Mathematical modeling analysis of intratumoral disposition of anticancer agents and drug delivery systems.

Expert Opin. Drug Metab. Toxicol. Downloaded from informahealthcare.com by University of Connecticut on 04/02/15 For personal use only.

Review

1.

Introduction

2.

Properties of the solid tumors that affect drug transport

3.

Mechanisms of drug/DDS disposition in the tumor tissue

4.

Mathematical models of intratumoral drug/DDS disposition

5.

Summary of publications that applied mathematical models to investigate of intratumoral drug/DDS disposition following systemic or local administration

6.

Discussion of selected publications that used mathematical models to analyze intratumoral disposition of anticancer drugs/DDSs

7.

Conclusion

8.

Expert opinion

Mathematical modeling analysis of intratumoral disposition of anticancer agents and drug delivery systems Hen Popilski & David Stepensky† Ben-Gurion University of the Negev, Department of Clinical Biochemistry and Pharmacology, The Faculty of Health Sciences, Beer Sheva, Israel

Introduction: Solid tumors are characterized by complex morphology. Numerous factors relating to the composition of the cells and tumor stroma, vascularization and drainage of fluids affect the local microenvironment within a specific location inside the tumor. As a result, the intratumoral drug/drug delivery system (DDS) disposition following systemic or local administration is non-homogeneous and its complexity reflects the differences in the local microenvironment. Mathematical models can be used to analyze the intratumoral drug/DDS disposition and pharmacological effects and to assist in choice of optimal anticancer treatment strategies. Areas covered: The mathematical models that have been applied by different research groups to describe the intratumoral disposition of anticancer drugs/ DDSs are summarized in this article. The properties of these models and of their suitability for prediction of the drug/DDS intratumoral disposition and pharmacological effects are reviewed. Expert opinion: Currently available mathematical models appear to neglect some of the major factors that govern the drug/DDS intratumoral disposition, and apparently possess limited prediction capabilities. More sophisticated and detailed mathematical models and their extensive validation are needed for reliable prediction of different treatment scenarios and for optimization of drug treatment in the individual cancer patients. Keywords: anticancer drugs and drug delivery systems, intratumoral drug disposition, mathematical modeling, optimization of drug treatment Expert Opin. Drug Metab. Toxicol. [Early Online]

1.

Introduction

In order for an anticancer drug to be effective, it must successfully reach the tumor cells at the site of action. Over the recent years, significant progress has been made in management of cancer disease, and novel drug delivery systems (DDSs) have been developed in order to target the chemotherapeutic drugs to the tumor cells, increase their efficacy and reduce the extent of their side effects [1]. For detailed description of the individual types of DDSs, we refer the readers to the recent reviews on this topic [2-4]. Unfortunately, the currently available anticancer drugs/DDSs possess limited clinical effectiveness and safety. It appears that only a small fraction of the administered drug is capable of exerting the pharmacological effect at the intended site of action in the tumor, and the overall targeting efficiency of the drugs/DDSs to the tumors (i.e., the ratio of drug concentrations in the tumor tissue vs other tissues) is low [1,5,6]. In addition to limited tumor accumulation of the anticancer drugs/ DDSs, it emerges that their inefficient intratumoral distribution and low 10.1517/17425255.2015.1030391 © 2015 Informa UK, Ltd. ISSN 1742-5255, e-ISSN 1744-7607 All rights reserved: reproduction in whole or in part not permitted

1

H. Popilsky & D. Stepensky

Article highlights. .

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The morphology of solid tumors is complex and many factors related to the composition of the cells and tumor stroma, vascularization and drainage of fluids affect the local microenvironment within a specific location inside the tumor. The intratumoral drug/drug delivery system (DDS) disposition in the tumor is non-homogeneous and its complexity reflects the differences in the local microenvironment. Mathematical models can be applied to describe and analyze the drug/DDS intratumoral disposition and pharmacological effects. In this review, the models proposed by different research groups and their major properties are reviewed. Currently available mathematical models appear to neglect some of the major factors that govern the drug/ DDS intratumoral disposition and apparently possess limited prediction capabilities. More sophisticated and detailed mathematical models and their extensive validation are needed for reliable prediction of different treatment scenarios and for optimization of drug treatment in the individual cancer patients.

This box summarizes key points contained in the article.

permeability to the ‘deep’ parts of the solid tumor is a major factor that limits the treatment efficiency, increases the risk of adaptation of the tumor cells to drug effects and contributes to evolvement of drug resistance (treatment relapse) [7-9]. To overcome these limitations, several strategies have been devised [10-13], including i) the application of external targeting signals (e.g., local magnetic field, heating or radiofrequency ablation); ii) the use of promoter drugs that are given in combination with the anticancer drug/DDS to improve their uptake by the tumor cells (e.g., angiotensin II that modulates tumor blood flow, histamine that modify the barrier function of tumor vessels, losartan that overcome stromal barriers, etc.) [14-16]; and iii) the application of multistage DDS that change their size on deposition in the tumor and can reach its deeper parts [17,18]. However, it is not yet clear as to what extent these and other approaches alter the intratumoral disposition (i.e., distribution and elimination) of drugs/DDSs and whether they increase the effectiveness of the anticancer treatments. Mathematical models can be applied to analyze the complex processes of drug/DDS intratumoral disposition and pharmacological effects, and these models can serve as a powerful tool for increasing the effectiveness of anticancer treatments [19-21]. The objective of this review is to describe the currently available mathematical models that analyze the intratumoral disposition (i.e., distribution and elimination) of anticancer drugs/ DDSs, to reveal the advantages and disadvantages of these models and their potential application for design of anticancer therapies. This review does not address the systemic pharmacokinetics (absorption, distribution, metabolism and 2

elimination) of anticancer drugs/DDSs and the efficiency of drug targeting of the tumor following systemic drug/DDS administration. Detailed description of these topics can be found in several excellent recent reviews [1,5,8,10,22,23].

