A Unified Mathematical Model for Nano-Liposomal Drug Delivery to Solid Tumors

Abstract

Nanoparticles such as liposomes allow more targeted drug delivery for improved efficacy and/or reduced toxicity in both passive (e.g. Doxil) or active (e.g. Thermo-Sensitive Liposomes or TSL) release forms compared to unencapsulated drug (i.e. conventional chemotherapy). Optimization and evaluation of these different drug delivery systems are experimentally challenging because of varying tissue parameters as well as limited avaiability of experimental data. Here, we present a novel unified mathematical model that can simulate various liposomal drug delivery systems and unencapsulated drug with a single set of equations. We use this model to evaluate the chemotherapy performance of free Doxorubicin (as drug), as well as various liposomal drug delivery systems: (1) passive liposomes (Doxil) (2) active-triggered Thermo-Sensitive Liposomes (TSL) with either intravascular (TSLi) or extravascular-triggered (TSLe) release. Furthermore, we implemented a more accurate expression to consider incomplete liposomal drug release. The proposed model matches experimental in vivo results in terms of maximum drug concentration in tumor. The simulations predict better overall performance for all liposomal delivery systems compared with free Dox. TSLe is shown to be more efficient for less permeable and perfused tumors than other systems. The optimal release rate is lower for TSLe and Doxil compared to TSLi. The performance of free DOX changes little for varying tumor characteristics such as perfusion and permeability.

I. Introduction

In the beginning of 20^th century, the Nobel laureate Paul Ehrlich described an effective therapy for cancer as a “magic bullet” that not only targets the tumor tissue, but also triggers the release of a toxin at the specific site [1]. Doxorubicin (Dox) as the most effective anthracycline is used for the treatment of solid tumors [2] such as lymphoma, genitourinary, thyroid, and stomach cancer [3]. Clinical use of conventional chemotherapeutics is often limited due to inadequate delivery of drug to the target tissue or due to harmful toxic effects on normal organs. Thus, a continuous search for more advanced drug delivery methods is of great importance [4].

Delivery of chemotherapeutic agents by dedicated nanoparticles has the potential to achieve high efficacy as well as low toxicity [5]. Liposomes are one of new drug delivery systems, consisiting of self-assembled colloidal particles that occur naturally and can be prepared artificially, as shown by Bangham and his students in the mid-1960s [6]. Liposomes are currently used for delivery of several drugs (e.g. Doxil®, Caelyx® [Johnson & Johnson, NJ,USA], and Ambisome® [Gilead, CA, USA] FDA-approved for clinic) [7] in spite of the totalfailure of the first negatively charged, medium-size OligoLamellar Liposomes (OLV) in the 80s [8].

Site-specific targeting and triggering are required features to fulfill Paul Ehrlich’s vision of a “magic bullet” for treatment of cancer [1]. However, none of the FDA-approved liposomes or lipid nanoparticles are yet coated with ligands or target moieties to deliver drugs to target tissues, cells or subcellular organelles [9]. Nevertheless, passive targeting has been exploited based on the Enhanced Permeability and Retention (EPR) effect in tumor tissue [10, 11]. The EPR effect arises from the fact that tumor cells must stimulate the rapid production of blood vessels in order to keep pace with ever-increasing oxygen and nutrient demands. The hastily grown neovasculature differs greatly from normal blood vessels, with an architecture characterized by poorly aligned endothelial cells, wide fenestrations, impaired functional cell receptors, wide lumens, and absent or abnormal smooth muscle layers, perivascular cells, and basement membranes [5]. As a result of these anatomical deficiencies, tumor blood vessels are irregularly shaped, dilated, and leaky, allowing for the extravasation of macromolecular drugs and nanomedicines into the tumor tissue. Furthermore, the tumor’s impaired lymphatic clearance of macromolecules and lipids from interstitial tissue prolongs the retention of these macromolecular species [11].

