemented to describe the saturable RES uptake rate; (4) LSA and uncertainty analyses with 1% increase in parameters; and (5) time- and size-dependent endocytosis/phagocytosis need verification by in vitro cellular uptake along with in vivo PK data
Lin et al.35	Gold NPs: 13 nm PEG-coated NPs; 16.1 and 18.4 nm citrate-coated NPs; 20 nm naked NPs; 15–20 nm gum arabic-stabilized NPs; 27 nm rhTNF-bound PEGylated colloidal NPs	IV	Mice (M), rats (M), pigs (NM) and humans (M/F)	(1) Permeability-limited PBPK model; (2) saturable RES uptake from the capillary blood included in the lungs, liver, spleen, kidneys, and rest of body and described by Hill function; (3) rats and pigs seemed more appropriate models than mice in animal-to-human extrapolation; and (4) the simulation approach may be applied to other NPs and provides guidance to design future translational studies
Aborig et al.140	Gold NPs: Experimental data: Curcumin-capped AuNPs: 19.6 nm (HD), PDI: 0.167, and −20 mV; EGCG-capped AuNPs: 25 nm (HD), PDI: 0.173, and −22 mV;	IV and IP	Mice (NM) and rats (NM)	(1) Lymphatic system-incorporated and perfusion-limited PBPK model; (2) LSA with ±10% change in parameters; and (3) applied in silico model to support experimental in vitro and in vivo findings, improving understanding and efficiency in preclinical development of large molecule drug products
	Independent data: 16 and 20 nm (TEM) citrate-coated AuNPs
Deng et al.141	Gold NPs: 13 and 100 nm PEG-coated AuNPs; 27 nm rhTNF-bound PEGylated colloidal NPs	IV	Mice (M) and humans (M/F)	(1) Permeability-limited PBPK model; (2) considered saturable RES uptake in blood, lungs, liver, spleen, and kidneys; (3) LSA and uncertainty analyses with 1% increase in parameters; and (4) the well-validated mouse and human PBPK models provided effective guides for nanomedicine delivery
Zhang et al.142	Gold NPs: c(RGDyC)-125I-Pt loaded NPs: 56.4 ± 3.0 nm (TEM), −26.9 ± 6.2 mV, and 2.9 × 1017 nm2/mg	IV	Tumor-bearing mice (NM)	(1) Permeability-limited and tumor-bearing PBPK model; (2) Hill function to describe saturable RES/tumor uptake; and (3) active-targeting NRs had higher tumor uptake and more effective tumor growth inhibition.
	c(RGDyC)-125I-Pt loaded NRs: 56.1 × 22.4 nm (TEM), −28.4 ± 4.4 mV, and 5.5 × 1017 nm2/mg
Zazo et al.192	AuNPs carrying stavudine (40 nm)	IP	Rats (M)	(1) Drug release kinetics from AuNPs was measured; (2) cellular uptake kinetics of AuNPs was determined in human macrophages; and (3) in vitro cellular uptake rate was converted to in vivo tissue uptake rate
Dubaj et al.19	PEG-coated AuNPs (PEG-AuNPs) with a core size of 10.5 ± 0.8 nm and hydrodynamic diameter of 13.1 ± 3.0 nm	IV	Rats (M)	(1) In vitro cellular uptake of PEG-AuNPs was studied in multiple cell lines (TH1, A549, Hep G2, and 16 HBE); (2) in vitro cellular uptake rates were extrapolated to in vivo uptake rates for different tissues; (3) significant differences were observed in the internalized amount of Au in individual cell lines compared to the corresponding tissues in vivo, with the highest found for renal TH1 cells and kidneys; and (4) caution should be exercised when extrapolating in vitro kinetic data to in vivo for NPs
Chou et al.143	Citrate-coated AuNPs: 1.4, 5, 18, 23, 80, and 200 nm	IV, oral, IT, and inhalation	Rats (F)	(1) Permeability-limited PBPK model; (2) saturable RES uptake from the capillary blood included in the lung, liver, spleen, kidney, and rest of body and described by Hill function; (3) traditional route-to-route extrapolation approach that is commonly used for small molecules is not applicable to NPs; (4) multiroute PBPK model for NPs should be calibrated using route-specific data; and (5) the final model was translated to a user-friendly web-based interface
Chen et al.146	ZnO2 NPs: Small: 10/62 nm (TEM/HD), −27.1 ± 1.4 mV	IV	Mice (M)	(1) Time- and size-dependent perfusion-limited PBPK model; (2) Hill equation to describe the time-dependent partition coefficients and elimination rate constants; (3) MAPE used to examine the prediction power of size-varied biodistributions; (4) LSA: 10% increase in parameters; and (5) caution is needed for 10 nm ZnO2 NPs for prolonged exposure in mice
	Large: 71/275 nm (TEM/HD), −19.3 ± 3.8 mV
Henrique Silva et al.193	SPIONs coated with oleic acid and mPEG2000 sized 168.9 ± 1.1 nm (HD) with surface charge of −18.2 ± 5.2 mV	IV	Mice (M) and humans (M)	(1) Permeability-limited PBPK model developed based upon in vitro assays and validated against mouse PK data; (2) extrapolated to simulate SPION dispositions in mans; and (3) in vitro mouse and human assays seemed to be an appropriate alternative to predict biodistribution and RES uptake of SPIONs in vivo
Chen et al.132	Superparamagnetic iron oxide nanoparticles coated by gold and conjugated with PEG (SPIO-Au-PEG NPs) (core size: 17.5 nm)	IP	Mice (M)	(1) measured the brain permeabilities of the NPs and determined concentrations of NPs in cerebral blood and brain tissue; (2) detailed description of transport of NPs to brain
Sweeney et al.150	Iridium NPs: 15 and 15–20 nm (CMD)	IV and inhalation	Rats (M)	(1) LNs-embedded permeability-limited PBPK model; (2) Bayesian MCMC simulation approach to assess parameter variability and uncertainty; and (3) this technique can potentially test various model structures at once, which helps identify the key factors governing TK for different NP types
	AuNPs: 20.3 ± 1.9 nm
	TiO2 NPs: 20–30 nm
Li et al.148	Ceria NPs:<200 nm (∼90%) fresh and aged nanoceria following bimodal log-normal distribution with peaks at 25 and 90 nm	Inhalation	Rats (M)	(1) Permeability-limited PBPK model; (2) saturable RES uptake considered in every compartment with majority of NPs captured; (3) LSA: 1% change in parameters; and (4) applicable to other NP inhalation studies with clearly defined experimental conditions
Carlander et al.130	Ceria NPs: IV: 5 and 31 nm citrate-coated; 5 nm citrate/EDTA-coated; 28 nm naked; 29 nm silica-coated NPs	IV, oral, IT, and inhalation	Rats (M/F/NM)	(1) Permeability-limited PBPK model; (2) saturable RES uptake considered in every compartment; (3) LSA with 1% change in parameters; (4) unable to describe non-IV biokinetics (oral, IT, and inhalation); and (5) confirmed size, dose, coating, and route dependency in biokinetics, providing the basis for future risk assessment without additional experimental data
	Oral: 7/13 nm (TEM/HD); 30 nm (SEM); 23/190 nm (TEM/HD) naked NPs; IT: 7/13 nm (TEM/HD) naked NPs
	Inhalation: 2–3 nm (TEM) naked NPs
Nonmetallic NPs
Péry et al.151	99mTc-labeled carbon NPs: 5–10 nm	Inhalation	Humans (M)	(1) Integrating imaging measurements to update parameters estimated using Bayesian MCMC framework in the perfusion-limited PBPK model; (2) uniform partition coefficients for free and bound 99mTc- carbon NPs to simulate biodistributions; and (3) modeling errors and uncertainties as well as interindividual variability were modeled
Dogra et al.152	Silica NPs:32/46 nm (TEM/HD), −5 ± 1 mV	IV	Rats with and without tumor (F)	(1) Lymphatic system-based transport considered for healthy tissues while a permeability-limited transport for a solid tumor; (2) RES uptake was included in the liver and spleen; (3) LSA and GSA with parameters perturbated within ±99% range; (4) NP size and degradation rate, tumor vascular porosity and fraction, and tumor blood viscosity were key factors for low tumor delivery efficiency; and (5) presented values for optimal tumor delivery efficiency
	55/69 nm (TEM/HD), −7 ± 1 mV
	93/113 nm (TEM/HD), −7 ± 1 mV
	142/162 nm (TEM/HD), −4 ± 0 mV

^aAbbreviations: BSA, bovine serum albumin; CMD, count median diameter; c(RGDyC), cyclo(Arg-Gly-Asp-d-Tyr-Cys) peptide; EGCG, epigallocatechin gallate; EDTA, ethylenediaminetetraacetic acid; F, female; GI, gastrointestinal tract; GSA, global sensitivity analysis; HD, hydrodynamic size; ICP-MS, inductively coupled plasma mass spectrometry; ID, initial dose; IP, intraperitoneal; IT, intratracheal instillation; IV, intravenous; LM, mercaptoundecanoic acid cross-linking with lysine; LNs, lymph nodes; LSA, local sensitivity analysis; M, male; MCMC, Markov chain Monte Carlo; MMD, mass median diameter; MPS or RES, mononuclear phagocyte system or reticuloendothelial system; NM, not mentioned; NPs, nanoparticles; NRs, nanorods; PEG, poly(ethylene glycol); Pt, cisplatin; rhTNF, recombinant human tumor necrosis factor alpha; SC, subcutaneous; SEM, scanning electron microscope; SPIONs, superparamagnetic iron oxide NPs; TEM, transmission electron microscopy.