Properties of the solid tumors that affect drug transport

2.

From the clinical point of view, solid tumors differ by the type of tissue from which the cancer originates (histological type), by location and size of the primary tumor and of the metastases. The composition of the individual tumor (Figure 1) is governed by numerous factors relating to the properties of the tumor and the stroma cells, expression of molecules and receptors on their membrane, composition of the extracellular matrix, vascularization and flow of blood, lymph and extracellular fluid, presence of inflammation and/or necrosis and so on. Due to these factors, there are dramatic changes in local microenvironment in different parts of the tumor, in the ability of the drug/DDSs to reach these parts and to exert anticancer effects [9]. Tumor cells tend to proliferate more rapidly than the normal cells, leading to increased demands of the tumor for oxygen and nutrients. On the other hand, growth of the blood vessels in many tumors lags behind the tumor growth. Hence, the newly formed blood vessels in the tumors have less tight intercellular junctions, discontinuous basement membrane, pores due to endothelial openings and irregular organization. As a result, the vessels walls are leaky and have high permeability, as compared to the vessels in the healthy tissue [24]. The lymphatic vessels in the tumor are also abnormal, leading to inefficient lymphatic drainage from the tumor. The abovementioned alterations in the vascularization and lymphatic drainage of the tumor affect the composition and patterns of flow of interstitial fluid and increase the interstitial fluid pressure in the tumor, as compared to the normal tissue. This leads to generation of gradient in the oxygen pressure and pH throughout the tumor from the blood vessels to the ‘deeper’ layers of tumor tissue (Figure 1). The regions that are remote from the blood vessels, where demand for oxygen exceeds the available supply, suffer from hypoxia and become necrotic (i.e., the necrotic core of the bigger tumors) [14,24]. Changes in the composition and flow of the interstitial fluid in the specific parts of the tumor are accompanied by alterations in the expression of specific receptors, pumps, transporters and other functional molecules on the surface of tumor and stroma cells [10,12,24].

Mechanisms of drug/DDS disposition in the tumor tissue

3.

Depending on the mechanism of action of a specific anticancer drug, we expect it to elicit the desired effect in the solid tumor via specific receptors on the surface of cancer or stroma cells (e.g., trastuzumab), intracellular targets inside the cells

Expert Opin. Drug Metab. Toxicol. (2015) 11(5)

Modeling of intratumoral drug disposition

Blood vessel


Tumor capsule

ity

id

d

, ia

P IF

ox

Avascular core

Cancer cell Stem cell

Lymphocyte Extracellular matrix

yp

H

Monocyte

g

ac

an

e

nc

ta

s si

re

ru

D

Fibroblast

Macrophage

Figure 1. Illustration showing the composition of the solid tumor and its complex microenvironment. The tumor is composed of cancer cells (that include the stem cells) that are embedded within the stroma composed of different cells and extracellular matrix. Within the tumor, increasing distance from the blood vessels is associated with increasing hypoxia, acidity and IFP, reduced drug accessibility and increased resistance to treatment. Reprinted from Danquah et al. 2011 [24] with permission from Elsevier. IFP: Interstitial fluid pressure.

Table 1. Pathways of drug/drug delivery system disposition in the solid tumor. Pathway Adsorption to the capillary wall Extravasation (can be enhanced in some tumors due to gaps between the capillary endothelial cells) Passive diffusion Facilitated transport (transporters, pumps) Convection Binding to the cells and/or extracellular matrix Uptake by the tumor and/or stroma cells Metabolism outside and/or inside the cells Lymphatic elimination Venous elimination

(e.g., doxorubicin and paclitaxel), neutralization of soluble targets (e.g., bevacizumab), effect on the capillaries permeability (e.g., TNF) and so on. Ability of the specific anticancer drug to interact with its intended target is governed by the drug/ DDS intratumoral disposition (the processes of drug/DDS distribution and elimination in the tumor tissue, Table 1). The composition of the tumor (Figure 1) and its unique microenvironment constitute a major challenge that restricts

transport of drugs and DDSs into the tumor and their intratumoral disposition [13,24,25]. Alterations in the vascularization, fluids drainage, cell surface properties and metabolic activity in the tumor tissue affect the intratumoral disposition of drugs/DDSs. For instance, drug/DDSs can passively diffuse in the tumor according to the concentration gradient, be transported through convection (movement with bulk of interstitial fluid), undergo metabolism and elimination with the lymph or venous blood and so on. High permeability of the blood vessels and inefficient fluid drainage can promote extravasation of DDSs and their retention in the tumor tissue (the enhanced permeability and retention [EPR] effect) [26]. Overexpression of efflux pumps (e.g., P-glycoprotein [P-gp], breast cancer resistance protein [BCRP] and others) and xenobiotic metabolizing enzymes can reduce the drug accumulation inside the tumor cells and diminish the treatment efficiency [27]. On the other hand, overexpression of specific receptors on the surface of tumor cells (e.g., HER-2, EGFR, etc.) can serve for active drug targeting of the tumor cells by the DDSs decorated with specific targeting residues [12,28]. Weakly acidic drugs can undergo ion trapping inside the cells (due to the higher pH inside the cells as compared to the extracellular fluid). In addition, lower pH of the extracellular fluid in the tumors


3


A.

B.

Krogh cylinder

C. Multilayer tumor spheroid

Tumor spheroid

Tumor center Blood flow

Tumor

Transition zone

Extravasation

Normal tissue

Tumor periphery

Interstitial transport

D.

Cells on 2-D vascular grid

E.