Doxil is a first-generation liposome that belongs to the family of Stealth liposomes, encapsulating the Dox as the anticancer drug. Doxil has long-circulating half-life owing to the coat of polyethylene glycol (PEG). The motivation for clinical use of Doxil, however, stems from reduced cardiotoxicity rather than from higher efficacy compared to unencapsulated Dox [8]. Furthermore, Doxil causes new side effects such as skin toxicity, including hand-foot syndrome and mucositis. It has been argued that these undesirable effects could be a consequence of the lack of an active release mechanism [12, 13].

In 1978, Yatvin and Weinstein introduced liposomes that released Neomycin and inhibited bacteria protein synthesis in vitro at specific temperatures [14]. These so-called Traditional Thermo-Senstiive Liposomes (TTSL) were able to release a hydrophilic drug when the temperature was increased a few degrees above physiological temperature. Following extravasation some hours after administration, the tumor area is heated so that the extravasated TSL (TSLe) releases the drug at the target site. The breakthrough in development of clinically-usable TSL formulations was the incorporation of Lysolipids (LTSL), as described by Needham et al. in 2000 [15] to lower the phase transition temperature and promote rapid drug release [16]. LTSL has undergone further pharmaceutical development by Celsion Inc. and is currently in clinical trials as ThermoDox® [17]. Recent findings suggest that the mechanism of drug delivery from combination of TSL and mild hyperthermia is better achieved through intravascular release [18]. In this intravascular release delivery paradigm (TSli), the LTSL encapsulating Dox are injected just prior to or during the hyperthermia treatment, with immediate release of their contents upon arrival in the heated tumor area (Fig. 1). The optimum release rate is still under investigation, with time constants ranging from a fraction of to several seconds [19, 20].

Fig. 1.

Targeted drug delivery with TSL: Fluorescent images showing unencapsulated Dox (red) in mice during heating (a) and after heating (b) TSL-Dox administration of 9mg/kg (unpublished results).

Although all liposomal delivery systems including Doxil, TSLe (extravasation), and TSLi (intravascular release) may lead to reduced side effects relative to free Dox, the drug dose delivered to the tumor or tumor cytotoxicity is still subject of investigation. Those studies confront many complicated and subjective factors such as, among others, lack of real-time masurement tools to track the liposomes (100 nm) and Dox (<1 nm) in different tumor compartments. Futhermore, many physical and physiological parameters, such as endothelial permeability, local plasma perfusion, protein binding, and liposomal release rate, influence pharmacokinetics as well as pharmacodynamics. One particular problem is that the optimization of drug release, both in vitro and in vivo, is a difficult task because the knowledge of drug pharmacodynamics in relation to drug delivery is often limited [4]. Accordingly, the development of a computer model that describes the PharmacoKinetics (PK) and PharmacoDynamics (PD) of antitumor agents could provide valuable information to evaluate the influence of various parameters and examine the efficiency of different liposomal formualtions and delivery methods.

PK-PD computational models of liposomal drug delivery systems are scarce in the literature. In 1999, Harashima et al. developed a compartmental model to predict the optimum conditions that maximize the antitumor effects of liposomal drug carriers [21]. Their model was developed just for passive liposomal delivery (e.g. doxil). In PD model, they considered the unbound or free Dox in the interstitial compartment as the cell-kill factor and neglected the saturation of transmembrane transport [22]. In addition, the back-diffusion of liposomes from the interstitial to the tumor plasma was ignored in the PK model. They did not consider free or released Dox separately in the tumor plasma compartment, but rather only considered liposomal (encapsulated) Dox. In 2000, El-Kareh et al. improved upon this prior PK model to evaluate Stealth liposomes and TSL against free Dox [23]. They used a simple temporal function to represent the liposomal drug in systemic plasma. This function was applied as an input to the differential equations representing tumor compartments. While they did not model intravascular release, this assumption is inadequate for intravascular release of TSL where the liposomal concentration of systemic plasma interacts dynamically with the release at tumor compartments. In short, the systemic plasma concentration of liposomes/drug should be modelled as a separate compartment (or differential equation) accounting for bidirectional interactions with the tumor compartments.

