Physiologically-based pharmacokinetic modeling of nanoparticles

Abstract

Nanoparticles are frequently designed to improve the pharmacokinetics profiles and tissue distribution of small molecules in order to prolong their systemic circulation, target specific tissue, or widen the therapeutic window. The multi-functionality of nanoparticles is frequently presented as an advantage but also results in distinct and complicated in vivo disposition properties compared to a conventional formulation of the same molecules. Physiologically-based pharmacokinetic (PBPK) modeling has been a useful tool in characterizing and predicting the systemic disposition, target exposure, and efficacy/toxicity of various types of drugs when coupled with pharmacodynamics (PD) modeling. Here, we review the unique disposition characteristics of nanoparticles, assess how PBPK modeling takes into account the unique disposition properties of nanoparticles, and comment on the applications and challenges of PBPK modeling in characterizing and predicting the disposition and biological effects of nanoparticles.

1. INTRODUCTION

The applications of nanoparticles in diagnosis and therapy have been increasing over the years.¹ In broad terms, nanoparticles are inorganic, organic or polymeric particles that have at least one size dimension in the nanoscale range (1–1000 nm).^1–3 Many nanoparticles have a core-shell structure with the active pharmaceutical ingredient (API) loaded in the core or the shell (liposomes, emulsions, and micelles), or have a matrix structure with the APIs blended in the polymeric matrix (nanocrystals and polyacrylamide nanoparticles).^3,4 The API can be loaded via physical entrapment or covalent attachment. Between 1970 and 2015, 359 applications for drug products containing nanomaterials were submitted to the US Food and Drug Administration (FDA)’s Center for Drug Evaluation and Research (CDER).¹ In terms of size, 40% of the submissions were less than 100 nm, 41% were between 100 and 300 nm, 10% were between 300 and 600 nm, and the remaining 9% were between 600 and 1000 nm.¹ In terms of particle types, liposomes were the most prevalent category (33%), followed by nanocrystals (23%), emulsions (14%), iron–polymer complexes (9%), and micelles (6%).¹ Regarding indications, 35% of the products were developed for cancer treatment, 18% for inflammatory/immune/pain disorders, 12% for infections, and the remaining 35% were used for other disease conditions and cosmetic, diagnosis, and nutrition purposes.¹ Li et al. recently reviewed the nanoparticle formulations in clinical trials or clinical use for cancer treatment.⁴

Nanoparticles can be engineered to have different properties such as size, shape, charge, and surface chemistry,² thus providing a versatile platform for modifying the delivery and pharmacological performance of the loaded drugs. From a formulation perspective, various nanoparticles have been used to modify the release profile and increase the apparent solubility and stability of APIs. In particular, nanocrystals are broadly used to enhance the oral dissolution of poorly water-soluble drugs in biopharmaceutical classification system (BCS) groups II and IV due to the high surface area to volume ratio.⁵ A high–capability micelle formulation was developed for paclitaxel to increase its apparent aqueous solubility from less than 1 mg/L to 45 g/L.⁶ From a pharmacology perspective, nanoparticles have been used to overcome drug resistance, prolong circulation, and improve the in vivo distribution of loaded APIs.^1,4,7–10 Due to their larger sizes, nanoparticles have slower or very limited renal clearance in comparison with small molecules. The surface of nanoparticles can be modified with hydrophilic polymers, such as poly(ethylene glycol) (PEG), to further extend the circulation half-life (PEGylation)^11,12 or can be coated with targeting ligands to increase selectivity to a target organ, such as the brain.¹³ As a specific example, Doxil^®, a PEGylated liposome of doxorubicin and the first FDA-approved nanodrug, greatly enhances circulation time and tumor accumulation, allowing a higher accumulated dose and significantly reduced cardiotoxicity compared to doxorubicin.^11,14

It is critical to evaluate the altered drug disposition to understand the exposure–efficacy relationships and address safety concerns. Nanodrugs could prompt new toxicity concerns due to the altered disposition of the APIs. For example, Doxil^® increases doxorubicin deposition in skin, resulting in a lower single-dose maximal tolerable dose (MTD) than that for doxorubicin (50 mg/m² every 4 weeks or 12.5 mg/m²/week versus 60 mg/m² every 3 weeks or 20 mg/m²/week).¹¹ In addition, the nonintended biological effects of the nanoparticle compositions should be determined. Safety concerns related to chronic exposure to non-biodegradable materials and increased penetration of biological barriers, such as the blood-brain barrier or the placenta, were raised in the FDA guidance for industry “Drug Products, Including Biological Products, that Contain Nanomaterials” (December 2017).

In this regard, physiologically–based pharmacokinetic (PBPK) modeling is one quantitative support tool for assessing nanoparticle hazards recommended by the Organization for Economic Cooperation and Development (OECD) and the new European Union regulatory framework, Registration, Evaluation, and Authorization of Chemicals (REACH).¹⁵ This tool has been well accepted by the pharmaceutical industry and regulatory agencies (FDA, the European Medicines Agency, and the Ministry of Health, Labor and Welfare of Japan) in PK modeling and simulations for various types of drugs.^16–18 PBPK modeling, with its distinctive separation of physiology- and drugdependent information, has become a viable option to provide a mechanistic understanding of the influential factors and sources of PK variability, which is thus helpful in predicting drug exposure at various clinically relevant scenarios. When combined with pharmacodynamic (PD) models relating exposure at target tissues to pharmacological effects, PBPK modeling can be used to predict efficacy and toxicity.¹⁶ PBPK models have been applied for many types of nanoparticles, including carbon nanoparticles,¹⁹ polymeric nanoparticles,^20,21, nanocrystals,^22–26 silver nanoparticles,^27–29 liposomes,^30,31 gold/dendrimer composite nanoparticles,³² and others.³³

The challenge of studying the disposition of nanodrugs relates to their structural and functional complexity; various particle properties, such as composition, size, shape, charge, and surface chemistry, affect particle interaction with the biological system.¹ Here, we review the dispositional characteristics of nanoparticles compared to small molecules, assess how PBPK models take into account the unique disposition features of nanoparticles, and comment on the application and challenges of PBPK modeling in characterizing and predicting the disposition and biological effects of nanoparticles.

2. PHYSIOLOGICALLY–BASED PHARMACOKINETIC MOEDLING

The concept of PBPK modeling was first introduced by Teorell as early as 1937.³⁴ A brief introduction to generic PBPK models is presented here. Readers can refer to a PBPK tutorial published by Jones et al¹⁶ for more details. Figure 1A shows the model structure of a generic PBPK model. Unlike mammillary models,³⁵ PBPK models use individual tissues in the body as building blocks or compartments. Typically, the main tissues of the body, namely, brain, gut, heart, kidney, liver, lung, spleen, muscle, and adipose tissues, are included.¹⁶ The remaining tissues of the body are often grouped into a remainder/carcass compartment if they are not from the organ of interest, and certain tissues can be ignored if they are not significant in terms of mass balance. Tissues with similar kinetics can also be lumped together to simplify the model (e.g., minimal PBPK model).³⁶ Similar to the physiological systems, all tissue compartments in PBPK models are connected by the circulating blood system and sometimes the lymphatic system.¹⁶ Drug clearance should be defined in drug-metabolizing tissue compartments, such as the liver and kidney.

Figure 1.

A generic PBPK model (A) and two types of tissue model structure (B). Qi: blood or plasma flow; k_p: tissue partitioning coefficient, namely concentration ratio between tissue and blood at steady-state; PS: membrane permeability coefficient; CL_hep: hepatic clearance; CL_renal: renal clearance.

As shown in Figure 1B, each tissue compartment is described by either a perfusionlimited or permeability-limited model.¹⁶ Perfusion-limited models assume that the drug in the tissue can reach quick distribution equilibrium with the drug in the circulation system, indicating that the drug can easily penetrate tissue cell membranes with blood perfusion as a limiting step. Permeability-limited models consider tissue cell membranes as diffusional barriers to the studied drug, and the tissue cell membranes divide the tissue into intracellular space and extracellular space.^16,37 Active drug transporters on cell membranes could also be modeled by incorporating uptake/efflux transporting mechanisms.^16,37 The model parameters associated with uptake/efflux transporters, like affinity and capacity, are usually derived from cell-based assays and are often adjusted using empirical in vitro-in vivo extrapolation (IVIVE) approaches.