F. Advanced compartmental

Simple compartmental Extracellular fluid

Cell ta k e

Drug

Release

Release

Cell

DDS Extravasation

Drug

Blood vessel Intracellular vesicle Nucleus Paramagnetic complex Liposome

B

Up

Extravasation


Local binding/metabolism

D

A From the central circulation

DDS

C E

F

Extracellular space

Plasma

Figure 2. Illustrations showing examples of the models used for describing intratumoral drug/DDS disposition. The models highly simplify the composition of the tumor and its microenvironment (Figure 1) and differ in their assumptions regarding the shape, composition and spatial homogeneity of the tumor tissue. A and D are reproduced from Thurber and Weissleder, 2011 [66], and Frieboes et al. 2014 [71], respectively. F is reproduced from Delli Castelli et al. 2010 [82] with permission from Elsevier. 2-D: Two-dimensional; DDS: Drug delivery system.

can be used as a trigger for drug release from the specially designed pH-sensitive DDSs. Due to the complex composition of the tumor (Figure 1) and multiple intratumoral disposition pathways (Table 1), the drug/DDS distribution in the tumor is highly non-homogeneous. Majority of the DDSs that reach the tumor tissue accumulate in proximity to the blood vessels, and only a small fraction of the DDSs penetrates to the ‘deep’ layers of the tumor [9,25]. A gradient of the drug intratumoral concentrations is formed with gradual decrease in drug concentrations with increasing distance from the local blood vessels. For a drug that acts on the intracellular targets (e.g., doxorubicin and paclitaxel), inefficient permeability to the ‘deep’ parts of the tumor will be a drawback since it limits the therapeutic efficiency of anticancer treatment. Coadministration with promoter drugs [14] or use of techniques that increase tumor permeability (ultrasound, local heating) can enhance the efficiency of these effector drugs. On the other hand, these promoter drugs and techniques can reduce efficiency of the effector drugs that act on the capillaries permeability or on the soluble targets (see above). Furthermore, drug release from the DDS (e.g., doxorubicin release from the liposomes) can enhance the anticancer effect if it takes place intracellularly (or in the ‘deep’ parts of the tumor) but can reduce the drug efficiency if it takes place in the bloodstream or outside the cells that are located close to the blood vessels. 4

From the abovementioned considerations, intratumoral disposition of anticancer drug/DDSs evolves as a critical factor for the treatment effectiveness. Thus, substantial accumulation of the drug in the tumor tissue following administration may be insufficient for induction of effective anticancer effects. Detailed analysis and understanding of the processes of drug/DDS intratumoral disposition is required to identify the barriers for drug penetration to the site of desired action in the tumor and to enhance its accumulation there.

Mathematical models of intratumoral drug/DDS disposition

4.

Mathematical (in silico) models can be applied to investigate the intratumoral drug/DDS disposition. To this end, the major factors that govern the intratumoral drug/DDS disposition (Table 1) are encoded in the form of mathematical equations, based on explicit assumptions regarding the dimensions, shape and composition of the tumor tissue. In addition, equations that describe the disease progression (e.g., expression levels of the metabolic enzymes and efflux pumps), effects of promoter drugs and techniques (e.g., antivascular drugs and ultrasound) and pharmacodynamic activities of the applied anticancer drug can be incorporated into the mathematical model.




The resulting set of mathematical equations can be applied to investigate the intratumoral pharmacokinetics and pharmacodynamics of anticancer drug/DDSs. For this purpose, the mathematical equations are encoded into the general purpose (e.g., MATLAB, Mathematica, R, SAS) or specialized (e.g., Monolix, Phoenix WinNonlin, ADAPT) software packages [29,30] and are investigated using these packages. The specific goals of this modeling-based investigation can be description and analysis of experimental data and/or prediction of experimental results [31]. Specifically, mathematical model can be used to consolidate information from different sources (e.g., in vitro and in vivo experimental data on permeability and elimination of a specific anticancer drug). It can be used for analyzing the existing data (e.g., identification of the key factors that govern the intratumoral drug/DDS disposition, such as extravasation and convection efficiency). Moreover, mathematical models can be used in predicting the impact of individual factors on drug/DDS disposition and effectiveness (increased expression of efflux pumps or metabolizing enzymes, application of promoter drugs, etc.). Due to the complex nature of the intratumoral drug/DDS disposition processes (see Section 3), their mathematical analysis is usually based on simplified models with a limited set of mathematical equations that depict these processes. In order to be useful for the purpose of the individual study (data description, analysis, and/or prediction), the structure of the applied mathematical model should reflect the key processes that govern intratumoral drug/DDS disposition, taking into account the study’ experimental setup (e.g., tumor type, stage and location).

Summary of publications that applied mathematical models to investigate of intratumoral drug/DDS disposition following systemic or local administration

5.

Different types of mathematical models can be applied to describe the processes of intratumoral drug/DDS disposition. Several examples of such models are shown in Figure 2. For the detailed description of the specific models, equations, modeling approaches and outcomes, the readers are referred to the individual publications that applied mathematical modeling to analyze intratumoral disposition of drugs/DDSs (summarized in Table 2). Krogh cylinder (Figure 2A) assumes that the tissue is composed of cylinders with the blood vessel in their center that is surrounded by tissue. The local blood concentration is determined by the intercapillary distance (i.e., the diameter of the cylinders), blood velocity, permeability of the vessel wall and additional parameters, such as extent of drug protein binding in the blood/plasma. Other types of models assume spherical shape of the tumor (Figure 2B) and radial (two- or three-dimensional) gradient of drug/DDS concentrations in it due to the diffusion and