In 2012, Gasselhuber et al. developed another compartmental model, evaluating all liposomal delivery systems, particularly for intravascular fast-release liposomes (TSLi) [19]. Although in their model, the drug release from TSL was considered as a bi-exponential function, the compartmental release rate should be revised to be valid for compartment models as demonstrated in the next section. In effect, the release term in the tumor vasculature was overestimated for fast-release delivery. Liu and Xu studied thermo-sensitive liposomal delivery of Doxorubicin to tumors using a mathematical model for intravascular release mode in 2015 [27]. Using peak tumor intracellular drug concentration, they showed the most sensitive parameters are liposomal release rate, tumor blood perfusion, and vascular drug permeability. However, they did not consider the liposomal diffusion into tumor EES that is here shown to be an important factor. In addition, the area under curve can more accurately represent the therapeutic effects rather than the peak intracellular concentration, though the latter is well correlated with cardiotoxicity [22, 28].

The above-referenced prior studies [19, 21, 23] did not use a single unified set of modeling equations, but employed various equation sets with different approximations depending on the drug delivery system. A unified model for modeling of various drug delivery systems based on the same equations has several advantages: (1) it can be adapted to new drug delivery systems without the need to develop a new model; (2) it simplifies direct comparison between different delivery systems; and (3) it may allow more direct cross validation of the model accuracy through comparing the results with experimental data.

In this paper, we develop a new multi-compartment model with a unified set of equations that are applicable to all examined delivery systems. Additionally, we develop a new expression to account more accurately for incomplete liposomal drug release. We used the proposed PK model to predict the drug concentrations in different tumor compartments considering model parameters based on experimental data for mice. In addition, different liposomal delivery systems are here simulated and compared in terms of therapy efficiency with regard to varying parameters, including TSL release rate, tumor perfusion and permeability to Dox. Finally, the effects of release rate on both desired tumor cytotoxicity and systemic toxicity are evaluated.

II. Methods

A unified structural in-silico model is presented to mathematically describe the PK of different drug delivery systems, including free (unencapsulated) Dox, Doxil (Stealth liposomes), extravascular release TSL triggered following to extravasation (TSLe), and intravascular release TSL with either slow- (STSLi) or fast- (FTSLi) release time constants (Table 1). Here, Dox has been used as the anticancer chemotherapeutic drug for all drug delivery systems. Dox is administered in free or liposomal form via intravenous bolus injection. All thermosensitive liposomes (TSLe, STSLi and FTSLi) have the same heat exposure of 42°C for 30min [35]. For intravascular release (FTSLi and STSLi), hyperthermia is applied immediately after TSL administration, whereas for TSLe, the tumor tissue is exposed to hyperthermia (at 42°C) only 24 hours after adiminstration. This delay allows the liposomes to leak out of the vasculature and accumulate in the interstitium owing to the EPR effect. STSLi and FTSLi are associated with considerable leakage, respectively α₃₇ = 10% and α₃₇ = 30% at body temperature due to the specific properties of the Lyso-liposomes used for fast release [36]. Furthermore, the liposomes do not entirely release the drug in response to the hyperthermia trigger, but α₄₂ = 80% for STSLi and FTSLi [19] (where α_xx corresponds to the fraction of drug released at temperature xx).

TABLE I.

Different drug delivery systems that are here modelled with proposed in-silico model. Leakage and Release time constants (τ₃₇ and τ₄₂) are shown as well as the final percentage of liposomes leaking α₃₇ (at 37°C) or releasing the drug in response to the heating trigger α₄₂ (at 42°C).