Table 4. Summary of Selected Physiologically Based Pharmacokinetic Modeling Studies for Organic Nanoparticles^a.

Refs	Physicochemical properties of administered NPs	Routes	Modeling species and sex	Main features and applications for the developed models
Polymerics
Wenger et al.153	PAA and PAA-PEG NPs with log-normal distribution and particle size of 30.9 ± 14.2 nm (HD) and ζ potential of −2.31 ± 0.77 mV	IV	Rats (M)	(1) Saturable RES uptake incorporated into the perfusion-limited semi-PBPK model; (2) used nonlinear ordinary least-squares regression method to assess model performance; and (3) comparisons made for biodistributions of different polymers with or without PEGylation
Li et al.154	PLGA NPs: 133.5 nm, −54.2 mV, PDI: 0.489	IV	Mice (F)	(1) Both perfusion- and permeability-limited PBPK models were compared to confirm that permeability-limited PBPK model was more appropriate for PLGA NPs; (2) LSA with 1% increase in parameters; (3) established relationships between NP properties and kinetic parameters for PLGA-mPEG495 model validation; and (4) implications include toxicity assessment and design of nanocarriers for imaging and drug delivery
	PLGA-mPEG495 NPs: 114.8 nm, −6.2 mV, PDI: 0.245
	PLGA-mPEG256 NPs: 97.4 nm, −5.9 mV, PDI: 0.385
	PLGA-mPEG153 NPs: 79.0 nm, −4.7 mV, PDI: 0.245
	PLGA-mPEG61 NPs: 67.0 nm, −5.2 mV, PDI: 0.292
	PLGA-mPEG34 NPs: 57.5 nm, −4.3 mV, PDI: 0.347
Li et al.129	PAA-PEG NPs with size of 30.9 ± 14.2 nm (HD) and surface charge of −2.31 ± 0.77 mV	IV	Rats (M)	(1) Permeability-limited PBPK model with RES uptake included in all tissues; (2) LSA: 1% change in parameters; and (3) phagocytic cells in richly perfused organs (e.g., lungs, heart, liver, spleen, bone marrow, and kidneys) capture the majority of NPs (83%) until saturation at 120 h
Gilkey et al.194	DEX-loaded and fluorescent dye labeled block copolymer NPs with size of 110 (TEM) and 124–127 nm (HD) and PDI of ∼0.05	IV	Mice (F)	(1) Perfusion-limited PBPK model; (2) LSA with ±20% change in parameters; and (3) proposed PBPK model can be an excellent tool for dosing optimization in acute leukemia treatment
Rajoli et al.195	Long-acting antiretrovirals loaded copolymer-based formulations for HIV treatment	IM and oral	Humans (F/M)	(1) Perfusion-limited whole-body PBPK model considering drug release from the IM depot and intestinal and first-pass metabolism; (2) the validated model helps identify optimal IM dosage to achieve an appropriate therapeutic concentration; and (3) provided a paradigm for the design of long-acting formulations for anti-HIV treatment
Shalgunov et al.196	92, 102.8, 108, and 118 nm (HD) VCR-loaded PLA-PEG NPs	IV	Tumor-bearing mice (F)	(1) Permeability-limited and tumor-bearing PBPK model incorporating drug release kinetics; (2) various partition coefficients considered for NPs and free drugs in different tissues; and (3) therapeutical effects need to be considered for this NP-based formulation because the systemic toxicity of API can affect both API and NP biodistributions
Jung et al.155	Flurbiprofen-loaded PCL NPs with size of 197.0 ± 0.8 nm, PDI of 0.154 ± 0.005, and a surface charge of −30.4 ± 1.0 mV	Oral	Humans (F/M)	(1) Permeability-limited PBPK model considering two-step absorbing process: diffusion via mucosa and permeation through the epithelial membrane; (2) in silico human PK prediction based on in vitro drug release assays; (3) LSA with ±10% change in drug release rate; and (4) dispersion releaser technology with higher detection sensitivity facilitates formulation design and quality control
Li et al.131	mPEG-PCL NPs	IV	Tumor-bearing mice (NM)	(1) Both perfusion- and permeability-limited tumor-bearing PBPK models were compared confirming perfusion-limited model was more appropriate; (2) LSA with 1% increase in parameters that determined maximum MPS uptake and release rates were key factors for NP biodistribution; and (3) successfully simulated in vivo fate of polymeric NPs traced by environmentally responsive near-infrared dye
	mPEG5k-PCL9.09%: 80/77.5 nm (TEM/HD), PDI: 0.12, −2.4 mV
	mPEG5k-PCL28.57%: 80/84.9 nm, PDI: 0.15, −3.4 mV
	mPEG2k-PCL28.57%: 200/207.6 nm, PDI: 0.06, −4.1 mV
	mPEG5k-PCL9.09%: 200/201.9 nm, PDI: 0.05, −2.8 mV
	mPEG5k-PCL28.57%: 200/213.7 nm, PDI: 0.03, −3.6 mV
Liposomes
Qin et al.197	Two dual-labeled PEG-conjugated liposomal NPs (long-circulating and temperature-sensitive) with 64Cu chelated on the shell and with hydrophilic near-infrared dye loaded in the core (∼100 ± 15 nm)	IV	Tumor-bearing mice (F)	(1) Perfusion-limited semi-PBPK model incorporating imaging measurements into the hybrid PK and PBPK model; (2) tumor vascular permeability estimates facilitate clinical translations; and (3) long-circulating liposomes had higher potential for therapeutic application
Kagan et al.156	Liposomal drug amphotericin B (AmB) (AmBisome) with mean diameter of <100 nm	IV	Mice (F), rats (M/F), and humans (M/F/NM)	(1) Dual PBPK model with drug release kinetics incorporated in liposomal phase while nonliposomal drug disposition is permeability-limited process; (2) a non-Michaelis–Menten mechanism included in liver and spleen to describe saturable MPS uptake of liposomes; and (3) the modeling framework has potential for optimizing AmBisome human therapy and investigating pathophysiological factors controlling AmB PK/PD
Lu et al.198	DTX-loaded FA–PEG-PCHL-modified liposomes (111.6 ± 9.6 nm)	IV	Mice (M/F), rats (M/F), and humans (M/F)	(1) Both perfusion- and permeability-limited PBPK models were compared and selected perfusion-limited model; (2) majority of estimated parameter values and predictions were withing bootstrap range and 90% CI, indicating model stability and adequate simulation; and (3) capable of considering both sex and dosage amount differences
Dendrimers
Opitz et al.199	Imaging dendrimers chelated with Gd3+ binding to the IGF-1 peptide analog	IV	Tumor-bearing mice (F/NM)	(1) Permeability-limited and tumor-bearing PBPK model; (2) a PBPK model for peptide–oligonucleotide chimeras in mice translated from a rat PBPK model; and (3) capability and potential for rational and optimal experimental design to improve clinical translation across species
Nanocrystals
Shono et al.158	125 mg nanosized aprepitant with mean particle size of 120 nm	Oral	Humans (M/F)	(1) Perfusion-limited PBPK model incorporating in vitro dissolution kinetics; and (2) the developed in silico model simulated adequately micronized and nanosized aprepitant under fasted and fed conditions
Dong et al.162	SNX-2112 nanocrystals with size of 203 nm, surface charge of −11.6 mV, and PDI of 0.153	IV	Rats (M)	(1) Perfusion-limited PBPK model with drug release kinetics; (2) wet-media milled nanocrystals enabled systemic delivery of insoluble drug SNX-2112; (3) cosolvent-like kinetics accounted for rapidly drug release; and (4) due to nanocrystal uptake, drug accumulated significantly at initial time points in the liver and spleen
Siccardi et al.159	Efavirenz SDNs (10% efavirenz + 75% polymer + 15% surfactant) with size, ζ potential, and PDI of 280 ± 254 nm (HD), −10.7 ± 21.8 mV, and 0.33 ± 0.19, respectively	Oral	Human cell lines (NA)	(1) Permeability-limited whole-body PBPK model with in vitro to in vivo extrapolation accounted for absorption and permeation processes; (2) SDNs had enhanced intestinal permeability and accumulation in different cell lines; and (3) valuable platform for SDN design and empirical method to select NPs with optimal properties and compositions
Kumar and Singh160	Carvedilol-loaded silk fibroin-casein NPs with optimized size of 240 ± 285 nm (SEM) and ∼350 nm (TEM) as well as surface charge of −22.3 mV	Oral	Rats (M) and humans (M/F)	(1) Whole-body PBPK model with in vitro NP dissolution established in GastroPlus; (2) sensitivity analysis was done using PSA module in GastroPlus to determine the impact of carvedilol release; and (3) enhanced bioavailability of optimized NP formulation with sustained effect demonstrates the potential of using silk fibroin as an alternative carrier for drug delivery
Litou et al.161	EMEND: 200 nm aprepitant tablet formulations	Oral	Humans (M/F/NM)	(1) Permeability-limited minimal PBPK model; (2) adequate prediction using PBPK model coupled with in vitro assay after oral administration at fasted and fed states; (3) the importance to consider gastric residence time and permeability-solubility interplay when predicting the absorption of poorly soluble API; and (4) this approach supported drug development by promoting rational formulation design and fewer, smaller, but equally robust clinical trials
Hybrid NPs
Mager et al.182	Au/PAMAM dendrimer nanodevices with sizes 5 nm (positive, negative, and neutral), 11 nm (negative), and 22 nm (positive)	IV	Tumor-bearing mice (M)	(1) Semi-PBPK model mixed with permeability-limited whole body PBPK model; (2) nontumor-bearing PBPK schematic was employed to calibrate with tumor-bearing mouse PK data; (3) 3-D GSA to determine the parameter sensitivity; and (4) identified the distributive properties of variously charged nanodevices using a PBPK model
Carlander et al.163	CTAB-coated gold NR with size of 56 × 13 nm (TEM) and 29.6 mV	IV	Rats (M)	(1) Generic permeability-limited PBPK model structure applied for inorganic and organic NPs; (2) NP-specific parameters estimated and optimized based on the generic model; (3) LSA with 1% change in parameters; and (4) dose and phagocytosis have profound impact on NP biokinetics and must be considered
	TiO2 NPs with size of 63 nm (HD) and surface charge of −43 mV
	31 nm (HD) PAA and PAA-PEG NPs with surface charge –2.3 mV
Cheng et al.164	Inorganic NPs: Gold (HD: 2.7–175.6 nm), silica (HD: 6.8–255.3 nm), iron oxide (HD: 16–139.1 nm), and other NPs (HD: 32–438.3 nm)	IV	Tumor-bearing mice (M/F/NM)	(1) Permeability-limited PBPK model; (2) successfully predicted tumor delivery efficiency of 376 types of NPs in tumor-bearing mice; (3) low tumor delivery efficiency estimates of ∼0.7% ID from publications in 2005–2018; (4) critical factors determining tumor delivery kinetics included ζ potential, cancer type, and tumor model; (5) low tumor delivery efficiency in AuNPs due to low distribution and permeability coefficients; and (6) provided a long-term strategy to facilitate future design of nanomedicines based on PBPK modeling perspectives
	Organic NPs: Dendrimer (HD: 5–277 nm), hydrogel (HD: 50–241.7 nm), polymeric (HD: 4–456.5 nm), liposome (HD: 12.4–386 nm), and other NPs (HD: 2.5–402 nm)
Price and Gesquiere18	Antibody: 10 nm IgG	IV, oral	Mice (M/F/NM), rats (M/F), and primates (NM)	(1) Lymphatic system-incorporated and perfusion-limited whole-body PBPK model; (2) integrated in vitro assay with computational fluid dynamic model to translate in vitro kinetics to whole-body simulations; (3) successfully validated across different NP types, routes, and species; and (4) perspectives to help refine, reduce, and replace animal testing during the drug development process
	QDs: 3.5 nm MPA-coated CdTe, 4.2 nm CdTe/CdS, 21.3 nm silica-coated CdSeS, and 7–25 nm LM-coated CdSe/ZnS
	Metallic NPs: 27.6 and 32 nm PEG-coated AuNPs, 31.5 nm PEG-coated and 66.1 nm PHEA-coated SPIO NPs
	Polymerics: 30.9 nm PAA–PEG NPs, 106 nm PS–PEO NPs, 112 nm 111In-PGA NPs, and 197 nm SA-PLGA NPs