degradation of the drug molecules. In some cases, effects of additional processes are incorporated in the model (such as drug internalization by the cells, rate of tumor cells divisions, tumor porosity, etc.), or multilayer nature of spheroids is assumed (Figure 2C). Non-uniformity of tumor tissue can be described by models composed of cells and blood vessels (Figure 2D) that can grow, move and degrade according to a set of predefined rules. Such models can incorporate processes of cell proliferation, migration, adhesion to other cells and extracellular matrix and cell death due to the apoptotic and oncotic processes. Compartmental models can also be applied for analyzing intratumoral drug/DDS disposition (Figure 2E and F). Such models assume existence of kinetically homogeneous compartments in the tumor tissue (such as drug concentrations in the capillaries, extracellular or intracellular fluids, within specific intracellular organelles, etc.) and transfer of drug/ DDS between these compartments (usually by the processes with first-order kinetics). Some of the compartmental models also incorporate the processes of drug release from the DDS and its degradation in the tumor tissue. The schemes shown in Figure 2 cover only some of the approaches that can be used to describe the intratumoral drug disposition following systemic or local drug/DDS administration. These schemes do not necessarily reflect the complexity of the specific model that can focus only on selected individual processes or describe the complex interplay between the tumor cells, matrix and drug/DDS using dozens of parameters and equations. Some models depict only the local drug/DDS pharmacokinetics (the processes of intratumoral drug/DDS disposition). Other models describe the systemic pharmacokinetics (time course of drug/DDS concentrations in the systemic circulation and in peripheral tissues) and/or pharmacological effects of the drug treatment (e.g., cell death and reduction in the tumor dimensions). Moreover, some models describe pharmacokinetic and pharmacodynamic effects of drug/DDS administration in combination with additional treatment modalities (local heating, radiofrequency ablation, ultrasound, etc.). Table 2 contains the summary of the publications that applied mathematical modeling to analyze intratumoral disposition of drugs/DDSs, and in some cases their systemic pharmacokinetics and the magnitude of pharmacological responses are described. This table is organized according to the experimental setup (in vitro settings, systemic or local drug/DDS administration), and, within each group, the publications are organized by the publication date. It can be seen that many publications analyzed the intratumoral drug/ DDS disposition following systemic administration, whereas focal administration was studied somewhat less extensively (Table 2). Tumor spheroids and Krogh cylinder were applied more frequently to analyze the intratumoral drug/DDS disposition in comparison to the other types of mathematical models. For the description of the specific model, its complexity and modeling outcomes, the readers are referred to the


5

6

3-D model of tumor spheroids composed of the interstitial fluid, cell surfaces and intracellular space 2-D tumor model with tumor (central and peripheral regions) surrounded by a layer of normal tissue 3-D tumor spheroids with three components (interstitium, cell surface and intracellular space)


Macromolecule agents

Doxorubicin

-

1-compartmental for the liposomal drug and 2-compartmental for the released free drug

Fluorescent marker

-

-

-

-

-

Administration

Solution or liposomes

Solution

i.v.

-

Cationic liposomes -

Liposomes

-

NP

Millirods

Formulation

In vitro studies of drug release from the millirods. Simulations of millirod implantation into a thermoablated tumor tissue In vitro study

Notes

-

Simulations only. Effects were simulated based on a cell kill kinetic model

Penetration of 40 nm NPs in SiHa multicellular spheroids Incubation of HSV In vitro experimental settings with HSTS26T cells and simulations or their tissue culture medium Effect of hyperthermia on drug tumor disposition and cell killing was analyzed using simulations Penetration of In vitro studies with liposomes in quantitative image analysis human pharynx FaDu cells spheroids Effects of the tumor vascular hydraulic conductivity and heterogeneity on the effectiveness of drug treatment were analyzed

-

Disease/data for validation

2-D: Two-dimensional; 3-D: Three-dimensional; CEA: Carcinoembryonic antigen; EPR: Enhanced permeability and retention; HSV: Herpes simplex virus; i.v.: Intravenous; MPS: Mononuclear phagocyte system; NP: Nanoparticle; pEGFR: Phosphorylated epidermal growth factor receptor; PK: Pharmacokinetics.

Wu et al. (2014) [57]

Doxorubicin

Herpes simplex virus

-

-

-

3-D model of NP diffusion into tumor spheroids (of spherical or non-spherical shape)

Drug(s)

Doxorubicin, carmustine

Mathematical model of the rest of the body

2-D model of drug transport and uptake by thermo-ablated tissue and non-ablated tissue

Mathematical model of the tumor

Tumor with three regions (proliferating, hypoxic, necrotic), preexisting vascular grid, tumor growth, angiogenesis, blood flow and lymphatic drainage Systemic administration Harashima et al. Compartmental model with tumor capillaries, interstitium (1999) [58] and cells as separate compartments

Wientjes et al. (2014) [56]

Zhang et al. (2009) [55]

Mok et al. (2009) [54]

Goodman et al. (2008) [53]

In vitro settings Qian et al. (2002) [52]

Source

Table 2. Summary of publications that applied quantitative models to analyze drug disposition in solid tumors.




Blood and MPS parts

Probability modeling of NP uptake by the MPS and its accumulation in the tumor due to the EPR effect

Paclitaxel

Solution

NP

Solution

Doxorubicin -

Solution

Solution

Recombinant immunotoxins

Paclitaxel, 5-fluorouracil

Solution

Solution

Solution

Formulation

VEGFR2 mAb

Doxorubicin

-

Drug(s)

i.v.

i.v.

i.v.

i.v.

i.v.

i.v.

i.v.

i.v.