PBPK models are parameterized with both physiological parameters (such as the tissue volume, tissue blood flow, and abundance of metabolizing enzymes and transporters) and drug-specific parameters (such as the clearance and tissue partition coefficient k_p), which together define drug disposition within the physiological system.¹⁶ The drug-specific parameters are frequently scaled from a variety of in vitro systems.^16,37 For instance, drug clearance from the liver can be scaled based on in vitro measurements using recombinant enzymes, liver microsomes, or hepatocytes.^16,17 The drug-specific k_p, which is defined by the tissue/blood concentration ratios at steady state, can be estimated using in silico methods based on both tissue composition and drug physicochemical characteristics, such as lipophilicity, charge, and protein binding.^16,37 Such IVIVE is integral to PBPK modeling, allowing prediction of the plasma and tissue concentration–time profiles without in vivo studies.¹⁶ IVIVE is helpful in the early stage of drug discovery when a large number of drug candidates need to be screened. Another advantage of PBPK modeling is that by separating drug-specific parameters from physiological parameters, the sources of PK variability can be better identified. Thus, PBPK modeling can simulate PK variability across sub-populations based on the distribution of patients’ physiological parameters.¹⁶ The mechanistic nature of PBPK models allows them to extrapolate the PKs to different disease states (e.g., liver and kidney dysfunction,^38,39 obesity), special populations (e.g., the elderly,⁴⁰ the pediatric population,^41,42 pregnant women,⁴³ carriers of genetic polymorphisms), and different species.^16,44 For these reasons, PBPK models have earned their popularity as a powerful tool for drug development from lead screening to late clinical evaluations.

In terms of implementation, PBPK models are typically expressed as differential equations describing the mass balance based on appropriate assumptions or simplifications of drug disposition processes. As mentioned above and shown in Figure 2, the initial parameter values can be obtained from in vitro measurements, literature reports, or initial estimates based on simulations. Typically, physiological parameters are fixed to a set of commonly used values and drug-specific parameters are to be optimized by fitting the model against all available experimental data. The mass transfer assumptions and optimized parameter values could be verified against an independent dataset that was not used for parameter optimization. Developing PBPK models is an iterative process and the model is open to further update once a new set of data becomes available. The verified PBPK model can then be used to predict drug PK and tissue distributions at different scenarios, such as different dosing regimens, special populations, or across-species.

Figure 2.

Implementation of PBPK models.

3. DISPOSITION OF NANOPARTICLES

The successful application of PBPK modeling requires the understanding of drug disposition. Here, we summarize the disposition characteristics of nanoparticles.

3.1. Absorption

To date, the majority of nanoparticles are intended for intravenous (IV) injection and many are designed to extend circulation or target a specific organ through the circulation system. In the present work, we focus on the disposition of nanoparticles post absorption. The absorption of nanoparticles is briefly discussed here. Upon non-IV injection, nanoparticles face two competitive processes: pre-absorption clearance and absorption.² Pre-absorption clearance includes local degradation and direct removal from the body in different forms, such as feces in the case of oral, nasal, and pulmonary administration. Local degradation at the administration sites can release the encapsulated APIs, and the released APIs follow the disposition pathways of small molecules. Upon oral administration, the nanoparticles that evade pre-absorption clearance can enter the blood and lymphatic systems by crossing the unstirred water layer⁴⁵ and the epithelium of the gastrointestinal (GI) tract.² For pulmonary exposure, nanoparticles deposited in the lungs can be exhaled, removed via mucociliary clearance to the GI tract, or sequestered and degraded by macrophages. The remaining nanoparticles can be absorbed after crossing the mucus and lung epithelium cells.^2,46 Due to their large sizes, nanoparticles administered via subcutaneous, intramuscular, intradermal, and intraperitoneal injections are mainly absorbed through the highly permeable lymph vessels or via macrophage and subsequent cell trafficking into regional lymph nodes.^2,33

3.2. Distribution

Upon entering systemic circulation, nanoparticles encounter physical and biological stresses that may change their properties and affect their deposition.⁹ The nanoparticles are immediately subject to an environment different from their formulation buffers in temperature, pH, ion strength, composition, and sheer stress. Physical processes such as dilution in the blood, diffusion of nanoparticle composition, and changes in polymer physicochemical properties can occur, altering particles’ colloidal stability and frequently resulting in the aggregation, swelling, or dissolving of some nanoparticles. Chemical or enzymatic degradation of nanoparticle components can also occur. Moreover, depending on the particle surface properties, numerous serum proteins, also called “opsonins”, can deposit onto the surface of nanoparticles or enter the nanoparticles to form dynamic protein coronas (biocoronas), a process known as opsonization.⁴⁷ Thus, the composition and physicochemical properties of nanoparticles are altered upon entering the systemic circulation. These changes (charge, size, shape, elasticity, and composition) determine the particle colloidal stability and drug release,⁴⁸ as well as the nanoparticles’ interaction with cells and their tissue distribution.^49–51

Opsonins act as recognition ligands for the activation and amplification of the complement system and facilitate opsonized nanoparticle uptake by the MPS phagocytic cells in the mononuclear phagocytic system (MPS), a family of cells consisting of bone marrow progenitors, blood monocytes, and tissue macrophages.^2,9,47,51,52 Notably, accumulations of nanoparticles in the liver and spleen are generally high, largely due to sequestration by the Kupffer cells residing in liver sinusoids and the macrophages residing in the spleen marginal zone and red pulp.^2,9 Kupffer cells make up 80–90% of the total macrophage population in the body.⁵³ It is estimated that 30–99% of nanoparticles from the bloodstream will accumulate in the liver.⁵³ Many particle properties, such as size, charge, shape, and elasticity, affect biocorona composition and the interaction with the MPS. It is commonly believed that larger nanoparticles are recognized more readily by the MPS than smaller nanoparticles and that cationic nanoparticles show the strongest macrophage uptake, followed by anionic nanoparticles and neutral nanoparticles.⁹ Charged nanoparticles can adsorb serum proteins, resulting in increased hydrodynamic size and stronger interaction with macrophages.⁵³ Many research groups have also reported that the MPS preferentially internalizes rigid, spherical nanoparticles over their soft, rod-shaped counterparts.⁵³ Only nanoparticles that escape MPS sequestration and renal clearance (discussed below) have the opportunity to distribute to non-MPS tissues.⁵⁴ To reduce MPS sequestration of nanoparticles, researchers typically modify the surface of nanoparticles with neutral and hydrophilic polymers, such as PEG,⁵⁵ or with zwitterionic polymers, such as poly(carboxybetaine) and poly(sulfobetaine),⁵⁶ or camouflage them with erythrocyte membranes⁵⁷ to reduce opsonization and recognition by macrophages (stealth feature). These strategies extend the circulation half-life and increase the chance for nanoparticles to accumulate in the target tissues.⁹ Other methods, such as pre-coating with specific proteins and modulation of particle surface chemistry, have been proposed to alter the protein corona composition so as to reduce or enhance the interaction with phagocytic cells (e.g., nanoparticles for delivery of vaccines) and to shift the in vivo disposition.⁴⁸

Apart from MPS sequestration, the amount of nanoparticles that accumulate in tissues is further determined by the tissue blood supply, the vascular permeability, and the interaction of nanoparticles with tissue resident cells.² Normal intact endothelium has an effective pore size of about 5 nm.⁵⁸ Nanoparticles larger than 5 nm in diameter can barely penetrate tissues with normal or tighter endothelium, such as the muscle and brain. Nanoparticles less than 60 nm may have easy access to tissues with fenestrated endothelium, such as gland tissue, digestive mucosa, and kidney tissue, as well as tissues with discontinuous endothelium, such as liver tissue, spleen tissue, and bone marrow.^2,59 In mice biodistribution studies, nanoparticles were found primarily in liver and spleen tissues, to a minor extent in kidney and lung tissues, and minimally in heart, muscle, and brain tissues.^2,59 Within the tissues, nanoparticles may reside in the extracellular space, adhere on the surface of tissue-resident macrophages or somatic cells, or enter the interior of cells.² It is worth noting that interaction with cell membranes and cell uptake of nanoparticles in tissues is affected by the biocorona gained during extravasation process and in the tissue interstitial fluid, which could be different from the biocorona formed in the blood circulation. The nanoparticles in the extracellular space can be drained into (smaller than 6 nm) or ferried by macrophages (larger than 6 nm) into regional lymph nodes.^2,60