^aAbbreviations: API, active pharmaceutical ingredient; CI, confidence interval; CTAB, cetyltrimethylammonium bromide; DTX, docetaxel; F, female; FA-PEG-PCHL, folate-poly(PEG-cyanoacrylate-co-cholesteryl cyanoacrylate); GSA, global sensitivity analysis; HD, hydrodynamic size; HIV, human immunodeficiency virus; ID, initial dose; IGF-1, insulin-like growth factor 1; IM, intramuscular; IV, intravenous; LM, mercaptoundecanoic acid cross-linking with lysine; LSA, local sensitivity analysis; M, male; MPA, 3-mercaptopropionic acid; mPEG, monomethoxy poly(ethyleneglycol); MPS or RES, mononuclear phagocyte system or reticuloendothelial system; NA, not available; NM, not mentioned; NPs, nanoparticles; PAA, poly(acrylamide); PAMAM, poly(amidoamine); PCL, poly(caprolactone); PDI, polydispersity index; PEG, poly(ethylene glycol); PGA, poly(glycolic acid); PHEA, poly(2-hydroxyethyl aspartamide); PLA, poly(lactide); PLGA, poly(lactic-co-glycolic) acid; PS-PEO, poly(styrene-b-ethylene oxide); QDs: quantum dots; SDN, solid drug nanoparticle; SEM, scanning electron microscope; SPIO, super paramagnetic iron oxide; TEM, transmission electron microscopy; VCR, vincristine.

Figure 3.

Schematic of PBPK model structures for nanoparticles. (A) A typical PBPK model for the nanoparticle itself only. (B) A PBPK model for the nanoparticle and its associated drug that is released from the nanoparticles. In the plasma and tissue of (B), drugs can be released from the nanoparticle carriers and then distributed throughout the body via blood flow. This figure was created with BioRender.com.

Inorganic Nanoparticles

Quantum Dots

A PBPK model is described in Lin et al.¹²⁵ for QD 705 in mice following intravenous injection. At that time, the PBPK modeling methodology for NMs just started to be explored. In this study, based on the most commonly used model structure for small molecules, the authors used a perfusion-limited model structure for all compartments to simulate the tissue distribution of QD in mice. However, unlike small molecules whose extent of tissue distribution is described using a constant parameter tissue:plasma partition coefficient (or tissue:blood partition coefficient), the authors used a nonconstant parameter termed “tissue distribution coefficient” to describe the ratio of the affinity of QD to a given tissue over that to blood and considered this parameter varied with time depending on the respective instantaneous concentrations in the individual tissues and blood as well as the microenvironment at the tissue site. The authors calculated the tissue distribution coefficient value using the AUCtissue/AUCblood method for a specific time period, resulting in a time-dependent tissue distribution coefficient for each tissue. The authors found that the time-dependent tissue distribution coefficient for each tissue was sigmoidal in shape when plotting against time, and this sigmoidal curve could be described using a modified Hill eq 10:

10

where DC_m is time-dependent distribution coefficient of QD for each tissue, DC_max is the maximum DC, DCT₅₀ is the time reaching half of the maximum DC (h), T is the experimental time (h), In is the initial value for DC, and n is the Hill coefficient, which dictates the overall shape of the curve. The use of a time-dependent distribution coefficient for NPs, rather than a partition coefficient as a constant represents one of the major differences in PBPK modeling between small molecules and NPs. The method of time-dependent distribution coefficients was later adapted and further developed by Lin et al.¹²⁶ for AuNPs (described below).

Based on the PBPK model for QD 705 in mice by Lin et al.,¹²⁵ Lee et al.¹²⁷ developed a flow-limited PBPK model using the same QD 705 data set from Lin et al.¹²⁵ and applied this model to test its predictive ability across different types of QD (e.g., QD 705, 525, 800, and 521), species (mice and rats), and routes (intravenous and intradermal). The authors reported that the PBPK model that was calibrated based on the data for QD 705 was not able to adequately describe the distribution for other types of QD in the same or other species following the same intravenous administration or other routes of exposure. Therefore, the authors concluded that more complex PBPK models are needed to better describe the biodistribution of QD in vivo following different types, species, and administration routes. Later on, a more complex permeability-limited PBPK model developed by Liang et al.,¹²⁸ based on a more mechanistic PBPK model from Lin et al.,¹²⁶ was able to adequately simulate the biodistribution of different types of QD following different routes of administration (intravenous and subcutaneous) in mice and rats.

Gold Nanoparticles

In the field of PBPK modeling for NPs, AuNPs are the most extensively modeled type of NPs. This is in part because gold is inert and very stable and does not metabolize or degrade in the body (so it is not needed to consider complex metabolism and dissolution processes) and analytical methods are available to accurately quantify the biodistribution of gold in the body. This also makes AuNPs an ideal example NM to explore PBPK modeling methodology for NMs.

Among the published PBPK models for AuNPs, Lin et al.¹²⁶ reported a PBPK model for PEG-coated AuNPs in mice. In this study, the authors systematically evaluated the performances of 14 different model structures for two sizes of AuNPs (a relatively small size of 13 nm and a relatively larger size of 100 nm). These structures included the traditional perfusion-limited model, traditional membrane-limited model, perfusion-limited model with time-dependent distribution coefficients, membrane-limited model with time-dependent distribution coefficients, perfusion-limited model with endocytosis of NPs in liver and spleen described using a linear equation, membrane-limited model with endocytosis of NPs in liver and spleen described using a linear equation, membrane-limited model with endocytosis of NPs in liver, spleen, kidneys, and lungs described using the Hill function, etc. The authors concluded that the model structure with the best predictive ability was the membrane-limited model with endocytosis of NPs in liver, spleen, kidneys, and lungs described using the Hill function.