Administration

Notes

Size-dependent uptake of liposomes by the MPS in Yoshida sarcoma-bearing rats Dynamic contrastenhanced magnetic resonance imaging data from breast cancer patients

-

-

Data of human tumor xenografts grown on nude mice

Tumor heterogeneity was identified as a critical parameter that affects tumor response to the treatment

-

The model included 2 types of tumor cell population, cell movement and intracellular PK and pharmacodynamics of the immunotoxins Optimal drug diffusion coefficient was analyzed using simulations Simulations only

U87 glioma tumor Simulations only. Combined in the mouse brain effects with radiotherapy were also simulated

Simulations only for tumors composed of one or two types of cells. Effects were simulated based on a cell kill kinetic model H2981 tumor xen- Effects were simulated based ografts in mice on a cell kill kinetic model

-



2-compartmental

2-compartmental or 3-compartmental 3-compartmental

Tumor spheroid growth model with compartmental cell cycle model and energy metabolism Tumor cord model

Venkatasubram- Tumor composed of Krogh tumor vessel cords with cell anian et al. growth, angiogenesis and (2010) [40] metabolism processes

Venkatasubramanian et al. (2008) [62] Eikenberry (2009) [63] Hara et al. (2010) [64]

Chen et al. (2008) [39]

Kohandel et al. (2007) [61]

3-D model of drug diffusion 3-compartmental into tumor spheroids. 3-compartment model for drug dynamics within the tumor (extracellular space, intracellular fluid and the cells nuclei) 3-D model of drug diffusion into tumor spheroids composed of tumor cells and the vascular network Tumor composed of spherical units with central blood vessels

Jackson (2003) [60]

2-compartmental


3-D model of drug diffusion into tumor spheroids


Jackson and Byrne (2000) [59]

Source

Table 2. Summary of publications that applied quantitative models to analyze drug disposition in solid tumors (continued).



7

8


Compartmental model (extracellular space, cell surface-bound, intracellular free and bound drug) Tumor with three regions (proliferating, hypoxic, necrotic), preexisting vascular grid, tumor growth, angiogenesis -

2-compartmental

3-compartmental with heart as a separate compartment 2-compartmental

Nanocells and liposomes

Solution

Formulation

Brentuximab--vedotin antibody--drug conjugate Doxorubicin

Antibody to CEA

Doxorubicin

SC HT-29 and A431 tumors in mice

i.v.

Porous plateloid silicon particles

Solution

i.v. (retroorbital)

i.v.

Image analysis of antibody intratumoral distribution based on tumor sections and immunofluorescence Cell distribution model of tumor growth inhibition and cell death

Effect of magnetic field on macrophage infiltration into a tumor was analyzed using simulations Rate-limiting steps for accumulation of drugs from different classes in the tumor were analyzed Simulations only

Effect of drug release kinetics on the treatment efficiency was analyzed using simulations

Simulation only for the intratumoral drug concentrations

Notes

SC B16 melanoma Intravital microscopic analysis in mice of particle accumulation in the tumors. Proliferating cells exposed to threshold drug concentrations were assumed to die instantaneously

SC CEA-positive human colorectal cancer LS174T in nude mice L540cy and Karpas299 tumors in mice

-

-

Published preclinical and clinical data on drug concentration in peripheral blood mononuclear cells Lewis lung carcinoma or BL6/ F10 melanoma in mice


i.v.

i.v.

i.v.

Administration

Solution, i.v. PEGylated or temperaturesensitive liposomes Solution i.v. (retroorbital)

Macrophages Solution (loaded with magnetic NP), cyclophosphamide Doxorubicin, cetux- Solution imab and others

Doxorubicin, combretastatin and their combination

Gemcitabine

Drug(s)


van de Ven et al. (2012) [70]

Shah et al. (2012) [69]

Gasselhuberetal. Compartmental model with tumor plasma, interstitium and (2012) [67] cells as separate compartments Rhoden and Krogh cylinder model of tumor Wittrup vasculature (2012) [68]

Thurber and Weissleder (2011) [66]

Owen et al. (2011) [42]

3-D model of tumor spheroids 1-compartmental composed of tumor cells and the vascular network, delivery of nanocells and liposomes and drug release from them 2-D lattice of cells with an 1-compartmental embedded vascular network. 4 layers: vascular, diffusible species, cellular, subcellular. Krogh cylinder model of tumor vasculature

Kohandel et al. (2011) [65]

2-compartmental


Compartmental, detailed intracellular PK model


Battaglia and Parker (2011) [41]

Source




3-layer spherical tumor, finite element method

2-D model of tumor growth, angiogenesis, flow and NP accumulation Compartmental model with vascular and extravascular compartments 2-D tumor model with tumor surrounded by a layer of normal tissue



-

-

-

2-D model of drug diffusion and elimination in the brain

Compartmental model with 3 compartments

2-D and 3-D models of drug diffusion, convection and elimination in the brain

Fung et al. (1998) [76]

Port et al. (1999) [77]

Fleming and Saltzman (2002) [78]

Carmustine

Carmustine, 4-hydroperoxycyclophosphamide, paclitaxel Floxuridine

Carmustine

Carmustine, dextran, iodoantipyrene

Wafer

Solution

Polymer implant

Polymer implant

Polymer implant

Solution

Solution or thermosensitive liposomes

Solution

NPs

Formulation

Drug concentrations in the brains of rabbits

Intraperitoneal ovarian tumor in mice

Mice with intracerebral glioblastoma -

-


Drug concentrations in the brains of Fisher rats with 9L gliomas Intra-brain Drug concentra(implant) tions in the brains of cynomolgus monkeys Intratumoral SC Morris hepatoma 3924A in rats Intratumoral Carmustine con(following resec- centrations in the tion of majority animal brain of the tumor) following wafer implantation

Intra-brain (implant)

Intra-brain (implant)

Intraperitoneal

i.v.

Oral

i.v.