To date, the majority of nanoparticles have been designed to improve the tumor accumulation and tumor selectivity of small molecules. Small molecules can penetrate normal tissues freely and have no selectivity towards tumor cells, resulting in a high volume of distribution and dose-limiting off-target toxicity.⁵⁹ The predominant principle for nanoparticles targeting a tumor is the enhanced permeation and retention (EPR) effect. Unlike most normal blood vessels, the blood vessels in tumors harbor large interendothelial cell gaps ranging from 100–4700 nm, and lymphatic drainage is usually deficient.^9,59 Combining enhanced permeation and reduced drainage, nanoparticles preferentially accumulate in tumors. Tumor accumulation of nanoparticles has been found to be up to 27-fold more than that of small molecular drugs.⁵⁹ While there is no consensus on the optimal particle size for the EPR effect, nanoparticles with a hydrodynamic diameter between 10 and 100 nm are usually sought to avoid renal clearance and to exploit the EPR effect.⁶¹ Nanoparticles with longer circulation half-lives have a better EPR effect. However, according to a 10-year survey (2006–2016) by Wilhelm et al., only 0.7% (median) of administered nanoparticles are found in solid tumors.⁹ It is now recognized that the EPR effect varies with tumor type, size, and stage.^2,9,59,62,63 In addition, due to the simultaneous ERP effect of plasma proteins and local inflammation, many tumors can exhibit 10- to 40-fold higher interstitial fluid pressure compared to normal tissues, creating a pressure gradient against the extravasation of nanoparticles by convection.^9,64,65 Such high interstitial fluid pressure also drives the outward interstitial flow to peripheral normal tissues and the nearest sentinel lymph nodes, further removing nanoparticles from the tumor tissues.^9,65 Therefore, for very permeable tumors (e.g., openings larger than 200 nm), the high interstitial fluid pressure prevents effective EPR effect for particles of all sizes; for less permeable tumors, the pressure difference across the vessel well benefits the extravasation of small particles less than 60 nm but excludes large particles.⁶⁶

Within a tumor, the distribution of nanoparticles is heterogeneous. This is due to the uneven distribution of blood vessels and the poor penetration of nanoparticles across the extracellular matrix.⁶⁵ The tumor vascular density is generally higher in the tumor’s advanced margin and lower in the center, forming a necrosis core that nanoparticles can barely reach.⁹ In addition, having extravasated the blood vessels, the nanoparticles need to navigate through a tortuous extracellular matrix, a semi-solid barrier consisting of interconnected collagen fibers and proteoglycans, to reach tumor cells.^9,59 The general consensus is that larger and charged nanoparticles tend to be restricted to the immediate vicinity of tumor blood vessels, while smaller and neutral nanoparticles have a higher chance of penetrating deeper into the tumor.^9,62[ref] Apart from being trapped by the extracellular matrix, nanoparticles can be sequestered by macrophages which generally have higher uptake capacity than tumor cells. Taken together, the percentage of nanoparticles that reach the tumor cells is likely much less than 0.7%.⁹ Note that the stealth PEG coating used to reduce opsonization and uptake by macrophages paradoxically reduces the uptake by tumor cells.^59,67 To increase tumor cell uptake, nanoparticles could be decorated with active targeting ligands^67–69 or cell penetrating peptides⁷⁰ to increase the selectivity to tumor cells.⁵⁹ Yet, it is likely that the addition of targeting ligands compromises the stealth feature and increases the particle size, resulting in a shorter circulation half-life and a slower diffusion rate.⁶¹ Moreover, targeted nanoparticles with high avidity are less likely to penetrate deep into the tumor due to the “binding site barrier”,^71,72 a consequence of strong binding to the tumor cells expressing the cognate receptors near the blood vessels.⁶¹ While we focus on the general distribution behaviors of nanoparticles, interested readers are encouraged to refer to a comprehensive review⁶⁶ discussing the desirable particle size, shape, and surface charge for maximizing the delivery of nanoparticles to solid tumors.

3.3. Metabolism

In a broad sense the metabolism of nanoparticles can be defined as any process that results in the loss of nanoparticles in their original form,² including aggregation, dissolving, opsonization, drug release, and so on. Here, we narrow down the definition to drug release and the degradation of nanoparticle compositions. Drug release is critical, as the temporal and spatial release profiles of APIs directly determine the efficacy and toxicity of nanodrugs. APIs can be released prematurely in the blood circulation due to altered particle stabilities and interactions with circulating monocytes. Undesirable drug release and tissue toxicities can also occur after the nanoparticles are decomposed in the lysosomes of MPS cells in certain tissues (notably liver and spleen tissues). Many particles are designed to release the drug in response to tissue-specific or external stimuli, such as acidic pH, elevated temperature, or enriched enzymes in the tumor microenvironment.^73,74 The release kinetics are both API- and nanoparticlespecific, depending on the API solubility, the diffusion rate through the nanoparticle matrix, and the erosion rate of the nanoparticles in the aqueous or intracellular environment.⁷⁵ For chemically conjugated drugs, the release rate also depends on the stability of the chemical bond in the biological environment.⁵⁹ Ideally, the nanoparticles should release the drugs only at target sites and be removed once all drugs are released.

Toxicity could arise from the un-intended accumulation of nanoparticle excipients. The processing within macrophages depends on the composition of nanoparticles.⁵³ Many inorganic nanoparticles (silver, gold, iron oxide, quantum dots, carbon, silica, and others) are very stable and persist for a long time in the body, whereas organic nanoparticles (liposomes, micelles, polymer conjugates, and others) are biodegradable and the degradation products can be excreted through urine or bile.^2,9,53 Long persistence of nanoparticles in macrophages results in fusion of macrophages, forming foreign body giant cells, and ultimately dense fibrous capsules.⁵⁹ Malignancy has also been observed as a result of phagocytosis and prolonged inflammation associated with nanoparticles.⁷⁶ The retention of metal nanoparticles also interferes with diagnostic imaging modalities, including X-ray imaging, magnetic resonance imaging, ultrasound, single photon emission computed tomography, and position emission tomography.⁶⁰

3.4. Excretion

The major organs responsible for nanoparticle excretion are the kidneys and the liver.^2,53,58,60 Longmire et al.⁶⁰ and Zhang et al.⁵³ provided an overview of the renal clearance and liver clearance of nanoparticles, respectively. In general, solid spherical nanoparticles smaller than 5.5 nm in hydrodynamic diameter (after opsonization) can be efficiently eliminated in urine through glomerular filtration.^58,60 While nanoparticles larger than 8 nm typically are not subject to glomerular filtration, nanoparticles within the intermediate range of 6–8 nm can still be filtered if the charge favors the interaction with the negatively charged glomerular capillary wall.⁶⁰ Hepatobiliary clearance is the main mechanism for excretion of nanoparticles that do not undergo renal clearance.⁶⁰ Note that only nanoparticles evading the phagocytic Kupffer cells can access the hepatocytes. Circulating nanoparticles can leak through the liver sinusoidal fenestrae (up to 150–200 nm) or transverse across the sinusoidal endothelial cells to reach hepatocytes.⁵³ It has been shown that hepatocytes favor the uptake of positively charged nanoparticles over their negatively charged counterparts.⁵³ Once processed by hepatocytes, the intact nanoparticles or their degradation products can be excreted in bile. Hepatobiliary excretion is usually slow for nanoparticles (ranging from hours to months),⁵³ making renal clearance the efficient route for nanoparticle elimination.⁶⁰ Nanoparticles that are not degradable by macrophages or that are incapable of renal clearance can be modified with targeting ligands for hepatocyte uptake to increase their hepatobiliary clearance.⁵³