Drug Delivery System	Tumor Targeting	Release Trigger	Liposome type	Leakage: τ37 [sec] α37 [%]	Release: τ42 [sec] α42 [%]	Treatment Method
Free Dox	Passive (EPR)	-	-	-	-	Bolus i.v.
Doxil	Passive (EPR)	None	Stealth (1st Gen.)	454.4 hours 100%	as leakage as leakage	Bolus i.v.
TSL-extravasation (TSLe)	Passive (EPR)	Heat	Traditional (TTSL)	454.4 hours 100%	4 hours 100%	Bolus i.v. + heating after 24h for 30min
Slow-release TSL (STSLi)	external localisation	Heat	Lyso-TSL (LTSL)	600 sec 10%	120 sec 80%	Simultaneous bolus i.v. and heating for 30min
Fast-release TSL (FTSLi)	external localisation	Heat	Lyso-TSL (LTSL)	120 sec 30%	3.3 sec 80%	Simultaneous bolus i.v. and heating for 30min

A. Compartmental Release Rate

In the context of compartment modeling, the compartmental release rate of liposomes at temperature T and time t, ${\text{Rls}}_{\text{T}}^{\text{lip}}(t)$, should be defined for each compartment such that: ${\text{dC}}_{\text{lip}}(\text{t})/dt=-{\text{Rls}}_{\text{T}}^{\text{lip}}(t)\cdot {\text{C}}_{\text{lip}}(\text{t})$, where C_lip(t) and T, and lip represent respectively the liposomal Dox concentration, temperature at each compartment and the liposome type. In the case of non-complete release (α_T ≠ 100%) as for Lyso-liposomes, the time varying concentration of liposomal Dox is: C_lip(t) = 1 − α_T + α_T. exp(−R_Tt) such that R_T and α_T are the (characteristic) liposomal release rate and final release percentage at temperature T (see Table 1). In the previous studies [19, 21, 23], the (constant) characteristic R_T was directly used in the compartment equations instead of ${\text{Rls}}_T^{lip}(t)$ which is not accurate in cases when α_R ≠ 100% (e.g. fast release case). Assuming a time-varying release rate as ${\text{C}}_{\text{lip}}(\text{t})={\text{C}}_0\cdot \text{exp}\left[-{\text{Rls}}_{\text{T}}^{\text{lip}}(t)\cdot \text{t}\right]$, the compartmental release rate ${\text{Rls}}_T^{lip}(t)$ would be (see Supplementary Material for mathematical proof):

$${\text{Rls}}_T^{lip}(\text{t})=\frac{{\alpha }_T\cdot R_{\text{T}}\cdot e^{-R_{\text{T}}t}}{1-{\alpha }_T+{\alpha }_T\cdot e^{-R_{\text{T}}t}}$$ (1)

The relation leads to ${\text{Rls}}_{\text{T}}^{\text{lip}}(\text{t})={\text{R}}_{\text{T}}$ only if α_T = 100%. This time-varying release rate (1) can be used for every liposome type in the compartment equations such that a unified compartment model for all liposomal delivery systems can be obtained in spite of previous works.

B. Drug uptake in Tumor Cells

An unresolved challenge in experimental therapeutics is how to quantitatively compare different drug administration protocols or delivery strategies for a specific cytotoxic agent [37]. Although the PK properties of the drug may be well defined, the best way to couple the PK to the PD effect remains elusive [37]. The principal term in determining cell damage is typically the molar drug accumulation history inside the tumor cells [38]. Intracellular drug accumulation is a complex process including drug uptake, retention, distribution, and efflux from the cell [31]. Based on the experimental data in [34], the model proposed by El-Kareh et al. is here used for describing the cellular uptake of Dox (or intracellular concentration) $C_{dox,i}^T$, from EES, $C_{\text{dox},e}^T$ as follows [23]:

$$\underset{\mathit{\text{cellular uptake from membrance}}}{\underbrace{K_{3ci}\cdot \left[K_{1ci}\cdot C_{dox,\text{e}}^T+\frac{K_{2ci}\cdot C_{dox,e}^T}{K_{ici}+C_{dox,e}^T}-K_{5ci}\cdot C_{dox,i}^T\right]}}$$ (2)

The coefficients in (2) were obtained by fitting the model to experimental data from [34] using Least-Square optimization.