Administration

Penetration distance of the drug from the implant into the surrounding tissues was measured Penetration distance of the drug from the implant into the surrounding tissues was measured Penetration distance of the drug from the implant into the surrounding tissues was measured [19F]-NMR spectroscopy analysis of intratumoral drug concentrations A review manuscript which summarizes PK of carmustine implant

Multiscale model that included multiple drug transport pathways (e.g., lymphatic drainage and absorption from the peritoneum)

Indirect response model of drug effect on the pEGFR levels in the tumor -

Simulations only

Notes


-

2-D model of drug diffusion and elimination in the brain

-

Paclitaxel

Doxorubicin

3-compartmental (for solution), 2-compartmental (for liposomes) 3-compartmental

-

Drug(s)

Gefitinib


2-compartmental

-

Fung et al. (1996) [75]

Local (focal) administration 2-D model of drug diffusion Strasser et al. and elimination in the brain (1995) [74]

Au et al. (2014) [43]

Zhan and Xu (2013) [73]

Sharma et al. (2013) [72]

Frieboes et al. (2013) [71]

Source




9

10


Wafers (14 mm diameter 1 mm thickness)

PEGylated or pH- Intratumoral sensitive liposomes

Carmustine

Paramagnetic marker

VX2 carcinoma implants into liver of rabbits -

VX2 carcinoma implants into liver of rabbits

-

-

-


Effect of radiofrequency ablation on the drug diffusion was determined experimentally and confirmed using simulations Drug distribution from single versus multiple implants was analyzed using simulations Penetration depth of the individual drugs and effect of the anti-angiogenic treatment were analyzed using simulations Effect of convective flow on exposure to the drug was analyzed using simulations

Effects of the surgical excision on the flow of the interstitial fluid and drug distribution were analyzed using simulations Effect of the drug release kinetics (linear vs double burst) on the penetration depth and therapeutic index were analyzed using simulations Effect of the drug release rate from the microspheres on the treatment efficiency was analyzed using simulations

Notes

SC B16 melanoma MRI data was used for in mice analysis

Intratumoral Carmustine pene(following resec- tration in monkey tion of majority brain of the tumor)

Wafers (14 mm diameter 1 mm thickness)

Intratumoral (following resection of 60% of the tumor)

Intratumoral

Intratumoral

Intratumoral

Intratumoral

Intratumoral

Administration

Carmustine, paclitaxel, 5-fluorouracil, methotrexate

Implants

Implants

Microspheres

Wafers

Wafers

Formulation


Delli Castelli et al. (2010) [82]

Airfin et al. (2009) [35]

3-D model of brain tumor and other parts of brain (based on the MRI data) that includes diffusion, reaction and convection processes 3-D model of brain tumor and other parts of brain (based on the MRI data) that includes diffusion, reaction and convection processes Compartmental model of extracellular and intracellular spaces

Airfin et al. (2009) [34]

Doxorubicin

Doxorubicin

-

-

Paclitaxel

-

3-D finite element model

Tumor spheroids composed of Krogh spheres with drugreleasing microspheres in their center. Drug concentrations in the interstitial fluid and intracellular space were analyzed 1-D cylindrically symmetric model and 3-D finite element model

Weinberg et al. (2008) [33]

Weinberg et al. (2007) [32]

Tzafriri et al. (2005) [81]

Etanidazole

3-D model of the cavity, tumor and normal tissue based on the MRI-based imaging of brain tumor geometry

Tan et al. (2003) [80]

Drug(s)

Etanidazole


2-D model of the cavity, tumor and normal tissue based on the MRI-based imaging of brain tumor geometry


Tan et al. (2003) [79]

Source





2-D model of the cavity and brain tissue based on the MRIbased imaging of brain geometry Semi-physiologically based 1-compartmental biopharmaceutical model of prostatic tissue (6 prostate tissue and 6 prostate vascular compartments)


Administration

PLGA nanofiber Intratumoral disks or submicron fiber disks, PLGA microspheres entrapped in hydrogel matrices Wafers Intratumoral

Formulation

2-hydroxyflutamide Injectable formula- Intratumoral tion based on sodium carboxymethyl cellulose and calcium sulfate

Paclitaxel, carmustine

Paclitaxel

Drug(s)

Patients with prostate cancer

-

BALB/c nude mice with intracranial human glioblastoma (U87 MGluc2)


Clinical study. Drug disposition in the prostate tissue following intratumoral injection at different doses, local exposure to the drug following oral administration, and in vitro--in vivo correlations of the developed formulation were analyzed using simulations

Drug distribution in human versus rat brain was analyzed using simulations

Intracranial tumor growth and therapeutic efficacy were analyzed using the noninvasive bioluminescence imaging

Notes


Sjogren et al. (2014) [44]

Torres et al. (2011) [83]

Ranganath et al. 1-D equation that includes (2010) [36] diffusion and reaction, but not convection processes

Source





11


B.

Non-ablated tumor 3.5

Doxorubicin conc., mol/m3

Doxorubicin conc., mol/m3


A.

3 2.5 2 1.5 1 0.5 0

0

1 2 3 Distance from implant center, mm

4

Ablated tumor Tumor boundary

Implant boundary

3.5 3 2.5 2 1.5 1 0.5 0

0

1

2

3

4

5

Distance from implant center, mm

Figure 3. Analysis of intratumoral disposition of doxorubicin from biodegradable implants in non-ablated and radiofrequency-ablated liver tumors is shown. VX2 carcinoma cells have been implanted into the liver of rabbits and were allowed to grow for 12 days without treatment A. or for 18 days with subsequent radiofrequency ablation B. and doxorubicin-containing implant was inserted into the center of the tumor. Four days later the tumor was removed, sectioned and the radial drug distribution from the implant (the data points) was determined using fluorescent imager and quantitate image analysis. Modeling analysis of doxorubicin analysis and elimination (the solid line) has been performed using a onedimensional cylindrically symmetric model. Reprinted from Weinberg et al. 2007 [32] with permission from Elsevier. conc.: Concentration.

specific publications that are summarized in Table 2. In this review, we provide a short description of results from two selected publications to illustrate the applicability of mathematical models for analysis of intratumoral disposition of anticancer drug/DDS and of the resulting pharmacological effects.