3.5. PK variability of nanodrugs

As the MPS plays a central role in nanoparticle disposition, the PK of nanoparticles is more variable than that of their small molecule counterparts.⁷⁷ A meta-analysis of nine liposomal anticancer agents showed that liposomes have a higher inter-patient variability than their small molecule counterparts.⁷⁸ The inter-patient variability in exposure (plasma AUC) of PEGylated liposomal CKD602 can be as high as 20- to 100-fold.^78,79 The meta-analysis also showed an inverse linear relationship between the clearance and inter-patient variability of liposomes but not of small molecules, indicating that the high PK variability of liposomes is resulted from the clearance mechanism.^77,78 Relatively smaller inter-patient variability was observed for PEGylated liposomal CKD602 at high doses when the clearance capacity of MPS was likely saturated.^78–80 The MPS function also explained clearance variabilities of some liposomal drugs. It is speculated that the slower clearance of liposomal anti-cancer drugs in the geriatric population and in females could be explained by the age- and gender-related decreases in MPS function.^81–84 Moreover, in tumor-bearing mice, tumor accumulation and the antitumor activity of liposomal drugs have been associated with the presence of phagocytic cells in tumors.⁸⁵ The phagocytic cells in xenograft tumors have been found to be responsible for the metabolism and subsequent drug release of nanoparticles.⁸⁵ PEGylated liposomal CKD602 has higher clearance in patients with liver tumor metastasis, which is the opposite for small molecules.⁸¹ The unexpected higher clearance of nanoparticles in patients with tumor metastasis could be explained by more activated macrophages in the liver triggered by local inflammation. Interestingly, the interaction between the MPS and nanoparticles is bidirectional.⁸⁶ Namely, nanoparticles can exert toxicity on the MPS via intracellularly released cytotoxic drugs.⁸⁶ As a result, a reduction in MPS function after prior treatment leads to increased exposure in subsequent treatment cycles, raising toxicity concerns for drugs with a narrow therapeutic window.⁸¹

The high inter- and intra-patient variability in the MPS clearance of liposomal anticancer agents makes it difficult to predict patients’ responses.⁷⁸ In addition, there are species differences in MPS, and standard allometric scaling based on body weight has failed to extrapolate the clearance of liposomal drugs in preclinical animal models (mice, rats, and dogs) to humans.⁸⁷ To mitigate the risk, Caron et al. proposed measuring the function of the MPS as a phenotypic probe to predict the clearance of nanoparticles.⁸⁸ A good correlation of the phagocytic capacity of circulating monocytes/dendritic cells and their production levels of reactive oxygen species with the clearance of three PEGylated liposomal anticancer agents has been found across species (mice, rats, dogs, and humans) and across human individuals.⁸⁸ Therefore, measuring a patient’s MPS function prior to the initiation of treatment and before each new cycle of nanoparticle anticancer drugs may be valuable for personalized therapy.

For PEGylated nanoparticles, variability in clearance and distribution could also arise from the accelerated blood clearance (ABC) phenomenon.⁸⁹ The ABC phenomenon refers to the enhanced clearance of a second dose of PEGylated nanoparticles when administered within a certain time interval from the administration of the first dose.⁸⁹ After intravenous or subcutaneous injection of the first dose, the PEG on the particle surface cross-links immunoglobulins on B cells in the splenic marginal zone or in the regional lymph node, triggering the secretion of anti-PEG IgM (the induction phase).^89–91 In the effectuation phase, circulating anti-PEG IgM can bind the subsequent dose(s) of PEGylated nanoparticles, activate the complement system, and trigger complement receptor-mediated endocytosis of the nanoparticles by liver and spleen macrophages, as well as drug release and hypersensitivity reactions.^{89,90,92–94} The second dose could be the same nanoparticles as the first dose or new PEGylated nanoparticles.⁹⁵

The occurrence and magnitude of ABC phenomenon depends on the dosing interval, the number of doses, the dose of the nanoparticle and encapsulated drug, the drug cargo, and the particle properties.^89,96 The dose interval is critical for the occurrence of ABC phenomenon and was found correlated with the production of IgM.⁸⁹ The ABC phenomenon became less pronounced after the third dose, at high induction doses or at high subsequent doses, or when the drug cargo was cytotoxic.⁸⁹ These effects were explained by the saturation of the uptake capacity of Kupffer cells, the cytotoxicity to Kupffer cells and B cells, or the induction of immune tolerance in response to accumulating nanoparticles and drugs.⁸⁹ Although most PEGylated nanoparticles can induce ABC phenomenon, the magnitude is affected by the particle properties, such as particle size and PEG-surface density, via modulating the circulation time of the initial dose, the conformation of PEG presented to B cells, and the binding of anti-PEG IgM to subsequent nanoparticles.^89,95 Due to the immunological nature, species differences in the induction of ABC phenomenon have been observed.⁸⁹

The ABC phenomenon poses challenges in clinical applications of PEGylated nanoparticles that require multiple doses.^89,97,98 Although chemotherapies at high doses may not manifest the ABC phenomenon,⁹² concerns could rise in view of the growing interests in using metronomic chemotherapy for management of cancer resistance.⁸⁹

4. PBPK MODELING OF NANOPARTICLES

The interaction between nanoparticles and the biological system as described above is complicated and extremely difficult to replicate in vitro.⁴⁷ Small variations in particle properties or the biological system, such as MPS function, can significantly alter the systemic disposition of nanoparticles.⁸² By jointly integrating the influential biological factors and the nanoparticle-specific properties, PBPK modeling can be very useful in predicting the PK and tissue exposure, as well as the efficacy and toxicity of nanoparticles. Here, we review several application examples of PBPK modeling and a few PBPK model structures for nanoparticles.

4.1. Application examples

4.1.1. Predicting the dose-effect relationship

Howell and Chauhan developed a PBPK model to test and optimize the therapeutic benefit of anionic, PEGylated liposomes to treat amitriptyline (AMI) and bupivacaine (BUP) overdoses in humans.³⁰ The liposomes were able to sequester AMI and BUP from serum and thereby reduce distribution of the drugs to the heart and brain, two tissues with toxicity concerns.⁹⁹ The PBPK model was developed for AMI and BUP rather than the liposomes, as the authors assumed that the liposomes remained within the blood compartments until elimination. The PBPK model was used to simulate the changes in AMI and BUP exposure and the efficacy of the liposome therapy. The dispositions of AMI and BUP were modeled using a generic PBPK model with 15 compartments, perfusion-limited tissue distribution, and clearance from the liver and kidneys. The time-varying sequestration of AMI and BUP by the liposomes in blood was modeled using an empirical equation describing the clearance of liposomes from blood and a linear relationship established in vitro relating the drug liposome/blood partition coefficients and liposome concentrations. The PBPK model was further used to simulate the reduction of AMI and BUP exposure (AUC and C_max) in plasma, the heart, and the brain upon liposome administration. The predicted heart concentration was combined with a PD model correlating the drug concentrations with the alteration of heart functions to predict the reversal of heart functions after liposome administration. The effects of the liposome dose and the time lapse between overdosing and liposome administration on therapeutic efficacy were also explored. As the authors pointed out, neglecting the tissue distribution (especially in the liver) of liposomes may result in underestimating the hepatic clearance of bound AMI and BUP; however, the bias could be partially reduced by using the total drug plasma concentration in calculating hepatic extraction.

4.1.2. Inter-species/population translation

Lu et al. developed a PBPK model to compare the differences in the disposition of docetaxel between a small molecule formulation and a folate-modified liposomal formulation in rats after IV administration and to predict mouse and human PKs.³¹ The PBPK model included arterial blood, venous blood, lung, brain, heart, spleen, liver, intestine, kidney, muscle tissues, a remainder compartment, and clearance from the liver and intestines. The final model assumed a perfusion-limited structure model for each compartment. Interestingly, the authors applied non-linear mixed-effect modeling and studied the effect of sex and formulation, two categorical covariates on k_p and clearance. In comparison to the small molecule formulation, the liposomal formulation was found to have higher k_p for lung, kidney, and muscle tissues and lower k_p for brain, spleen, and liver tissues. The higher kidney k_p for the liposomal formulation was ascribed to the expression of folate receptors in the kidneys. Additionally, female rats had higher k_p for heart tissues but lower intestinal clearance. The interaction between formulation and sex was not studied. The model was developed based on rat data and was then successfully used to predict mouse and human data using species-specific physiological parameters and allometric scaling for clearance.

4.1.3. Formulation development of nanodrugs

Rajoli et al. applied PBPK modeling to address the feasibility of developing monthly intramuscular injectable nanodrugs (solid drug nanoparticles) for antiretrovirals.¹⁰⁰ Poor patient adherence greatly hinders the efficacy of current oral formulations of antiretrovirals, which necessitates lifetime daily dosing. It is anticipated that monthly nanodrugs can greatly improve patient adherence and treatment efficacy. However, it is unclear if the long-acting formulations can maintain effective antiretroviral concentrations between dosing intervals. A PBPK model was developed based on clinical data of the oral formulation for each antiretroviral using a compartmental absorption and transit model published earlier¹⁰¹ and was then extended with an intramuscular depot compartment to simulate drug release and absorption. The full PBPK model was subsequently verified against an existing antiretroviral nanodrug. Through simulations, the authors were able to optimize the combination of dose and release rate for each of eight antiretrovirals to maintain therapeutic plasma concentrations for the entire dosing interval. The feasibility of a monthly nanodrug for each antiretroviral was confirmed when the optimized dose was within the dose limits for intramuscular injections. However, the feasibility of the optimized nanodrugs requires experimental confirmation.