Compared to the earlier study by Lin et al.,¹²⁵ the study by Lin et al.¹²⁶ also used the parameter “distribution coefficient” and further suggested that the other parameter “partition coefficient” is not applicable to NPs because this parameter represents the ratio of concentrations of a dissolved compound between two phases at thermodynamic equilibrium, but NPs do not form solutions, but colloidal dispersions, which are thermodynamically unstable. However, unlike Lin et al.,¹²⁵ which used a time-dependent tissue distribution, Lin et al.¹²⁶ used a constant value for the distribution coefficient for each tissue. Instead, Lin et al.¹²⁶ proposed that endocytosis of NPs from the blood circulation into tissue cells is the major mechanism that determines the tissue distribution profiles of NPs, and this process can be described with the Hill equation:

11

where K_{up_t} (per h) is the uptake rate parameter of NPs by phagocytic cells in an organ t at a particular time T, T (h) is the simulation time, K_{max_t} (per h) is the maximum uptake rate parameter in the organ t, K_{50_t} (h) is the time reaching half of K_{max_t}, and n_t (unitless) is the Hill coefficient. This approach was created based on in vitro cellular uptake studies (as described above), earlier PBPK models for other types of NPs,^11,12,129 and after comparing the model simulation results of 14 different model structures.¹²⁶ This modeling strategy is now commonly used for the PBPK modeling of NPs.^130−132

The PBPK model for AuNPs in mice was later extrapolated to rats, pigs, and humans.³⁵ In this study, the authors performed interspecies extrapolation from each of the animal species (i.e., mice, rats, and pigs) to humans and found that rats and pigs might be more appropriate animal models than mice in animal-to-human extrapolation of the PK of AuNPs and the dose and age should be considered in this extrapolation. The final optimal human model was derived from the rat model. This group of researchers also collected literature data on in vitro toxicity of AuNPs in primary human liver and blood cells and in vivo toxicity in rats^133−135 and then calculated human equivalent doses associated with these in vitro and in vivo toxicity end points. Human equivalent doses are a crucial component in the derivation of a reference dose for chronic exposure to a xenobiotic in humans.

The PBPK model for AuNPs in rats and humans³⁵ provides a basis for further development of toxicity and risk assessment approaches for NPs. The same group of researchers conducted extensive in vitro toxicity assays of AuNPs using different types of primary human cells (i.e., hepatocytes, umbilical vein endothelial cells, renal proximal tubule epithelial cells, and keratinocytes).^136−139 Based on their PBPK model, in vitro toxicity studies, and additional in vivo toxicity data from the literature, they developed a probabilistic risk assessment approach for AuNPs.¹³ A schematic of this approach is provided in Figure 4. This method can be used to calculate human equivalent doses associated with reported points of departure from in vitro and in vivo toxicity, to predict the potential risk of exposure to AuNPs under different scenarios, and to perform IVIVE and animal-to-human exposure.

Figure 4.

A flowchart of a probabilistic risk assessment approach for nanoparticles. E_min and E_max: minimum and maximum fractional cell death; EC₅, EC₁₀, and EC₅₀: exposure concentration causing 5%, 10%, and 50% maximum cell death, respectively; n: Hill coefficient; PDF: probability density function; CDF: cumulative distribution function; NOAEL and LOAEL: no and lowest observed adverse effect level, respectively. Reproduced with permission from ref (13). Copyright 2018 Taylor & Francis.

Besides the PBPK models by Lin and colleagues mentioned above, multiple other groups have also published PBPK models for AuNPs. Bachler et al.¹² integrated in vitro and in silico methods to evaluate the pulmonary translocation and biodistribution of AuNPs using a PBPK model. The authors determined the translocation kinetics across the lung epithelial tissue barrier using human and mouse alveolar epithelial cell’s exposure to AuNPs of various doses (25–200 ng/cm²) and sizes (2–80 nm) for different time periods (0–72 h). They found that the translocation kinetics of AuNPs across human and mouse lung epithelial cells was similar. The translocation fraction was inversely proportional to the size of the particle and was independent of the applied dose (up to 100 ng/cm²). The authors incorporated the measured in vitro translocation kinetics into a PBPK model, and the derived PBPK model accurately predicted the in vivo biodistribution of AuNPs following inhalational or intratracheal instillation in rats. The authors concluded that this combined in vitro and in silico PBPK modeling approach has the potential to replace short-term animal studies that aim to assess the pulmonary absorption and biodistribution of NPs and to serve as a screening tool to identify NPs of toxicity concern.

Similar to the earlier models by Lin et al.¹²⁶ and Bachler et al.,¹² recent PBPK models for AuNPs by other groups^140−143 had similar model structures, in which there was an “endocytosis” subcompartment in each major RES tissue compartment that accounted for the mechanism of active cellular uptake of NPs by phagocytic cells. The common conclusion from these studies was that cell-mediated endocytosis played an important role in the biodistribution of NPs in the body. Note that the model by Chou et al. was translated to a user-friendly web-based interface so that readers can apply this web interface to run PBPK simulations to predict the concentrations of AuNPs in plasma and tissues following different routes of administration.¹⁴³

Silver Nanoparticles

PBPK modeling for AgNPs is different from AuNPs as dissolution and metabolism need to be considered. Bachler et al.¹¹ developed a PBPK model for both ionic silver and AgNPs in rats and humans following oral, dermal, and inhalational exposures. In this model, two possible pathways were considered for the metabolism of AgNPs: (1) AgNPs dissolve and release soluble silver species, which are then connected to the ionic silver PBPK model; and (2) AgNPs are directly transformed to silver sulfide particles as a storage depot in the tissues. The model was calibrated with data from intravenous exposure studies and validated with independent data following other routes of exposure (dermal, oral, and inhalation). The final model was able to successfully simulate the biodistribution of both ionic silver and 15–150 nm AgNPs (those that were not coated with substances designed to prolong the circulatory time, such as PEG) following different routes of administration in all cases examined in their study. This model was applied to assess the exposure and risk of ionic silver and AgNPs based on five exposure scenarios, including dietary intake, oral uptake of AgNPs released from food boxes, dermal uptake of AgNPs released from T-shirts, oral uptake of emitted AgNPs from a throat spray, and occupational exposure. The results showed that in all consumer product scenarios, the silver levels of all organs were around or below the dietary background (except the lung silver level after the usage of the colloidal silver throat spray), indicating that the risk of adverse health effects from exposure to AgNPs in consumer products was very small. However, in the occupational exposure scenario, the predicted soluble silver levels in different organs exceeded the background levels by a large margin with the levels in the bone marrow and lung above levels where adverse effects have been reported, indicating a significant risk for the occupational exposure population.

Titanium Dioxide Nanoparticles

Bachler et al.¹² also developed a PBPK model for titanium dioxide NPs (15–150 nm) in rodents following different routes of exposure (intravenous, oral, and dermal) with a similar model structure as their earlier model for AgNPs.¹¹ The major difference was that dissolution was not considered as titanium dioxide NPs are very stable in the body.¹² Route-to-route extrapolation from intravenous to oral or dermal was done by including an intestinal absorption fraction or percutaneous absorption fraction to account for the orally absorbed and dermally absorbed doses, respectively. These fractions were estimated by fitting to oral and dermal data sets from the literature using the PBPK model. The oral PBPK model was then used to assess the internal exposure and potential risk following dietary exposure to titanium dioxide NPs for the German population divided into six age classes (toddlers [1–2 years], other children [3–9 years], adolescents [10–17 years], adults [18–64 years], elderly [65–74 years], and very elderly [75 years and older]) on the basis of three main consumer product groups: (1) foods and beverages, (2) drugs and dietary supplements, and (3) toothpastes. The results showed that the estimated daily median intake was between 0.5 and 1.0 mg/kg for all age groups except the age group “other children” (∼2.0 mg/kg). The estimated 95th percentile titanium organ levels after oral ingestion were <170 ng/g for all age groups and in all organs. This value is much lower than levels where toxic effects have been reported in most in vitro studies.^12,144,145 The authors concluded that the risk from oral exposure to titanium dioxide NPs via the studied main consumer products was small for the German population.

Other Metal Nanoparticles

Besides the above-mentioned models, PBPK models have been developed for other inorganic NPs, such as zinc oxide,¹⁴⁶ iron oxide,^132,147 ceria,^130,148 iridium,^149,15099mTc-labeled carbon,¹⁵¹ and silica¹⁵² NPs. Readers are referred to Table 3 and the original manuscripts of these models for further details.