C. Efficacy and Toxicity

Many parameters having influence on the (beneficial) tumor cytotoxicity are very hard or impossible to measure. The therapeutic efficiency of drug is often described by PD models. On the other hand, the cytotoxic effects of the drug often occur long after the exposure of the cell to the drug [37]. The efficacy of drugs are usually modelled using nonlinear PD models in terms of different cell uptake amounts such as the peak or Area-Under-Curve (AUC) values of the intracellular concentration [22, 28]. To avoid any PD modelling error due to the lack of appropriate in vivo experimental data, here we report the AUC and peak values of the drug concentration in the intracellular compartment as a surrogate for PD model rather than explicitly implementing a PD model. Both AUC and peak values of drug intracellular concentration have been shown to correlate with cell viability [22, 28]. To assess adverse side effects, systemic toxicity is here evaluated. The toxicity of each delivery system depends on the amount of unbound Dox circulating in the systemic plasma to which the non-target healthy tissues are continuously exposed. To study toxicity, the AUC and peak values of Dox concentration are evaluated in the systemic plasma for each delivery system.

D. Unified Compartment Model

The same unified compartment model is here used for all delivery systems. The quantities of liposomal and released (or free) Dox are tracked in the body using distinct compartments (see Fig. 2). The concentrations of liposomal drug C_lip and free Dox C_dox are the corresponding variables. As shown in Fig. 2, the tumor plasma has not been considered as a separate compartment for liposomal Dox primarily because of large spatial variations along the tumor vasculature particularly for intravascular release system. This non-uniform distribution of liposomes directly determines the supply of Dox to the tumor tissue. Avoiding any spatial complexity in the differential equations, the liposomal Dox of tumor plasma can be estimated at the incoming and outgoing plasma and applied to the adjacent compartments (see Supplementary Material). Invoking the compartment model in Fig. 2 and also the compartmental release rate in (1), the system equations can be described for all the compartments. The respective differential equations are presented in Supplementary Material.

Fig. 2.

Schematic of the proposed compartment model. Concentration of the liposome (right in brown) and Dox (left in blue) are modelled in separate compartment equations. Non-tumor tissues include systemic plasma (S) and Lumped Tissue (LT) compartments. Tumor consists of plasma (p), extra-cellular extra-vessel Space (e) and intracellular (i) compartments. Liposomes in tumor plasma (cloud) have no separate compartment since its effect is incorporated in the neighbouring compartments’ equations. TS and TT represent the Systemic (37°C) and Tumor Temperatures (either 37°C or 42°C during hyperthermia) respectively.

E. Parameters of the Compartment Model

As mentioned in the preceding subsection, the system equations are presented in the Supplementary Material for describing the PK of liposomal Dox delivery. To simulate and validate that model, all the parameters were either adopted from the literature, or calculated by fitting to experimental data from mouse studies. An equivalent Dox dosage of 9 mg/kg was assumed for all drug delivery modalities as used in [19]. It is known that a considerable fraction of un-encapsulated Dox is bound to proteins in the plasma as well as in the interstitial space (>70%) [39]. This bound fraction has been approximately accounted for by using the apparent volume of distribution for systemic plasma ${\text{V}}_{\text{p},\text{dist}}^{\text{S}}$. The physiological parameters such as compartment volumes were extracted from mice experiments as listed in Table 2 [24–26].

TABLE II.

Physiological parameters of experimental mice used in the compartment model adopted or calculated from the data in [24–26].