4.1.4. In vitro-in vivo correlation (IVIVC)

Jung et al. used PBPK modeling to correlate in vitro drug release with plasma concentrations after the oral administration of nanocrystal flurbiprofen.²⁴ To simplify the model, they used a two-compartment model (plasma and peripheral) to describe the distribution and elimination processes after absorption. For the absorption process, a more mechanistic PBPK model was used. The GI tract was divided into two compartments, stomach and intestine. The fraction of drugs released in the stomach and the intestine was estimated separately using in vitro biorelevant release tests. The drug released from the formulation during the GI transition needs to diffuse into the unstirred water layer and then permeate the intestinal barrier to reach the plasma. The PK data of a reference tablet formulation were used to optimize the drug diffusion rate across the unstirred water layer and the intestinal barrier and the systemic distribution and elimination parameters. The validated model was then used to simulate the PKs of a nanocrystal formulation by updating the model with the in vitro release profiles. In addition, the authors also compared two in vitro dissolution methods, a filter method and a dispersion releaser technology. It was found that plasma PK was more sensitive to the variability in drug release predicted by the dispersion releaser method than by the filter method. Therefore, the dispersion releaser technology and the PBPK model can be combined to screen drug formulations. Similarly, Shono et al. combined PBPK modeling and biorelevant release tests to investigate the size effect of nanocrystals on absorption and plasma PK.¹⁰² Moreover, by considering the difference in the GI emptying rate and fluid volume with and without food, the model was used to predict the effect of food on drug absorption and plasma PKs. Kumar and Singh correlated the Weibull in vitro release profile of a carvedilol loaded silk fibroin-casein nanoparticle with the plasma PKs after oral administration in rats using the built-in Advanced Compartmental Absorption Transit (ACAT) model and PBPK model in GastroPlus™.¹⁰³ The validated IVIVC was further used to simulate plasma PKs based on different in vitro release profiles. Other PBPK examples in oral absorption of nanodrugs can be found in references^23,26 and others.

4.1.5. Risk assessment

As nanoparticles are increasingly used in industrial and consumer products, workers and consumers can be exposed to them via various routes.¹⁰⁴ PBPK modeling has been applied to assess the risk of exposure to nanoparticles. Péry et al. applied PBPK modeling to solve the debate regarding whether nanoparticles can translocate from the lungs to the systemic circulation in humans.¹⁹ The model was developed based on technetium imaging data for healthy male volunteers dosed with an aerosol suspension of ^99mTc-labelled carbon nanoparticles. The imaging data could not determine whether the nanoparticles can translocate from the lungs to the systemic circulation because the radioactivity in the liver where most of nanoparticles were expected to accumulate overlapped with that in the lungs and because of the confounding free technetium. The PBPK model was able to trace the free technetium and small and large ^99mTc-labeled nanoparticles independently. Only the large ^99mTc-labeled nanoparticles were assumed to be incapable of translocation. A different partition coefficient was assumed for each technetium population. It was found that the model fitted the imaging data less well when assuming the small ^99mTc-labeled nanoparticles were unable to translocate from the lungs to the blood, meaning that the small nanoparticle can be absorbed into systemic circulation through the lungs. In addition, the model could predict the percentage of particles entering the systemic circulation post inhalation, which is of interest in the investigation of particulate air pollution.

Lankveld et al. characterized the kinetics of 20, 80, and 110 nm silver nanoparticles in rats after five daily IV doses.²⁸ As expected, the tissue distribution differed according to size; the 20 nm nanoparticles accumulated mainly in the liver, followed by the kidneys and the spleen, whereas the 80 and 110 nm nanoparticles distributed mainly to the spleen, followed by the liver and the lungs. The plasma and tissue silver concentrations for these particles can be described by a common PBPK model structure but with different kinetic parameter values. Note that silver nanoparticles can dissolve and release metallic silver in vivo and that the model predicted the total silver concentrations. The PBPK model included blood, liver, kidneys, spleen, and a remainder compartment. Clearance was introduced to the blood compartment to represent the overall clearance (renal and biliary). Except for the blood and remainder compartments, each tissue compartment was divided into two sub-compartments, one freely exchangeable with blood and one for irreversible incorporation in tissues reflecting the low biodegradability of silver nanoparticles. Using this PBPK model, the authors were able to estimate the amount of silver irreversibly retained in the tissues for risk evaluation.

Laomettachit et al. developed a PBPK-PD model to predict the liver toxicity of titanium dioxide (TiO₂) nanoparticles in humans.¹⁰⁵ A permeability-limited PBPK model was used, including venous blood, arterial blood, lungs, spleen, and two clearance organs, the liver and kidneys. The PBPK model was developed based on mice data, scaled to rats for validation, and then scaled to humans for prediction. To predict the liver toxicity of nanoparticles, a cell response model was established based on the in vitro relationship between intracellular nanoparticle concentrations and cellular toxicity. The effects of cell division on the dilution of intracellular nanoparticle concentrations and recovery from liver toxicity were also considered in the cell response model. The cell response model was combined with the predicted liver nanoparticle concentrations in the PBPK model to simulate the dose-liver toxicity relationship and the liver toxicity recovery time in humans after exposure to TiO₂ nanoparticles.

4.1.6. Nanoparticle property-disposition relationship

As nanoparticles are frequently used to change the biodistribution of encapsulated drugs, methods to predict the in vivo disposition of nanoparticles based on measurable particle properties would greatly guide the design and optimization of nanoparticles.²⁰ In order to establish such a quantitative relationship, the in vivo disposition should be quantified to facilitate comparison. Unlike the time-concentration curves, PBPK models can conveniently derive nanoparticle-specific PK parameters. Because PBPK models separately consider drug- and system-specific information, the derived PK parameters from PBPK models are relatively independent of study design, studied population, and altered physiology, which makes PBPK models good tools for the development of quantitative nanoparticle property-disposition relationships.²⁰

The diverse design principles and multi-functionality of nanoparticles make it difficult to build general quantitative relationships between nanoparticle properties and their in vivo performance, especially across multiple types of nanoparticles. Indeed, Sweeney et al. found that the PBPK model (Table 1) calibrated by iridium nanoparticles cannot predict the distribution of silver nanoparticles of similar size.²⁷ Due to the model’s complexity, Sweeney et al. applied Bayesian population analysis and Markov Chain Monte Carlo simulation rather than a deterministic approach to assess the parameter variability and uncertainty. The model was first applied to the data set of iridium nanoparticles (15–20 nm), and then a new dataset was added step-wisely to evaluate the impact of each additional data set on model performance and the precision of parameter estimates. Adding another dataset of iridium nanoparticles (15 ±1.6 nm) did not substantially alter the model performance; however, adding a data set of silver nanoparticles of similar size (20.3 ±1.9 nm) decreased the precision of parameter estimates. The iridium-derived model parameters were not able to predict the kinetics of silver nanoparticles.

Table 1.

Selected PBPK model structures for nanoparticles.