Organic nanoparticles

Polymeric Nanoparticles

PBPK models have been published for different types of polymeric NPs, including PEG-coated polyacrylamide (PAA-PEG) NPs,^129,153 poly(lactic-co-glycolic) acid (PLGA) with varied content of monomethoxypoly(ethyleneglycol) (mPEG) (PLGA, PLGA-mPEG256, PLGA-mPEG153, PLGA-mPEG51, PLGA-mPEG34),¹⁵⁴ poly(l-lactide)-poly(ethylene glycol) block-copolymer (PLA-PEG) NPs,¹⁵⁴ and polymeric methoxy poly (ethylene glycol)-poly(ε-caprolactone) (mPEG-PCL) NPs.^131,155 Among these studies, the model by Li et al.¹²⁹ is one of the most comprehensively described. In this study, the model consisted of 9 compartments: blood, lungs, spleen, liver, kidneys, heart, brain, bone marrow, and the rest of body. Within the blood and each tissue compartment, there were three subcompartments representing capillary blood, tissue, and phagocytizing cells. The model was built based on the assumption that upon entering into the blood circulation, NPs are distributed throughout tissues, and a fraction of the NPs entering each tissue is sequestrated by phagocytic cells. As the amount of captured NPs increases, phagocytic cells can eventually be saturated. The maximum uptake capacity was assumed to be organ-specific to reflect the variation in the density of phagocytic cells as well as differences in uptake capacity for different types of phagocytic cells in different organs. It was also assumed that NPs could re-enter the tissue after desorption from phagocytic cells by processes of exocytosis. This PBPK modeling strategy was adapted by Lin et al.¹²⁶ to develop a PBPK model for AuNPs described above, and this revised method is now commonly used in developing PBPK models for NPs.

Another representative PBPK modeling study for polymeric NPs is the one by Li et al.¹⁵⁴ for PLGA NPs. In this study, experimental data sets were available for 5 different types of PLGA NPs with different sizes (57.5–133.5 nm), ζ potentials (−4.3 to −54.2 mV), and number of PEG molecules per unit surface area (0–0.77). A PBPK model was developed and used to estimate biodistribution parameters for each type of these NPs. Multivariate regression analyses were performed to build the relationship between NP properties and biodistribution parameters (i.e., organ-specific diffusion coefficients, partition coefficients, and excretion coefficients). The resulting multivariate regression equations were used to estimate biodistribution parameters for a sixth type of PLGA NPs (PLGA-mPEG495) with different physicochemical properties, and the PBPK simulated concentrations were compared with observed data at 3 h after intravenous injection. The results showed that the PBPK-predicted values were close to experimental data for all tissues. This study demonstrates that it is possible to develop NP property-biodistribution relationship models to estimate biodistribution parameters and generate PBPK models for the same type of NPs with variations in physicochemical properties.

Liposomes

Liposomes are a common nanoparticulate formulation among the approved nanomedicines.¹ Kagan et al. developed a dual PBPK model to describe the disposition of both liposomal and nonliposomal formulations of amphotericin B in mice and rats following intravenous injection, and then extrapolated the model to simulate amphotericin B concentrations in human tissues.¹⁵⁶ The authors started with a PBPK model for a nonliposomal formulation of amphotericin B as this is an important prerequisite and provides a basis for understanding the PK of the liposomal particulate formulations.¹⁵⁷ The authors established a compartmental release model to simulate the release of amphotericin B from liposomes by fitting to plasma concentration–time profiles for total and nonliposomal drug following intravenous administration of the liposomal and nonliposomal drug in rats and humans. Next, a dual PBPK model was created for liposomal and nonliposomal amphotericin B. In this model, liposomal amphotericin B circulates throughout the body via blood capillaries and can undergo release to its nonliposomal form by a first-order rate constant. This release was assumed to take place in both plasma and tissues. The derived human PBPK model-predicted concentrations of liposomal and nonliposomal concentrations of amphotericin B well corresponded the observed plasma data in humans following intravenous injection of a liposomal formulation (AmBisome) and also correlated with limited concentration data in various tissues from autopsy reports. Since therapeutic success of amphotericin B depends on its concentration in target tissues, the authors concluded that this model can be potentially utilized to optimize therapeutic plans for AmBisome in humans and to investigate pathophysiological factors controlling amphotericin B PK and pharmacodynamics.

Nanocrystals

Nanocrystals (or nanosuspensions) refer to a submicron colloidal dispersion system of pure drug particles in water. Nanocrystals have many strengths in drug delivery, especially for drugs with poor solubility (those in Biopharmaceutics Classification System Classes II or IV). These strengths include: (1) a high drug load because the drug is suspended in a solid state; (2) no significant toxicity concern because only a small quantity of excipient (stabilizer) is used; (3) suitability for various administration routes, including oral, parenteral, pulmonary, and ocular pathways; and (4) great potential in passive targeting and provide a depot effect. Dong et al. developed a PBPK model for SNX-2112 (a potential anticancer drug with poor solubility in water) and its nanoparticulate formulation as nanocrystals. The PBPK model for the nonparticulate drug cosolvent is a standard perfusion-limited model; whereas in the PBPK model for the nanocrystals, a species of the nanoparticulate drug was included in the plasma, liver, and spleen. The in vitro release kinetics of the nonparticulate form of SNX-2112 from the nanoparticulate form was determined using the bulk-equilibrium reverse dialysis method and then described using a first-order rate constant. This model was built based on the assumption that uptake of nanocrystals only takes place in the liver and spleen (two major RES organs), thus the nanoparticulate form of the drug is only present in plasma, liver, and spleen. This assumption worked well in their model as their model was able to adequately simulate the available experimental data. However, this assumption is considered overly simplified as phagocytic cells also exist in other organs of the RES systems (i.e., lungs, kidneys, and bone marrow), and many PBPK modeling studies suggest the endocytosis of NPs should also consider other RES organs, such as kidneys and lungs.

PBPK models have also been developed for other nanocrystals.^158−161 Similar to the study by Dong et al.,¹⁶² these models were developed mainly to support the development and evaluation of nanoparticulate formulations of poorly soluble drugs. These studies consistently support that PBPK models can be a useful tool in correlating in vitro release kinetic profile of a poorly soluble drug with in vivo PK profile of the drug in either nanoparticulate or nonparticulate forms, thereby helping develop nanoparticulate forms of poorly soluble drugs.

Other Organic or Hybrid Nanoparticles

As described above, there are numerous different types of NPs, and the same type of NPs could have many variations in physicochemical properties (i.e., differences in sizes, ζ potential, surface coating). As such, it is impractical to develop an individual PBPK model for each type of NPs (this would be a daunting task) for safety and risk assessment purposes. Several research groups have been trying to develop generic PBPK models for multiple different types of NPs, including both inorganic and organic NPs. Carlander et al.¹⁶³ reported a generic PBPK model for four types of nondegradable NPs, including PEG-coated PAA, uncoated PAA, gold, and titanium dioxide NPs in rats following intravenous injection. The same model structure and physiological parameter values were applied, whereas NP-specific parameters were estimated by fitting to respective experimental time–concentration data in various tissues for each type of NPs. The final model adequately simulated the observed PK behavior of all four types of NPs despite substantial differences in the PK behavior and physicochemical properties. Similarly, Price and Gesquiere¹⁸ developed a generic PBPK model for many different types of NPs (i.e., QD, gold, silica, iron oxide, and polymeric NPs) in healthy animals of several species (i.e., mice, rats, and monkeys).

In addition, Cheng et al.¹⁶⁴ developed a generic PBPK model that is applicable for a variety of NPs in tumor-bearing mice. This model was based on their earlier model for AuNPs in healthy mice.^35,126 The authors extended their model to tumor-bearing mice by adding a tumor compartment. All physiological parameters and NP-specific parameters in nontumor tissues were kept the same as in the healthy mice. Only parameters related to the tumor compartment were estimated for each type of NPs by fitting to their respective data set from the literature (summarized and published as an Excel file termed Nano-Tumor Database).¹⁶⁴ The model was used to simulate distribution kinetics in the tumor tissue for different types of NPs and the performance was evaluated against 376 data sets. The model was able to adequate simulate >83% of the data sets (with a determination coefficient of R² > 0.75 in the regression analysis between predicted and observed data). The authors conducted multivariate regression analyses to determine the relationship of tumor delivery efficiency with physicochemical properties and targeting strategies and cancer types. It was found that low tumor delivery efficiency of ∼0.7% injected dose was associated with low distribution and permeability coefficients at the tumor site. This study demonstrates that PBPK modeling can be an effective tool to identify determinants associated with target tissue dosimetry in tumors or normal tissues.

In Vitro to In Vivo Extrapolation of Cellular and Tissue Dosimetry of Nanoparticles

IVIVE is a well-established approach for PBPK modeling of small molecules. The latest PBPK modeling guideline from Organisation for Economic Cooperation and Development (OECD) provides strategies to develop and evaluate or validate a PBPK model for a chemical only based on in vitro and in silico data, without relying on in vivo PK data for the model validation.¹⁶⁵ One approach to address the above-mentioned limitation to account for cell-mediated endocytosis of NPs in a PBPK model is to measure the cellular uptake and release kinetic parameters using in vitro cell models and then translate the in vitro results to in vivo based on cell type and cell density of individual organs. Some studies have successfully implemented this approach. For example, Bachler et al. determined the translocation kinetics of AuNPs across human and mouse lung epithelial cells and then incorporated this parameter into a PBPK model; the model-predicted pulmonary translocation and biodistribution correlated with the observed data well.¹⁶⁶

Price and Gesquiere¹⁸ measured the in vitro uptake, degradation, and release kinetics of different NPs in different cells. A step-by-step process of the IVIVE in this study is illustrated in Figure 5. The overall simulation was based on a three-compartment model that included the medium compartment, cell membrane compartment, and cell space compartment as described in detail above. The in vitro cell kinetics was translated to whole-body simulations based on the in vitro assay results using a computational fluid dynamic model to predict NP distribution into individual tissue cells. In the body, the transport of a NP from the blood supply to cells of a particular tissue was simulated through the computational fluid dynamic theory primarily obtained from a hydrodynamic simulation of solutes through cell membrane pores captured by modeling a reflection coefficient, σ. When the reflection coefficient approaches 0, NPs can enter the pathways to interstitial space or cell space through the cell membrane. When it approaches 1, the membrane pore excludes the NP and it remains outside the pores. Variable reflection coefficients were calculated using a fluid dynamic model to account for differences in NP size, animal species, and capillary membrane pore diameters. The final model was validated against 15 preclinical data sets, which included different dosing scenarios (0.029 to 64.3 mg/kg body weight), NP types (polymer, QD, metal, and antibody), and sizes (4 to 197 nm diameter) in mice, rats, and nonhuman primates. The authors concluded that this in vitro- and in silico-based PBPK modeling strategy provides a viable platform to refine, reduce, and replace animal testing in nanomedicine development and safety evaluation.