Symb.	Description	Value		Symb.	Description	Value
Dose	Total Dox dose	0.18mg		${\text{V}}_{\text{p}}^{\text{T}}$	Volume of Tumor plasma	0.3 μl
Cdox.lip	Mass ratio of dox to liposome [dox/lip]	1/20		${\text{V}}_{\text{e}}^{\text{T}}$	Volume of Tumor EES	1.9 μl
${\text{V}}_{\text{p}}^{\text{S}}$	Volume of Systemic plasma	0.54ml		${\text{V}}_{\text{i}}^{\text{T}}$	Volume of Tumor cells	1.9 μl
${\text{V}}_{\text{p},\text{dist}}^{\text{S}}$	Apparent Volume of Systemic plasma	18.9ml		${\text{w}}_{\text{p}}^{\text{ST}}$	Plasma Perfusion per Tumor volume (ml)	16ml/min

To simulate the compartment model, the kinetic parameters (or microrates) of compartments are necessary for two types of particles: free or unbound Dox and the liposomal Dox considering all delivery systems of Doxil, TSLe, slow- and fast-release TSLs (STSLi and FTSLi). Table 3 demonstrates the kinetic parameters adopted from the experimental literature.

TABLE III.

Kinetic (Micro-rate) parameters for free (unbound) and liposomal Dox used in the compartmental modelling based on mice experimental data.

Liposomal Doxorubicin [25,29,30]
Systemic clearance of Liposomes: kelim,lip [1/hour]
Doxil	TSLe	Slow-release TSLi	Fast-release TSLi
0.0339	0.0339	0.8021	0.8021
Intracellular uptake (Endocytosis) Neglected	Permeability in tumor: vasculature-EES [m/sec] $P_{pe}^{lip}$=2E-9		Surface area per tumor volume: vasculature-EES [m2/m3] Spe,vas =2000

Free (unbound) Doxorubicin [22,31–33]
Systemic clearance [1/hour]	Permeability in tumor vasculature-EES: [m/sec]	Transfer rate: Systemic Plasma-Lumped Tissue [1/hour]
kelim.dox=7.56	$P_{pe}^{dox}$=2.45E-6	kSPLT=33.8	kLTSP=0.254

Tumor Intracellular uptake [19,34]
K1ci=2.251	K2ci=0.045 [kg/m3]	K3ci=2.Se-4 [1/sec]	Kici=5.29e-4[kg/m3]	K5ci=1

III. Simulations

To evaluate and compare the performances of different delivery modalities, the proposed compartment model (Fig. 2) was simulated in MATLAB. The parameters listed in Table 1, Table 2, and Table 3 are used as nominal or ‘x1’ values in the sensitivity analyses. As mentioned in the previous subsection, the effectiveness and toxicity were estimated for each delivery system based on peak/AUC values of unbound DOX in tumor cells and systemic plasma respectively.

To study and evaluate the performance of different drug delivery systems, the most critical and controllable parameters should be selected out of a wide range of parameters. Previous studies show that the most sensitive parameters are liposomal release rate, tumor perfusion rate (we have equivalently addressed the tumor plasma perfusion), and the permeability of tumor vasculature to the drug [27]. Thus, the proposed PKPD model was simulated with consideration of varying values of parameters including permeability to Dox (${\text{P}}_{\text{pe}}^{\text{dox}}$ in Table 3), tumor plasma flow $\left({\text{F}}_{\text{p}}^{\text{ST}}\right)$, and the release rate for each liposome (α_T, R_T in Table 1). The plasma flow ${\text{F}}_{\text{p}}^{\text{ST}}$ in tumor (Supplementary Material) was obtained by multiplying the tumor perfusion rate (${\text{w}}_{\text{p}}^{\text{ST}}$ in Table 2) by the total tumor volume (${\text{V}}^{\text{T}}={\text{V}}_{\text{p}}^{\text{T}}+{\text{V}}_{\text{e}}^{\text{T}}+{\text{V}}_{\text{i}}^{\text{T}}$ in Table 2).

IV. Results

Invoking the model shown in Fig. 2 and the corresponding system equations in Supplementary Material, the drug transport of Dox was calculated for different delivery systems including: free Dox, Doxil (Stealth liposomes), extravascular-release TSLe, and intravascular-release of FTSLi and STSLi (associated with fast- and slow-relase formulations respectively). Fig. 3 demonstrates the unencapsulated Dox concentration in the tumor compartments for different delivery modalities. The intravascular fast-release (FTSLi) mode shows a significantly higher amount of Dox concentration in all three tumor compartments compared with the other modalities.