Reference	Nanopartic	Dosing route	Species	Modeling entity	Tissues considered	Clearance mechanism	Tissue models
Kagan, Gershkovich et al. 2014106	AmBisome®	IV	Mice, rats	Liposomal amphotericin B (AmB) and free AmB	Plasma, liver, spleen, kidneys, GI tract, lungs, heart, and the remainders	Clearance of free AmB was defined in the liver, the kidneys, and the remainder compartment	In the spleen and kidneys: In others:
Li, Johanson et al. 201421	PEG-coated polyacrylamide nanoparticle	IV	Rats	Nanoparticle	Arterial blood, venous blood, lungs, bone marrow, brain, heart, kidneys, liver, spleen, and the remainders	Biodegradation was considered to be negligible. Clearance was defined in the kidneys and the liver.	In blood: In organs:
Lin, Monteiro- Riviere et al. 2016111	PEG-coated gold nanoparticles	IV	Mice	Nanoparticle	Arterial plasma, venous plasma, lungs, brain, spleen, liver, kidneys, and the remainders	Metabolism and dissolution were not considered. Clearance was defined in the kidneys and the liver.	In the brain and the rest of the body: In the lung, spleen, liver, and kidneys for 13 nm nanoparticles: For 100 nm nanoparticle:
Lin, Monteiro-Riviere et al. 2016112	PEG-coated gold nanoparticles	IV	Mice, rats, and pigs	Nanoparticle	Arterial plasma, venous plasma, lungs, spleen, liver, kidneys, brain, and the remainders	Metabolism and dissolution were not considered. Clearance was defined in the kidneys and the liver.	In the brain: In others:
Sweeney, MacCalman et al. 201527	Iridium nanoparticles, sliver nanoparticles, titanium dioxide nanoparticles	Inhalation	Rats	Nanoparticle	Arterial blood, venous blood, olfactory region, upper airways, alveolar region, interstitium, lymph nodes, GI tract, liver, spleen, kidneys, heart, brain, and the remainders	Clearance was defined in the kidneys and the liver.	In the liver: In others:

Therefore, we should establish nanoparticle properties–in vivo performance relationships within a particular type of nanoparticles. Such quantitative relationships, once well understood, even with a certain amount of uncertainty, can be readily integrated into PBPK models to make translations from nanoparticle physiochemical properties to their systemic PK performance, which will benefit the development and regulatory assessment of nanodrugs.³ Li et al. applied a generic permeability-limited PBPK model to fit the biodistribution data of five PEGylated poly(lactic-co-glycolic) acid (PLGA) nanoparticles of varied PEG contents, and then performed a multivariate linear regression to correlate three particle properties (size, charge, PEG contents) with biodistribution parameters.²⁰ A multivariate regression was chosen because the particle size and charge varied as PEG contents and these properties together define the particle disposition.²⁰ A moderate to good relationship was observed for each biodistribution parameter. This relationship was then used to predict the PBPK model parameters for a sixth PEGylated PLGA nanoparticle that was in the studied range of PEG contents. The simulated biodistribution for the sixth formulation was close to experimental measurements. As discussed by Li et al., such a nanoparticle propertydisposition relationship may not be applicable to nanoparticles of different chemical compositions.

However, even for the same nanoparticle type, a correlation between particle properties and PBPK model parameters may still be challenging. Mager et al. found no solid relationship between size and charge with PBPK model parameters for gold/dendrimer composite nanoparticles in mice.³² The model included nine compartments—blood, lung, heart, kidney, muscle, brain, spleen, liver, and a remainder compartment. Each tissue was divided into a vascular and an extravascular sub-compartment. For the brain, spleen, and liver, a unidirectional flow from the vascular space to the extravascular space was incorporated to account for the significant tissue accumulation and slow washout during the study time frame for these tissues. Other tissues were described using the generic permeability-limited structure model. The model was developed individually for four gold/dendrimer composite nanoparticles of different charges and sizes (5 nm positive, negative, and neutral, and 11 nm negative), and then extrapolated to a 22 nm positive particle based on the projected effect of size and charge on model parameters. The model did not adequately predict the distribution of the 22 nm positive particle in mice. Similarly, no clear relationships between nanoparticle size and the model parameters were found in the study by Lankveld et al. for 20, 80, and 110 nm silver nanoparticles (the model structure was discussed earlier).²⁸

Correlating in vitro particle properties with in vivo disposition is challenging due to the facts that the commonly measured particle properties (e.g., size, surface charge) are not the only determinants of in vivo disposition. Many other particle properties (e.g., surface chemistry, intra-liposomal ammonium sulfate concentration¹¹) are also critical and oftentimes highly inter-reliant. ^27,28 In addition, the in vitro particle properties do not well reflect the in vivo conditions, such as the formation of biocorona and aggregation. ^28,29,49 It is also worth noting that the protein corona changes with time, the biological environment (e.g., blood circulation and tissue extracelluar space), and species.⁴⁹ It has been shown that the model prediction of nanoparticle cell uptake could be improved when the protein corona composition in serum was included in a multivariate model correlating nanoparticle properties with cell uptake.^49,50

4.2. PBPK models for the unique disposition properties of nanoparticles

4.2.1. Dual PBPK models to describe the disposition of both nanoparticles and released APIs

The disposition of active drugs is controlled by the disposition of particulate drugs and in vivo drug release; the disposition of particulate drugs defines where the active drugs would be released. Therefore, ideally, the model should describe the free drug and particulate drug simultaneously. Amphotericin B (AmB) is a drug to treat fungal and parasitic infections. Kagan et al. developed a dual PBPK model to simultaneously describe the plasma and tissue concentrations of liposomal AmB and the released AmB.¹⁰⁶ The model was based on a previous PBPK model developed by the same group for a small molecule formulation of AmB.¹⁰⁷ The dual PBPK model contains eight compartments—plasma, liver, spleen, kidneys, GI tract, lungs, heart, and a remainder compartment. AmB was assumed to be cleared only in free form, and the clearance was assigned to the liver, the kidneys, and the remainder compartment. Each of these eight compartments contained a liposomal sub-compartment and a nonliposomal sub-compartment reflecting the released AmB. For the non-liposomal subcompartment, the extravascular space was assumed to reach quick equilibrium with the plasma (liver, heart, lungs, and GI tract), be permeability-limited (remainder compartment), or be divided into a permeability-limited extravascular compartment and a “deep tissue” compartment (spleen and kidneys). For the liposomal subcompartment, a linear uptake clearance was introduced to characterize the uptake of liposomes from plasma to tissue for most organs, except the liver and the spleen, where a non-linear uptake clearance was necessary to fit the data. Model structures of the organs can be found in Table 1 or the original publication.¹⁰⁶ Drug release was considered in plasma and the vascular and extravascular space of the tissues. To obtain the in vivo release rate of AmB, the authors first fitted the data of the small molecule formulation to a two-compartment model to calibrate the model parameters for free AmB and then estimated the burst release and the secondary first-order drug release in the central and peripheral compartments by fitting the plasma profiles of liposomal AmB. The estimated drug releases were introduced to plasma and each organ extravascular space in the PBPK model. The dual model was developed in mice and rats and was then successfully scaled to humans using allometric scaling. The model predictions of tissue concentrations in humans agreed well with autopsy reports.

Another dual PBPK model was developed by Dong et al. to characterize the distribution and in vivo drug release of a nanocrystal formulation of the anticancer agent SNX-2112 (203 nm, −11.6 mV) in rats after IV administration.²² A two-step strategy was employed. First, a generic perfusion-limited PBPK model was developed for the non-particulate drug using the PK data of a cosolvent formulation (a small molecule formulation) in rats. The model included plasma, lungs, heart, kidneys, spleen, intestine, and the liver as the only clearance organs. Second, processes describing the particulate drug were included. Uptake of the nanocrystal was included only in the liver and the spleen to account for their significantly higher uptake of the nanocrystal within the first hour compared to the cosolvent formulation. Drug release was considered in plasma, the liver, and the spleen and was described by first-order kinetics. Using the PBPK modeling strategy, the authors concluded that the nanocrystal can rapidly release the poorly soluble drug in vivo and presents minimal systemic risk associated with particulate injection.

4.2.2. A PBPK model with time-varying tissue partition coefficient

Chen et al. applied a perfusion-limited PBPK model to describe the tissue distribution of zinc oxide nanoparticles (10 and 71 nm) and soluble zinc nitrate.¹⁰⁸ Separate tissue partition coefficients were estimated for each size of the nanoparticles and the soluble salt. Instead of assuming a constant tissue partition coefficient which indicates that the tissue reaches equilibrium with the blood, Chen et al. integrated time-varying tissue partition coefficients in their model. This was because reaching equilibrium for nanoparticles in specific tissues may take from days to months. In particular, the partition coefficients for the brain, carcass, and most time points of the heart were sigmoidal against time, whereas other tissues followed monoexponential curve. Therefore, Hill function was used to describe the time-varying partition coefficients for the heart, brain, and carcass. Similarly, time-varying clearance from the GI, the liver, and the kidneys was described by another Hill function. It was also found that the kinetics of zinc oxide nanoparticles after seven days can be better described by replacing the partition coefficients and elimination coefficients with those estimated for the soluble zinc nitrate. This change was explained by the decomposition of zinc oxide nanoparticles into soluble salt after seven days.