Figure 5.

General overview of the Nano-IVIVE-PBPK modeling process. In vitro data on cell-nanoparticle interactions serve as the input for the IVIVE process, which delivers whole body predictions on nanoparticle predisposition. Clockwise from bottom left, in vitro data representative of the time evolution of cell-nanoparticle interactions are collected and are treated with an in vitro kinetics model to extract parameters describing interactions considered, such as rate constants for of adsorption, desorption, internalization, degradation, exocytosis, and other parameters. The example data at the top illustrate how such parameters can then be used to learn about, e.g., subcellular processes. These parameters can then be scaled for in vivo application in pharmacokinetic modeling to predict the ADME of the nanoparticles. An example for the liver is shown on the right of the scheme, illustrating reported in vivo data and the corresponding prediction of the pharmacokinetic model. Applying this approach for multiple tissues and organs allows for whole-body in silico predictive simulations. Adapted with permission under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/) from ref (112) and under a Creative Commons Attribution CC BY-NC license (https://creativecommons.org/licenses/by-nc/4.0/) from ref (18). Copyright 2020 The Authors, some rights reserved; exclusive licensee AAAS for ref (18). Copyright 2019 The Authors for ref (112). Cartoons are in the public domain at https://publicdomainvectors.org/.

However, not all studies successfully implemented this IVIVE of the cellular uptake and release kinetics. For example, Dubaj et al.¹⁹ determined the internalization and exclusion of PEG-coated AuNPs in several human cell lines (TH1, A549, Hep G2, and 16HBE), modeled these processes with first-order rate equations, and then translated the in vitro uptake and release parameters to in vivo using a perfusion rate-limited PBPK model. However, there were notable differences between the observed internalized amount of gold in individual cell lines and the corresponding tissues in vivo. The authors concluded that researchers should be cautious when extrapolating in vitro kinetic data to predict tissue biodistribution of NPs in vivo.

Challenges and Future Perspectives

Challenges and Future Perspectives on In Vivo Pharmacokinetic Study of Nanoparticles

NPs have been identified as promising candidates for many biological and biomedical applications. Because of their small particle sizes, versatile physicochemical properties, and ease of their fabrication involving bottom-up and top-down approaches, NPs have been widely used, more specifically, as a drug delivery system. Special properties arise due to nanosize, including a high surface-to-volume ratio has given us tremendous room to envision it as a multimodality system. Despite its vast potential benefit in the biomedical field, there are technical challenges to map their tissue bioavailability and toxicology. Commonly synthesized NPs are often characterized in the ranges of sizes, although unifying their size distribution and segregating uniform size is possible but results in low product yield for clinical application. While efforts have been made to study the size effect on tissue distribution and toxicity, less attention has been given to the systematic challenges arising due to the protein corona formation that significantly alters the NP’s PK and TK. The formation of protein corona over the surface of NPs in circulation affects NP uptake and diffusion throughout the surrounding tissue; hence, it is crucial to design experimental and mathematical approaches to account for these variations. Fluorescence probes have been used to account for the distribution of NPs in tissues, cells, and residence in plasma; whereas less attention is given to using multimodality techniques to map its distribution accurately. Therefore, future studies should be based on the question raised, such as distribution differences for NPs with different sizes, chemistry, and functionalities, which are central questions that apply to any enveloped NP systems and have parallels to the structural changes and interactions seen for many NPs, and so the results will clarify some of the underlying rules about how NPs interact with their host ligands to target intended sites, thereby helping assess the potential risk of different NPs with target site dosimetry.

Challenges and Future Perspectives on In Vitro Nanoparticle-Cell Interaction Kinetics Research

The in vitro NP kinetics work reviewed above demonstrates that in vitro studies allow for the discovery of pharmacological end points including NP uptake and removal with associated kinetics. Care should be taken however to ensure reproducible data are interpreted correctly.

First, studies by Kim et al. have shown that cellular uptake of NPs is affected by the cell cycle.¹⁶⁷ While uptake rates stayed consistent in different cell cycle phases, it was found that the dose of NPs internalized in each cell in a population was dependent on the phase of the cell cycle. It was also found that the absence of uptake saturation was related to cell division rather than the expulsion of NPs.

Second, corona formation can strongly alter the kinetics of NP-cell interactions. The effect of corona formation on NP-cell interaction kinetics was investigated for carboxylated polystyrene and silica NPs.¹⁶⁸ The authors reported that protein adsorption on the NP surface significantly reduced NP adhesion on the cell surface compared to the bare NPs for the case of nonspecific interactions. This was attributed to the lower surface energy of the NPs when they have a protein corona, illustrating the importance of the biological environment on NP-cell interaction kinetics. With the rational design of NPs, the formation and composition of protein corona can be controlled as demonstrated by Yang et al.¹⁶⁹ Changing the fraction of zwitterionic and charged lipids in liposomes was demonstrated to affect the amount and type of serum proteins adsorbed on the NPs in vitro. Moreover, the different liposome formulations also resulted in varying uptake kinetics. These findings suggest the potential to tune NP design to achieve desired cell uptake properties through the engineering of the protein corona.

Third, the effect of diffusion and sedimentation phenomena on cellular uptake of NPs was studied in depth by Cho et al. for AuNPs of varying shapes cultured in vitro under upright and inverted configurations.¹⁰⁰ The researchers concluded that for particles that sediment faster there was a greater difference in cellular uptake between the upright and inverted in vitro configurations, in particular where the ratio between the sedimentation and diffusion velocities is >1. Some AuNPs (54 nm nanospheres and 62 nm nanocages) did show back diffusion due to concentration gradient so this has to be considered carefully as well. In a later study, Bancos et al. further validated that assay preparation and conditions affect quantitative outcomes and thus need to be carefully considered.¹⁷⁰ Such reports indicate that some issues with unexpected or inexplicable results from in vitro assays can be traced to sedimentation of larger or dense NPs. For instance, Bancos et al. found that increasing the applied dose of AuNPs leads to increased uptake rates, and that certain media conditions could lead to particle aggregation.¹⁷⁰ Since sedimentation of dense particles typically results in a higher exposure dose of cells to NPs, not considering this effect could lead to unexpected or misunderstood results. A similar conclusion was made by Rischitor et al., noting that NP deposition occurs in two phases, where phase 1 is cellular uptake driven by NP-cell affinity and phase 2 is the deposition of NPs on the cell membrane.¹⁷¹ The authors also suggest that particle size and exposure duration play their part in these phenomena together with the liquid height in the wells for in vitro assays. Conversely, a recent study by Bitounis et al. for larger cellulose NPs demonstrates that only 26% of the applied dose could reach within 100 μm from the bottom of a well after 24 h, where cells would be located.¹⁷²

This was addressed in particular by Teeguarden et al.¹⁷³ and Hinderliter et al.¹⁷⁴ in their ISDD model built on the work by Limbach et al.¹⁷⁵ Gravitational sedimentation and particle aggregation were considered in the calculation of the true delivered NP dose, which can differ from the applied dose in an in vitro assay. The In vitro Sedimentation, Diffusion, Dissolution, and Dosimetry (ISD3) model extends this work to include particle dissolution.¹⁷⁶ Along similar lines, Cohen et al. propose a further study on particle-media interactions that NP dispersion protocols should be standardized to ensure that in vitro research shows less variation between studies.¹⁷⁷ This was more recently corroborated by Moore et al. when the authors reported that the method of applying AuNPs in vitro affected the fraction of NPs deposited on cells.¹⁷⁸ Note that the ISDD and ISD3 models^174,176 are mainly used to predict the delivered dose on the membrane of cells at the bottom of the cell culture well based on the physicochemical properties of NPs and the in vitro cell culture and exposure conditions. This is different from the in vitro cellular kinetic models reviewed in this article, which are mainly used to simulate uptake and release of NPs into and out of cells and can be used to predict intracellular dose of NPs. However, the connection of these two types of models is that the delivered dose predicted from the ISDD or ISD3 model can serve as a more accurately input dose parameter to the in vitro cellular kinetic model to predict intracellular dose.