Discussion of selected publications that used mathematical models to analyze intratumoral disposition of anticancer drugs/DDSs 6.

Weinberg et al. analyzed the transport of doxorubicin released from biodegradable implants into liver tumors [32]. The authors performed experiments in animals inoculated with tumor cells (VX2 carcinoma implanted into liver of rabbits) that undergo intratumoral insertion of cylindrical doxorubicin-poly(lactic-co-glycolic acid) implants (in some animals -- after radiofrequency ablation at the tumor site). Subsequently, the tumors were removed, fixed and sectioned, and the radial distribution of doxorubicin from the implants was determined using fluorescent imaging of the sections. The authors applied one-dimensional, cylindrically symmetric transport model that was generated using a finite element method to analyze the intratumoral disposition of doxorubicin [32]. The major parameters of this model were drug diffusion and elimination coefficients that were assumed to be constant in the non-ablated tumor and to change with the distance from the tumor center in the ablated tumor. The applied model was able to describe the radial distribution of 12

doxorubicin into the tumor tissue in non-ablated and radiofrequency-ablated liver tumors (Figure 3) and increased permeability of the drug into the tumor following the radiofrequency ablation that was attributed to increased diffusion and reduced elimination of the drug from the ablated parts of the tumor. Subsequently, the authors applied the obtained drug diffusion and elimination coefficients and a threedimensional model to simulate drug distribution from an implant in a larger tumor, with or without radiofrequency ablation [32]. It is not clear, though, whether these simulations reflect the intratumoral drug disposition in in vivo settings. In addition, due to the differences in the drug perfusion and convection in solid tumors of different origin, model-based predictions can be inappropriate for the common types of solid tumors (e.g., for the breast, prostate, lung and brain cancers). The authors published an additional manuscript that extended the developed mathematical model [32] and validated its predictions in the same experimental model (rabbits with VX2 carcinomas in the liver) [33]. There were substantial differences between the measured doxorubicin concentrations and the model-based predictions in ablated rabbit livers, either normal or with VX2 carcinoma tumors. Nevertheless, as the authors concluded, combination of model simulations with a small set of animal experiments appears to be a more efficient strategy for identification of an optimal implant strategy for treating large tumors, as compared to animal experiments alone without modeling and simulations [33]. At the time of preparation of this manuscript, the publication by Weinberg et al. that described the mathematical model of intratumoral drug disposition [32] has been cited



A. The 3-D geometry of the tumor, cavity and implants

C.

3e+01 7e+00 2e+00 4e-01 1e-01

Predicted mean drug concentrations in the cavity and in the remnant tumor Mean conc./therapeutic threshold


B. Simulations of paclitaxel concentrations in the cavity and in the remnant tumor

1.E+02

Carmustine Paclitaxel 5-FU MTX

1.E+01

1.E+00

1.E-01 Cavity

Tumor

Figure 4. Analysis of distribution of anticancer drugs that are released from implants into brain tumor and surrounding tissues is shown. 3-D tissue geometry was extracted from MRI of a brain tumor, was assumed to undergo resection of 60% of tumor volume and implantation of eight drug-contacting wafers in the cavity that was formed due to this surgical procedure. Drug concentrations in the cavity and remnant tumor were predicted using a 3-D model that accounted for the tissue geometry and diffusion/reaction/convection processes. A. On the left image, the geometry of the brain (gray) is shown, with ventricle (blue) and a tumor (red) located in temporal lobe. On the right image, the treatment domain is shown with eight wafers containing drug (light green with blue line), cavity (dark red) and remnant tumor (gray). B. Simulations of paclitaxel concentration gradient on the surface of the cavity and in its surrounding remnant tumor are shown. Lines A, B and C denote three domains with different characteristics of convective flow. C. Mean drug concentrations in the cavity and in the remnant tumor for several investigated anticancer drugs are shown. Reprinted from Arifin et al. [34] with permission from Elsevier. 3-D: Three-dimensional; 5-FU: 5-Fluorouracil; conc.: Concentration; MTX: Methotrexate.

20 times, in reviews and in experimental research papers (Web of Science search on 18 December 2014), and there were no additional attempts to validate the model-based predictions. Thus, it appears that this mathematical model [32] has undergone only partial validation in preclinical [33] but not in clinical settings. Arifin et al. analyzed the transport of several anticancer drugs released from polymeric wafers implanted into brain tumors [34]. The authors used magnetic resonance images to reconstruct the three-dimensional geometry of the brain with tumor (Figure 4A) and applied diffusion/reaction/ convection model, in which Darcy’s law was used to account for the convective contribution of the interstitial fluid, to study the intratumoral disposition of anticancer drugs. According to the model-based simulations, drug physicochemical properties had a major impact on its disposition in the brain [34]. As a result, paclitaxel concentrations near the implants and in the remnant tumor (Figure 4B) were higher than those of the other drugs (carmustine, 5-fluorouracil and methotrexate), leading to its ~ 100-fold higher effective concentrations in the remnant tumor. Model-based analysis

revealed that the convective flow significantly alters the paclitaxel penetration and that the drug penetrates insufficiently to the regions that are the nearest to the ventricles of the brain [34]. Thus, tumor size and location in the brain appear to be the major factors that affect the exposure of the tumor cells to the drug and the efficiency of drug treatment. The authors published an additional manuscript that extended the application of this model (simulations only) for carmustine disposition in the brain tumors [35]. In a subsequent publication, the authors analyzed the disposition and antitumor effects of paclitaxel-loaded poly(lactic-co-glycolic acid) discs in the human glioblastoma xenograft model in nude mice [36] but did not compare the obtained experimental data and the mathematical model-based predictions. At the time of preparation of this manuscript, publication by Arifin et al. [34] had been cited 11 times, mostly by reviews (Web of Science search on 18 December 2014), and there were no additional attempts to validate the model-based predictions. Thus, it appears that the modeling-based predictions from the study by Arifin et al. [34] have not been verified yet in preclinical or clinical studies.