4.2.3. PBPK models with tissue-specific MPS function

Few PBPK modeling studies have taken into account the critical role of MPS in nanoparticle distribution and sequestration.^{21,29,104,109} In the model of Li et al. (Table 1), the arterial blood and venous blood compartments were divided into blood and phagocytic cell sub-compartments, all tissue compartments were divided into capillary blood and tissue sub-compartments, and a phagocytic cell sub-compartment was incorporated within the tissue sub-compartment.²¹ The transport of nanoparticles from capillary blood to tissue was permeability-limited and constant for all tissues. The phagocytic uptake rate of nanoparticles was considered to be saturable and decreased linearly as the internal nanoparticle concentration increased. In addition, the phagocytic capacity and cell uptake rate were tissue-specific to reflect the differences in the tissue density of phagocytic cells and tissue physiology. The phagocytic cells could release nanoparticles back to the tissue sub-compartment. The model was fitted to the PK data of ¹⁴C-labelled PEGylated polyacrylamide nanoparticles (35 nm) in rats after IV administration. As expected, the MPS organs—spleen, liver, bone marrow, and lungs—had the highest phagocytic uptake capacity. The sequestration of nanoparticles by MPS accounted for 83% of the tissue exposure. The release rate of nanoparticles from phagocytic cells were orders of magnitude lower than the uptake rate. Unexpectedly, the remainder compartment also had high uptake capacity. This could suggest that the phagocytic uptake parameters also represented parallel mechanisms such as protein binding.²¹ This model was used to describe the role of MPS in nanoparticle tissue distribution. As discussed by Li et al., sequestration of nanoparticles by MPS may reduce toxicity to tissue cells; however, the MPS may also serve as an internal reservoir and slowly release the nanoparticles back to tissues. Safety concerns can also rise from long persistence of nanoparticles in macrophages as discussed in the Metabolism section.

With minor modifications, Carlander et al. extended the model of Li et al. ²¹ to three other non-degradable nanoparticles in rats: uncoated polyacrylamide nanoparticles (31 nm), titanium dioxide nanoparticles (63 nm), and gold nanorods (13 nm in diameter and 56 nm long).¹¹⁰ Instead of assuming all tissues have the same partition coefficient in the study of Li et al., the final model selected by Carlander et al. grouped the tissues into three sets—tissues with high (liver, spleen, and bone marrow), medium (rest of the body), and low permeability (brain). Nanoparticle-specific parameters, such as tissue partition coefficients, clearance rates into urine and feces, and phagocytic uptake rate, were fitted for each type of nanoparticles, while the numbers of tissue phagocytic cells were kept the same across all nanoparticles. Changes in the parameter settings did not significantly affect the fitting for PEGylated polyacrylamide nanoparticles compared to Li’s study. The model of Carlander et al. adequately described the temporal profiles of tissue and blood concentrations for all four nanoparticles after IV administration in rats. As expected, the nanoparticlespecific parameters (uptake rate and capacity) differed distinctly across nanoparticles. Consistent with the study by Li et al., the majority of the nanoparticles accumulated in phagocytic cells—71% of PEGylated polyacrylamide nanoparticles, 84% of uncoated polyacrylamide nanoparticles, 99.6% of gold nanorods, 100% of titanium dioxide nanoparticles. The doses of the latter two particles were too low to saturate the liver phagocytic cells, and consequently, these particles were mainly captured by liver phagocytic cells. Consistent with the stealth property, the PEGylated polyacrylamide nanoparticles had lower phagocytic uptake capacity and clearance rates.

Similarly, Lin et al. incorporated phagocytic uptake in the main MPS organs in their PBPK model for 13 nm and 100 nm PEGylated gold nanoparticles in adult mice.¹¹¹ To improve model parsimony, the distribution and permeability coefficients were kept the same, while endocytosis-related parameters for each size were optimized separately. The authors compared the phagocytic cell uptake of nanoparticles from tissue, as in the studies of Li²¹ and Carlander¹¹⁰, versus uptake directly from the blood (Table 1) and found that the 13 nm particle was better characterized by the former pathway and the 100 nm particle was slightly better characterized by the later pathway. This is consistent with the general belief that larger nanoparticles are more efficiently taken up by phagocytic cells and with the observation of shorter circulation half-life for larger nanoparticles. For consistency, the former pathway was selected for nanoparticles of both sizes. As expected, phagocytic uptake rates for the 13 nm particle were lower than those for the 100 nm particle. Lin et al. also found that the saturable tissue phagocytic uptake was better described by the Hill function than the linear equation used in the study of Li et al..²¹ The Hill function was able to capture the approximately 10 h delayed increase in tissue concentrations of the 13 nm nanoparticle, reflecting the delayed activation of phagocytic cells by smaller nanoparticles. The model sensitivity to physiological parameters was found to be size-dependent. The models was able to adequately predict the plasma and tissue concentrations of nanoparticles of the same type and similar size in mice—16 and 20 nm particles using the 13 nm model parameters and 80 nm particles using the 100 nm model parameters.

A follow-up study by Lin et al. explored the interspecies extrapolation of MPS endocytic parameters based on the above mice PBPK model¹¹¹ with minor modifications (Table 1).¹¹² The mice PBPK model was updated with the physiological parameters for rats, pigs, and humans. The tissue distribution and permeability coefficients of PEGylated gold nanoparticles in all species were kept the same as those in mice. The nanoparticle phagocytic parameters for rats and pigs were reestimated and validated with independent datasets. Because the nanoparticle dose in humans was significantly lower than that in rodents and the PK of nanoparticles was dose-dependent, a low-dose rat model with a new set of endocytic parameters was developed. For the pig model, the maximum endocytic rates and capacities in the Hill function were scaled to humans based on the relative density of phagocytic cells in the liver (a revised pig model). The authors then compared the translations of endocytic parameters from animals to humans. It was found that only the low dose rat model and the revised pig model could adequately predict the plasma nanoparticle concentrations in humans. The final selected human model was used to predict the toxic dose of the studied nanoparticle in humans by integrating the reported lowest toxic concentrations in rats, human tissue, and blood cells.¹¹² The model was also applied by Cheng et al. to estimate the toxic dose in humans using measured in vitro dose-response relationships in healthy human cells.¹¹³

More examples of toxicokinetics can be found. Bachler et al. applied PBPK modeling to estimate the absorption and systemic distribution of silver nanoparticles, TiO₂ nanoparticles, and gold nanoparticles post percutaneous, intestinal or pulmonary exposure.^29,104,109 In their PBPK model, tissue translocation of nanoparticles was permeability-limited and depended on the vasculature structure. The tissues were categorized into four types based on the vascular structure—non-sinusoidal nonfenestrated type (brain, muscles, lung, and heart), non-sinusoidal fenestrated type (skin, testes, kidneys, and intestines), sinusoidal reticuloendothelial type (liver and bone marrow), and sinusoidal non-reticuloendothelial type (spleen). Size-independent tissue translocation was assumed. Compared to the model of Li et al.,²¹ the MPS subcompartment was only incorporated into the main MPS organs, which were the liver, spleen, and lungs. MPS uptake was assumed to be size-dependent and non-saturable. However, it was found that MPS uptake contributed negligibly to tissue distribution at blood silver concentrations lower than 180 ng/g.²⁹ Bachler’s model was able to describe the tissue disposition of 15–150 nm sized nanoparticles in rats and humans for risk assessment. In particular, the model was able to distinguish the mucociliary cleared from biliary cleared particles in the feces to estimate the percentage of inhaled nanoparticles cleared without entering the human body, which was challenging to discern in animal studies.¹⁰⁴ Li et al. used a similar PBPK model to study the clearance and systemic distribution of combustion-generated cerium oxide nanoparticles in rats through inhalation.¹¹⁴

As mentioned earlier, nanoparticles tissue distribution is heterogeneous. Due to large sizes and charges, extravasated nanoparticles are largely restricted in perivascular areas within tissue interstitial space. Sometimes, tissue-associated macrophage could take up extravasated nanoparticles, serving as a reservoir of nanoparticles.^{21,29,104,109} Therefore, unlike small molecules, the tissue-average k_p term is often not appropriate to delineate the heterogeneous tissue distributions of nanoparticles. When data is available, considerations of tissue-specific factors, like MPS abundance^2,9,47,51,52 and basement properties of tissue vascular endothelial membranes,^2,59 are always preferred in PBPK models.