Fourth, Serpooshan et al. have made a case for the importance of considering the effect of cell sex on in vitro NP kinetics.¹⁷⁹ The researchers discovered that differences exist in NP uptake for human amniotic stem cells, with greater uptake of NPs by female cells. When this study was replicated with fibroblast the opposite effect of cell sex was found.

Challenges and Future Perspectives on PBPK Modeling and IVIVE of Nanoparticles

Unlike PBPK modeling for small molecules and macromolecules such as monoclonal antibodies for which the methodology is well-established,^180,181 PBPK modeling for NPs is an emerging field, and the methodology is still under development. While there are around 50 published PBPK models for different NPs, many of these studies use different methods. Based on existing studies, there are many differences in PBPK modeling methods between small molecules and NPs, as summarized in Table 5. All of these have caused a challenge for the field to progress and also raised a need to comprehensively evaluate different strategies to determine the best PBPK modeling methods for NPs. In this review article, several key points are discussed below along with our perspectives.

Table 5. Differences in Physiologically Based Pharmacokinetic Modeling between Small Molecules and Nanoparticles^a.

PBPK modeling considerations	Small molecules	Nanoparticles
Dose metrics and unit	In mass per unit volume, such as mg/L or mM/L for concentration of a chemical in plasma or mg/kg in tissues.	Usually the same as small molecules in mass per unit volume, but some studies showed that the dose metrics of particle number or surface area may be more appropriate for certain toxicity end points, such as lung inflammation.4
Absorption	Approaches to simulate extravascular absorption for small molecules are well established. Depending on the administration route and availability of data, it could be a simple first-order linear absorption or a more complex mechanistic model, such as the advanced compartmental absorption and transit model for oral absorption.	Approaches to simulate extravascular absorption of NPs remain to be established. In general, the absorption of NPs from extravascular routes is very low due to the size of the particle. This makes it difficult to collect time-dependent kinetic data to build a PBPK model for extravascular routes. Currently most PBPK models for NPs are only for IV route.
Model compartment structure	Both perfusion-limited and diffusion-limited model structures are appropriate depending on the physicochemical properties of the chemical (e.g., molecular weight, lipophilicity, and hydrophilicity).	Several studies compared perfusion-limited with permeability-limited model structures for NPs. Some concluded that permeability-limited model was more appropriate for NPs,126,154 whereas others showed that perfusion-limited model worked better.131,198 Most existing PBPK models for NPs are based on a permeability-limited model structure. Most of these models divided an organ into three subcompartments, including vascular space, phagocytic cells that actively uptake NPs, and the rest of the organ. However, the proper PBPK model compartmental structure for NPs remains to be developed and experimentally verified.
Circulation through body	For the majority of small molecules, systemic distribution occurs through the blood circulatory system.	Trafficking through the lymphatic system may also occur and dictate the structure of the PBPK model used. Note that for PBPK modeling of monoclonal antibodies, it is well accepted lymphatic system plays an important role in tissue distribution and should be included in a PBPK model.183 For NPs, many existing PBPK models for NPs did not include lymphatic systems, but a few did. Whether the lymphatic system should be included in a PBPK model for NPs remains to be investigated.
Plasma protein binding	Small molecules can undergo nonspecific protein binding (e.g., plasma albumin) by an equilibrium process. This can be described as a saturable process (e.g., association rate constant, dissociation rate constant, and maximal binding capacity).200 Alternatively, this can be simply described using a plasma protein binding percentage assuming this percentage remains the same across a wide dose range, which can be supported if experimental data are available.	NPs may associate with multiple proteins and other macromolecules in a dynamic fashion, sensitive to the environment, to form a biocorona that could be a primary determinant of biodistribution and elimination. Protein corona could be “hard” or “soft” depending on the affinity and interaction between NPs and proteins.4 Approaches to incorporate biocorona formation kinetics into a PBPK model for NPs have not been established.
Cellular uptake	In most cases, cellular uptake occurs by either diffusion down a concentration gradient or by classic molecular transporter systems (organic acid transporter system, p-glycoprotein, etc.) that are well described using saturable but reversible models (e.g., Michaelis–Menten).	Cellular uptake is via capacity-limited vesicular transport systems with charge and size specificity (e.g., phagocytosis, micro- and macro-pinocytosis, and membrane rafts). Release of NPs from cells via exocytosis should be considered when simulating the cellular uptake and release kinetics.
Tissue trapping	Small molecules which are week acids or bases get ion-trapped in tissues with different pH as only the noncharged moiety can diffuse across the membrane. This process can be calculated using the Henderson–Hasselbalch equation.	NPs have colloidal properties that result in aggregation or agglomeration depending upon local microenvironment (pH, ionic strength). NPs may change size and surface properties when they enter tissue sites or cellular compartments (e.g., lysosomes) which may influence their movement, often trapping on the side of membrane promoting aggregation.
First-pass effects	The pathway of absorption of a small molecule chemical into the body may have a major impact on its subsequent deposition by first-pass hepatic metabolism.	If an absorbed NP forms a tight association with a biomolecule (e.g., protein and surfactant) in the process of absorption, its subsequent deposition could be changed.
Metabolism or biotransformation	Common for small molecule chemicals, and are mediated by Phase 1 and/or Phase 2 enzymes. Metabolism is usually simulated using the Michaelis–Menten equation and can be measured using in vitro systems, and then extrapolated to in vivo using physiological scaling factors. If experimental data are not available, metabolism can also be described using a simple first-order equation provided that the drug doses are within the linear kinetic range.	Many “hard” NPs (e.g., metal and carbon) are relatively inert and often too large to be degraded or metabolized by enzymes. However, some metal NPs, such as AgNPs, can be metabolized to become silver ion. For organic NPs, especially polymers with a functional group on the surface, the functional group may be degraded by enzymatic process. Existing PBPK models for NPs only simulate the pharmacokinetics of the NP itself. The mechanism of metabolism of NPs remains to be investigated and the approaches to simulate NP metabolism remain to be established.
Excretion	Mainly through urinary, fecal, and biliary excretion routes and can be described using either first-order linear progress or saturable process depending on the mechanism and availability of data.	Excretion is generally slow for NPs due to long retention time in tissues, especially in tissues of the RES system (e.g., liver, spleen, lungs, and kidneys) and the size limitation. Usually small NPs (≤5–6 nm) are excreted through the kidney, and larger NPs are mainly excreted via biliary pathway.
IVIVE	IVIVE approaches are well established for small molecules. It is common to determine PBPK parameters in vitro, and then extrapolate to a whole-body PBPK model using physiological scaling factors. The latest PBPK guideline from OECD provides details on how to develop and evaluate/validate a PBPK model based on in silico and in vitro data only.	Most published PBPK models for NPs were built based on in vivo pharmacokinetic data. One recent study demonstrated it was possible to develop a PBPK model for NPs (e.g., quantum dots and polystyrene nanoparticles) based on in vitro cellular uptake data,18 and another recent study showed that there were notable differences between observed and simulated data from a PBPK model that was developed based on in vitro cellular uptake data for AuNPs.19 IVIVE approaches for NPs remain to be established.
Interspecies extrapolation	It is well accepted that PBPK models are a scientifically sound tool to perform interspecies extrapolation for small molecules.	Only a few studies have attempted to extrapolate PBPK models for NPs from animals to humans using the same approaches used for small molecules. However, the validity of the derived human model has not been extensively evaluated. One study showed that animal-to-human extrapolation of PBPK models for NPs may need to consider biocorona kinetics because there is a longer circulation time in humans than in small rodents.34
PBPK modeling guidelines	There are multiple guideline documents on how to develop, evaluate/validate, apply, and document PBPK models, such as EPA,201 WHO,202 and OECD.165	PBPK modeling guidelines that are specific to NPs are still not available. In fact, a recent study showed that some commonly used PBPK modeling approaches for small molecules, such as route-to-route extrapolation, may not be applicable to NPs.143
Regulatory acceptance	Multiple regulatory agencies (e.g., EPA, Health Canada, OECD, WHO) accept PBPK models in different areas, such as drug discovery and development and human health risk assessment of environmental chemicals.	The PBPK approach by itself is accepted by regulatory agencies in risk assessment of both small molecular chemicals and NPs.203 Several studies have shown that it is feasible to use PBPK models to assess the human health risk of a few types of NPs. However, formal use of a well-validated PBPK model to assess human health risk of a NP has not been reported by regulatory agencies.

^aAbbreviations: IVIVE, in vitro to in vivo extrapolation; NPs, nanoparticles; RES, reticuloendothelial system. This table is modified based in Table 7 from Lin et al.²⁶ and Table 1 from Riviere et al.²⁰⁴

Endocytosis

One major difference in PBPK modeling between small molecules and NPs is the consideration of endocytosis in organs of the RES system. The very early NP PBPK models did not consider this mechanism in part because of a lack of mechanistic information at that time.^125,127 Later on, most NP PBPK models included this mechanism, but with variations in how to account for this mechanism, with some studies considering endocytosis in all compartments,^129,130 some only considered it in major RES organs,^126,132 and some only considered it in liver and spleen.^162,182 Also, the mathematical equations of the endocytosis among these studies are somewhat different (e.g., first-order linear process vs saturable process). While the inclusion of cell-mediated endocytosis in organs of the RES system results in a more physiologically realistic model, it also requires multiple cellular uptake and release kinetics parameters for each organ, and these parameter values are dependent on the type of NPs. Experimentally determined values of these parameters are usually not available, thus these parameters have to be estimated, resulting in increased uncertainty of the model. To minimize this uncertainty, one research group did not include endocytosis mechanisms in their PBPK model for AuNPs.¹⁹ While a consensus approach remains to be established, most of the existing studies suggest that (1) cell-mediated endocytosis in organs of the RES system plays an important role in the biodistribution of NPs, and should be included in a whole-body PBPK model; (2) the uptake process appears to be better described as a saturable process; whereas the release process can be simplified as a first-order linear process; and (3) ideally the uptake and release rate parameters should be determined using in vitro or in vivo approaches to minimize the uncertainty of the PBPK model.