Fig. 3.

Concentration of unencapsulated Dox in tumor plasma, EES and cells versus time (logarithmic scale) for different delivery modalities: Free Dox (top-left), intravascular slow-release (top right) and fast-release (bottom-left) TSLi and the extravascular release TSle.

The disposition of Dox in tumor cells and plasma is shown in Fig. 4 for the free-Dox administration with varying values of tumor vasculature-EES permeability as well as tumor perfusion. The variation of parameters was considered around the nominal value x1 ranging from 1/20 to 20x the nominal value. As anticipated, the Dox uptake by tumor cells is reduced at lower permeability and perfusion. Similar temporal simulations are shown in Fig. 5 for fast-release TSLi (FTSLi). In contrast to free Dox, cellular drug uptake of FTSLi delivery is more affected by vasculature permeability in tumor. Also, the release rate highly impacts cell uptake. However, drug uptake by tumor cells is nearly insensitive to perfusion rate for FTSLi. According to Fig. 5, the Mean Residence Time (MRT) of the drug within tumor cells increases for lower permeability and perfusion rates.

Fig. 4.

The bio-disposition of free Dox in tumor plasma and cells versus time for varying (vascular) Dox permeability and tumor plasma perfusion.

Fig. 5.

Concentration of unbound Dox in tumor cells versus time for intravascular-release mode (FTSLi), sweeping different permeability, perfusion and release rate. Faster the release rate, more the cell uptake increases.

To quantitatively compare and evaluate the performance of different delivery systems, the peak and AUC of Dox concentration were calculated for both systemic plasma (as measure of toxicity) and tumor cells (as measure of tumor cytotoxicity or therapy effectiveness). We carried out the analysis for varying values of three parameters: vasculature permeability to Dox, plasma perfusion, and the liposomal release rate. The toxicity of each delivery system was assessed by the peak and AUC values of unbound Dox at the systemic plasma as shown in Fig. 6. Both the peak and AUC values of unbound Dox concentration in systemic plasma are consistently showing a low toxicity for Doxil (Stealth) and extravascular-release TSLe modalities. Although free Dox demonstrates the highest toxicity, the toxicity of fast-release TSLi exceeds free DOX at high release rates.

Fig. 6.

Systemic plasma: AUC (left) and peak (right) values of Dox concentration in systemic plasma for different delivery systems, with varying permeability (top), plasma perfusion (middle) and release rate (bottom) parameters around nominal value x1 (0.05–20x of nominal value).

Fig. 7 demonstrates the effectiveness of each delivery system in terms of the drug bioavailability at the target site (bio-phase) considering the intracellular concentration of unbound or released Dox in the tumor as the measure. At nominal parameter values (x1), free Dox performs better than Doxil, the only FDA-approved liposomal delivery system. Thermo-sensitive liposomes demonstrate comparable (for TSLe) or better (for STSLi and FTSLi) tumor cytotoxicity compared with the free Dox administration.

Fig. 7.

Tumor Cells: AUC (left) and peak (right) values of Dox concentration in tumor cells for different delivery methods with varying permeability (top), plasma perfusion (middle) and release rate (bottom) parameters around the nominal value x1 (0.05–20x of nominal value).

To more objectively compare different delivery modalities, it is necessary to evaluate the therapeutic index such that both the tumor cytotoxicity and adverse toxicity (or effectiveness versus side effect) are considered. As the drug concentration at the target site (tumor cells) is here used to assess the PD instead of percentage of tumor cells that are killed, the therapeutic index cannot be directly obtained. According to the cell uptake model (2) adopted from [22], the peak intracellular drug concentration is directly correlated with effectiveness or tumor cytotoxicity. In addition, we cannot assess the lethal or toxic doses due to the lack of experimental data. However, the AUC of drug concentration at systemic plasma is associated with some types of toxicity. Accordingly, the beneficial response (or tumor cytotoxicity) and adverse effect (or toxicity) are here separately discussed using the in silico modelling results, rather than calculating a therapeutic index.