5. SUMMARY, FUTURE DIRECTION, AND CHALLENGES

Nanoparticles have many distinct disposition characteristics compared to small molecule drugs that should be considered in developing PBPK models. 1) The physicochemical properties of nanoparticles change immediately upon entering the body. Their new identities in the biological environments (altered particle size, shape, composition, surface chemistry, formation of biocorona, elasticities, and so on) determine the biological fate. 2) Nanoparticles are recognized as foreign particles and are readily sequestered by circulating and tissue-resident phagocytic cells (the MPS). Therefore, the levels of phagocytic cells residing in individual tissues should be quantified to support the development of PBPK models. The metabolism of nanoparticles mainly occurs in the lysosomes of the MPS, whereas small molecules are frequently metabolized in the hepatocytes. Drug release following particle degradation could occur in circulation, in MPS-abundant organs, and in targeted tissues. Therefore, an abrupt release phase is often needed in the PBPK models to capture the immediately occurring concentrations of APIs.¹⁰⁶ Although largely ignored, nanodrugs may alter the metabolite profiles or activation of prodrugs (see FDA guidance for industry “Liposome Drug Products” published on April 2018 and reference¹¹⁵). Changes in the kinetics of drugs presented to transporters or enzymes (drug release) and in the disposition tissues and/ or cell types due to encapsulation in nanoparticles may result in saturation or alterations of metabolism pathways. As a result, the compositions, relative abundance, and kinetics of metabolites may be altered, raising efficacy or toxicity concerns. Although not explored, PBPK modeling and simulation can be used to simulate the changes in metabolite profiles and related biological effects of nanodrugs in comparison to small molecule drugs. 3) Due to their larger size, the tissue distribution of nanoparticles is often governed by convection rather than by diffusion.² Nanoparticles tend to accumulate in tissues with leaky vasculatures, such as the liver, spleen, kidneys, and some tumors. Similar to the PBPK models developed for protein drugs, PBPK models for nanoparticles should entail the lymphatic system to drive nanoparticle tissue distribution and recycle into the blood. In contrast, many solid tumors, due to the collapsed lymphatic system and irregular blood vessels, need to be modeled differently. Diffusion rather than convection across tumor blood vessels is often assumed, and no significant lymphatic recycling is present.⁶⁶ 4) Within tumors, nanoparticle distribution is usually heterogeneous, showing high perivascular distribution, which is associated with restricted diffusion through the extracellular matrix, high interstitial fluid pressure, and binding site barriers (for targeted nanoparticles). When of research interest, a reaction–diffusion model should be developed specifically for tumors in the PBPK models to describe the spatial distributions of targeted nanoparticles in tumors. The tissue spatial distribution of nanoparticles could be studied using many imaging techniques to support the model development. Although it is common in PBPK models for small molecules, there is no PBPK model ever developed for nanoparticles detailing the active transport or efflux mechanisms. The contribution of these uptake/efflux transport mechanisms in tissue distribution can be reflected in a compound parameter, tissue partition coefficients, particularly when the transport process is fast and the transport kinetics is beyond the scope of research interest. The schematic of a nearly complete PBPK model structure for nanoparticles is proposed in Figure 3, where we can find almost all the above-mentioned components.

Figure 3.

A specialized PBPK model for drugs encapsulated in nanoparticles. Compared to the generic PBPK model for small molecules as shown in Figure 1, this specialized model for nanodrugs consists of two layers of PBPK models for each of nanoparticles and released small molecules. The two layers are connected by drug release (orange arrows). Drug release is tissue-specific; the presence of phagocytic cells and local stimuli (e.g. low pH in tumors) affect drug release (thicker orange arrows). For small molecules, the PBPK modeling layer is the same as the model in Figure 1. For nanoparticles, due to the large size, tissue distribution is convection-driven (unidirectional blue arrows) in most tissues. One exception is the tumor, where passive diffusion is the major distribution mechanism because the high interstitial pressure in solid tumors limits effective convection. The enhanced accumulation of nanoparticles in the lung, spleen, and liver is associated with the leaky vasculature structures and MPS sequestration, which is depicted by the thicker blue arrows. High tumor accumulation of nanoparticles is ascribed to the EPR effect. This specialized model also includes the lymphatic systems for recycling nanoparticles (blue dashed arrows) from the interstitial space.

There are several challenges associated with the complexity of nanoparticle disposition that limit the application of PBPK modeling in nanoparticle disposition. First, in vivo conditioning of nanoparticles, such as biocorona formation, aggregation, and degradation significantly and dynamically change particles’ properties and disposition. Nanoparticles’ properties and the interaction with the body system are usually associated with the local microenvironment.⁴⁹ The tissue partition coefficients, cell uptake rate, and drug release rate are likely to be time- and tissue- dependent, presenting challenges in parameterizing their spatiotemporal disposition in PBPK models. Meanwhile, due to different biological conditioning,^49,116 model parameters calibrated against one administration route may not predict the disposition of another administration route.¹¹³ Efforts to include time-varying tissue partition coefficients have been made to reflect the slow distribution equilibrium and the in vivo degradation of a silver nanoparticle.¹⁰⁸ Empirical models correlating the nanoparticle biocorona in serum and the amount of cell association have been attempted.^49,50 Apart from complement adsorption and MPS sequestration, adsorption of some other serum proteins, such as apolipoprotein E, was found associated with additional clearance pathways mediated by the cognate receptors.¹¹⁷ The varying levels of these serum proteins and the expression of the cognate receptors could be a source of PK variability. Thus, the biocorona composition and the impact of adsorbed molecules on the disposition of nanoparticles should be assessed. It is critical for a mechanistic PBPK model to describe the interactions of nanoparticles with the serum proteins and the cognate receptors for prediction of PK variability.

Second, the body’s response to nanoparticles is also a dynamic and population-specific process. In particular, the MPS phagocytosis of nanoparticles is saturable, and tissue- and population-specific.^80,83,84,87 To this end, a Hill function have been used to describe the decreasing phagocytic uptake at high nanoparticle concentrations;²¹ tissue-specific phagocytic uptake has been modeled by updating the phagocytic uptake rate or density of phagocytic cells;²¹ interspecies scaling has been conducted based on the relative density of phagocytic cells in the liver;¹¹² and translational probes for the quantification of MPS functions have been proposed to monitor the function of MPS for precision dosing in different scenarios (populations and dosing cycles).⁸⁸ Interspecies differences in biocorona composition and evolution may also complicate the translation of PBPK models across species.^33,118

Third, nanoparticles are a heterogeneous population of individual particles that have different particle properties. The average particle properties may not predict the disposition of individual nanoparticles.¹¹⁹ Two formulations with the same average but distinct distributions of properties may exhibit largely different disposition behaviors.¹¹⁹ However, a reliable analytical method to characterize and track nanoparticles as individual nanoparticles in vitro and in vivo is lacking,³³ restricting our understanding in particle disposition and the establishment of correlation between particle properties and disposition. To date, nanoparticles have been tracked by encapsulated APIs, constituent elements (e.g., metal elements), or degradation products. Using constituent element concentrations to reflect nanoparticle levels is possible for stable metallic nanoparticles. However, for unstable nanoparticles, such methods would yield high bias because the particles constantly change in vivo, for example in terms of aggregation, degradation, and drug release. Moreover, it is important to separately measure the concentrations of free drug, protein-bound drug, and nanoparticleassociated drug in blood and tissues. These three populations have distinct physicochemical properties and disposition. However, reliable technologies to dynamically quantify these populations in blood and tissues are also missing.¹²⁰

PBPK modeling and simulation has become a critical tool to address a variety of drug development and evaluation questions, particularly for questions that cannot be readily assessed through clinical trials. PBPK models capture the best current understanding of the complex interplays between drugs and physiology and biology. As reviewed in this work, drugs encapsulated in nanoparticles have exhibited complex interactions with the physiological and biological systems, which makes PBPK modeling and simulation an excellent technology to integrate information about APIs, nanoparticles, systems, and their dynamic interactions. There are many considerations and challenges associated with the development of PBPK models for nanoparticles. This research filed remains in its infancy; however, along with the advent of novel analytical technologies, there will be a significant increase in the use of PBPK modeling and simulation to characterize and predict the disposition and biological effects of nanoparticles. The PBPK models summarized here can be updated when details about in vivo trafficking and the diffusion of nanoparticles become available.