In Vitro to In Vivo Extrapolation

Unlike small molecules of which the IVIVE methodology is well-established, the IVIVE strategies for NPs remain to be developed. Currently, there are only a few studies that have attempted to perform IVIVE for NPs, with some studies showing IVIVE is feasible for NPs,^18,166 whereas another study suggests that direct IVIVE may produce suboptimal results for NPs and thus may require further optimization of the methodology.¹⁹ Additional studies are needed to develop and validate IVIVE methodology for NPs. We term the process of “nanomaterial in vitro to in vivo extrapolation via physiologically based pharmacokinetic modeling” as Nano-IVIVE-PBPK (Figure 5). We think this framework is important for high-throughput screening of the target cellular and dosimetry as well as potential toxicity of different NMs based on their physicochemical properties. Specifically, in vitro cellular uptake and release studies should be performed in several major cell types in rodents and humans, and the in vitro kinetics should be modeled using approaches that are compatible with PBPK modeling, similar to our recent study.¹⁸ Next, IVIVE can be performed via a PBPK model, and the PBPK-simulated in vivo kinetic profiles in plasma and tissues should be evaluated and validated with in vivo PK data in rodents. Depending on the evaluation results, the IVIVE approach may be optimized as needed. Once the IVIVE approach is established in rodents, it can be extrapolated to humans. This methodology development will require in vitro, in silico modeling, and in vivo PK studies using the same type of NPs to avoid compounding by other variables and uncertainty. Once the methodology is developed for one NP, subsequent evaluation with other NPs should be done to develop a general methodology for NPs.

Route-to-Route Extrapolation

Route-to-route extrapolation is a standard application for PBPK modeling of small molecules. For NPs, several studies have performed route-to-route extrapolation, typically from intravenous to an extravascular route, by assuming all parameters remained the same and only estimating the route-specific parameter (e.g., oral absorption fraction or rate constant). This is a default approach for small molecules, but whether this approach is suitable to NPs remains to be tested. Upon entering into the body following different routes of administration, NPs will be in contact with different biological fluids, resulting in different protein coronas that lead to different PK properties. Therefore, the default route-to-route extrapolation approach for small molecules may not be applicable for NPs. To test this, one recent study developed a PBPK model for different sizes of AuNPs following different routes of administration, including intravenous, oral, inhalational, and intratracheal exposures in rats.¹⁴³ The in vivo data sets following different routes of exposure were collected from the same laboratory using the same measurement method and same NPs, making them an ideal data source to assess the route-to-route extrapolation of NPs. The authors compared the model predictive performance between two approaches: a traditional approach that is commonly used for small molecules by directly extrapolating the model from intravenous to other routes of exposure and a proposed route-specific approach for NPs to calibrate the model using route-specific data. The authors found that the multiroute model derived from the proposed route-specific approach had a much more accurate prediction than that from the traditional approach. This study demonstrated that the traditional route-to-route extrapolation approach for PBPK modeling of small molecules is not applicable to NPs. For NPs, multiroute PBPK models should be established using route-specific data. This study suggests that PBPK modeling methods for NPs could be quite different from small molecules, and a more systematic evaluation of PBPK modeling methods for NPs is needed.

Another question is whether the lymphatic system and protein corona formation kinetics are needed and how to incorporate them into a PBPK model for NPs. It is a standard practice to include the lymphatic system into a PBPK model for macromolecules, such as monoclonal antibodies.¹⁸³ It is known that the lymphatic system plays an important role in the biodistribution of NPs. However, among existing NP PBPK models, only a few included the lymphatic system.^18,140 Many PBPK models have acceptable model predictive performance without including the lymphatic system (Tables 3 and 4). The performance of a PBPK model with and without the lymphatic system has not been systematically evaluated, and this should be done in order to figure out whether it is necessary to describe the lymphatic system in a PBPK model for NPs. Regarding protein coronas, it is well-recognized that protein corona formation plays an important role in the PK of NPs, but protein corona formation kinetics has not been incorporated into existing NP PBPK models. This is in part because of the complexity of describing protein corona formation in vivo and the differences in the formation kinetics between in vitro and in vivo systems. This remains a critical barrier to be explored in future studies.

Generic PBPK Models

For PBPK modeling of small molecules, it is a general trend to develop generic models that work for multiple chemicals as it is impractical to develop an individual PBPK model for each chemical given a large number of chemicals in commerce whose safety and risk need to be evaluated. This trend also applies to NPs. A few studies have tried to develop generic PBPK models for several types of NPs.^{18,150,163,164} However, these models were calibrated using particle-specific data sets, thus it still requires an in vivo data set for each type of NPs to be included in the model framework. For small molecules, it is common to build quantitative structure–property/activity relationship (QSPR or QSAR) models to estimate parameters for chemicals based on chemical structures. For NPs, Li et al.¹⁵⁴ have demonstrated that it is feasible to build multivariate linear regression equations to predict model parameters for a different PLGA NP based on physicochemical properties. Similarly, in our recent study, we constructed several multivariate linear regression equations that can be used to predict PBPK model parameters (e.g., maximum uptake rate and capacity parameters) for AuNPs.¹⁴³ To build more robust models, machine learning and artificial intelligence approaches may be applied,^184,185 but this will require a larger amount of training data to avoid overfitting of the model.

Interspecies Extrapolation

Finally, one of the reasons PBPK modeling is very useful in small molecular chemical risk assessment is its ability to extrapolate PK and toxicity data from animals to humans. Several NP PBPK models have attempted to conduct extrapolation from one species to another using the traditional approach for small molecules.^{12,18,35,140,141} The study by Lin et al.³⁵ suggested that rats and pigs may be more suitable animal models for extrapolating the PK of AuNPs to humans. In addition, the PBPK model in mice by Aborig et al.¹⁴⁰ was extrapolated to rats by adapting physiological parameters to represent rats, and the resulting rat model adequately simulated PK and tissue distribution data of citrated-capped AuNPs in rats. These studies support the potential of interspecies extrapolation of NM PK using PBPK modeling. Nevertheless, interspecies extrapolation of PBPK modeling remains to be evaluated for other types of NPs, and the exact methodology remains to be developed.

Summary of How This Article Furthers the Knowledge and Understanding of the Field

This review article summarizes the state of the art and the research challenges in the field of in vivo PK, in vitro cellular kinetic modeling, and in vivo PBPK modeling as well as IVIVE of NPs. We share our perspectives on how to address these challenges. We also propose an emerging research paradigm for NPs, i.e., Nano-IVIVE-PBPK (Figure 5). All of this information will help us better understand the progress, challenges, and potential solutions in this field, which will further help design future studies to ensure in vitro and in vivo studies are useful within the Nano-IVIVE-PBPK framework to move this field forward. In the long term, the research strategies described in this article will be instrumental to refine, reduce, and replace animal experimentation in the field of nanoscience.

Glossary

Vocabulary

Target cellular and tissue dosimetry

the amount or the concentration of a substance, such as a chemical and a nanoparticle that reaches the cell or the target organ in the body and directly interacts with target cellular components to induce responses at the molecular, cellular, organ and/or whole-body level

in vitro to in vivo extrapolation (IVIVE)

a process of extrapolating data, such as pharmacokinetic and toxicity data that are generated using in vitro cell culture systems to in vivo conditions

nanomaterial in vitro to in vivo extrapolation via physiologically based pharmacokinetic modeling (Nano-IVIVE-PBPK)

an emerging research paradigm in which cellular uptake kinetic parameters of a nanoparticle are collected using in vitro model systems, and then these parameters are extrapolated to in vivo and incorporated into a whole-body physiologically based pharmacokinetic model to simulate the biodistribution of the nanoparticle in the body

pharmacokinetics

the science of studying the rate and extent of absorption, distribution, metabolism, and excretion processes of a drug and its metabolites in the body as well as the factors that control the time course of these processes using experimental or mathematical modeling approaches

physiologically based pharmacokinetic (PBPK) modeling

a computational simulation process that describes the absorption, distribution, metabolism, and excretion of a substance, such as an environmental chemical, drug, and nanoparticle and/or its metabolites in an organism based on interrelationships among key physiological, biochemical, and physicochemical determinants using mathematical equations

risk assessment

a process to characterize the nature and magnitude of potential risks of exposure to a chemical, a nanoparticle or other stressors on human health

toxicokinetics

similar to pharmacokinetics, but it is focused on toxicants, instead of drugs

The authors declare no competing financial interest.