V. Discussion

We demonstrate the proposed unified model as a novel means to test various drug delivery systems in cancer therapy, some of which have been verified previously and some of which should be validated by further animal and clinical studies. To validate and evaluate the performance of the proposed model, the predicted Dox concentration in tumor tissue is compared with the in vivo experimental results in [40] as well as the simulation results of previous modeling method in [19]. Fig. 8 shows the comparative results for Dox concentration in tumor tissue. The proposed in silico model in most cases fits the in vivo experimental data more closely than the previous mathematical model in [19] does.

Fig. 8.

Maximum Dox concentration at tumor tissue for different drug delivery systems: Free Dox, Doxil, Traditional TSL (TTSL) and Lyso-TSL (LTSL). The results compare the proposed method to in vivo experimentation results in [40] as well as to previous in silico model of [19]. The proposed method demonstrates a better fit for the experimental data compared with the previous mathematical model.

According to the proposed in-silico model, the delivery systems are characterized and compared as following:

• Free Dox: Tumor cytotoxicity is not optimal for very low vascular permeability $\left({\text{P}}_{\text{pe}}^{\text{dox}}\right)$ in tumor to Dox. Moreover, the performance of conventional chemotherapy through free DOX administration remains nearly unchanged at various tumor perfusion rates. Systemic toxicity in terms of both the peak and AUC of Dox concentration in the systemic plasma, representing the side-effect of therapy, is nearly constant for varying permeability and perfusion rates as overall side-effect.

• Doxil and TSLe: Performance is better at higher vascular permeability and perfusion due to more efficient passive liposomal accumulation in the EES, which is the targeting mechanism of these modalities. Systemic toxicity remains nearly constant at low values regardless of tumor characteristics because of low leakage that the related liposomes are characterized with.

• TSLi: Both fast- and slow-release cases showed better beneficiary responses at lower tumor vascular permeability, with constant systemic toxicity. Lower perfusion permits released DOX to diffuse more into tumor EES owing to a longer transit time in the tumor vasculature. Systemic toxicity rises at higher tumor perfusion. However, a compound parameter like therapeutic index is required to have an overall evaluation of the TSLi performance.

The performance of different liposomal delivery systems is sensitive to varying release rates. In all liposomal modalities, the peak and AUC of unbound Dox concentration in systemic plasma as a surrogate for toxicity rise with higher release rate. Nevertheless, the increase of toxicity for Doxil and TSLe is more pronounced because the systemic plasma is exposed to the liposomal drugs over a longer period compared with the TSLi. In summary, all liposomal modalities improve tumor accumulation with higher release rates and outperform free Dox administration.

The overall efficiency of different modalities, considering both delivery efficacy (or tumor cytotoxicity) and adverse systemic toxicity, is superior for all liposomal modalities compared with free Dox. TSLe appears more efficient at low tumor permeability and perfusion rates than the other modes. Also, the TSLe and Doxil appear to be more efficient at lower release rates compared to TSLi, which shows better performance at higher release rates.

Importantly, the approximations used in system equations (Supplementary Material) may not be adequate for the TSLi case because the concentration gradient along the microvasculature is not considered, which may lead to unrealistic approximations. These approximations are acceptable and realistic for the other delivery modalities: free Dox, Doxil and TSLe. Therefore, it may be beneficial to implement additional equations for accurate modelling of spatial drug concentration gradients for TSLi, to replace the approximations in Supplementary Material.

In summary, we demonstrate a novel, unified model that can predict various liposomal drug delivery systems, and may with little or no modification be adapted to other nanoparticle drug delivery systems. We show that the effects of parameters such as tumor vascular permeability to Dox and perfusion rate highly depend on the drug delivery system. Since it is not feasible to study variation of such biological parameters in vivo, computer models are a valuable tool in characterizing drug delivery systems.