13



7.

Conclusion

Solid tumors have complex composition and are characterized by highly heterogeneous structure. At a certain location within the solid tumor, the local microenvironment is governed by the types of cells that populate it and their functional status (metabolic activity, secretion of cytokines, etc.), composition of the extracellular matrix, distance from the blood vessels and the existence of gaps in their walls, direction of the convective flow and numerous other factors. Within a specific tumor, this local microenvironment can change with time due to disease progression and due effects of treatments that the patient is receiving. The properties of local microenvironment in the tumor tissue affect the drug/DDS disposition and the efficiency of the anticancer therapy. To describe the intratumoral drug/ DDS pharmacokinetics and the resulting pharmacological effects, different types of mathematical models have been developed by several research groups. The specific models that were summarized in this review focus on selected factors that affect the tumor structure, local microenvironment and intratumoral drug/DDS disposition. All the developed mathematical models oversimplify the tumor properties and the processes of drug disposition within the tumors. Nevertheless, these models can serve as a useful tool for developing anticancer treatments based on the specific drugs and DDSs, alone or in combination with different treatment modalities (local heating, ultrasound, radiofrequency ablation, promoter drugs, etc.), and for the choice of optimal administration schedules of these treatments. 8.

Expert opinion

In the recent years, significant advances took place in the computer hardware and software, and in the theoretical and applied aspects of the mathematical modeling of the biological systems. Due to these factors, mathematical models are applied more frequently to analyze the pharmacokinetics of anticancer drugs/DDSs and the resulting pharmacological effects. These models can be very useful in developing anticancer treatments and the choice of their optimal administration schedules in case that they account for the major factors that govern the drug/DDS pharmacokinetics and pharmacodynamics and are properly validated. However, applicability and robustness of these mathematical models for development of anticancer drugs/DDSs and optimization of anticancer treatments is currently limited due to the following several major factors. Integration of local and systemic drug/DDS pharmacokinetics and pharmacodynamics

8.1

Successful clinical application of mathematical models in anticancer treatment requires models that take into account the drug/DDS pharmacokinetics and pharmacodynamics 14

both within the tumor and in other tissues/organs of the body. Currently, there is a controversy regarding the ability of the drug/DDSs to reach the intended site of action in the tumor following systemic administration (e.g., due to the EPR effect [6]), and the targeting efficiency (i.e., the ratio of drug concentrations in the tumor tissue versus other tissues) of the systemically versus locally administered anticancer drugs/DDSs [37,38]. In order to resolve this controversy, carefully planned experiments and analysis of the experimental data using multiscale models that account for the local versus global drug/DDS disposition and pharmacological effects are required. Indeed, such multiscale models were applied in several recent studies [39-44] and provided potential explanations for the factors that limit the drug targeting efficiency in the specific experimental settings. It should be noted that due to the nonlinear shape of the pharmacokinetic--pharmacodynamic curves [45], changes in the local exposure of tumor cells to the drug do not necessarily lead to alterations in the efficiency of local antitumor effects. Therefore, multiscale mathematical models that incorporate pharmacokinetic--pharmacodynamic parameters can provide more reliable estimates of antitumor drug/DDS efficiency, as compared to the estimation based solely on the tumor exposure to the drug/DDSs (e.g., on the pharmacokinetic data alone). Oversimplification of the properties of the local microenvironment and of the factors that affect the local drug disposition

8.2

All the mathematical models that were applied so far highly oversimplify the factors that affect the local and the systemic pharmacokinetics and pharmacodynamics of drug/DDSs. For instance, it is clear that the Krogh cylinder, spheroid and compartmental models (Figure 2) provide a very simplified description of the local microenvironment within the tumors. Many of these models appear to be merely theoretical, and apparently are not supported by the experimental data from preclinical or clinical studies. Moreover, these models may not capture some of the important factors that govern the intratumoral disposition of specific drugs, such as tumor cell heterogeneity [46], lipophilicity of doxorubicin that is important for the intratumoral disposition of the drug released from Doxyl liposomes [47] or ‘jamming’ of the binding sites that can limit the efficiency of DDSs decorated with specific targeting residues [8,23]. Binding and endocytosis of the drug/DDSs by the individual types of cells that compose the tumor tissue (tumor stem cells versus differentiated cells versus other cells), and the intracellular disposition in these cells (e.g., efficiency of doxorubicin delivery to the nucleus) can be also important for appropriate description of the drug/DDS disposition and pharmacological effects by the mathematical models. However, virtually none of the mathematical models that were applied so far account for these processes.




8.3

Dynamic changes of the tumor properties

The processes that govern the intratumoral drug/DDS disposition and the sensitivity of the cells to the treatment can undergo substantial alterations during the time course of the disease and due to the applied treatments [48,49]. For instance, genomic instability of tumor cells can lead to dramatic changes in the activity of the xenobiotic metabolizing enzymes and of the efflux pumps [50]. In addition, the first rounds of anticancer therapy can kill the drug-sensitive cells located in vicinity of the blood vessels in the tumor and affect the tumor size, permeability of the drug/DDSs to the ‘deep’ parts of the tumor, sensitivity of the remaining cells to the drug and so on. However, majority of the mathematical models neglect the effect of the treatment and of the tumor cells’ adaptation to the treatment [51] on intratumoral drug/ DDS disposition. Hence, these models may provide inaccurate predictions of the time course of the disease and of the treatment efficiency. Due to the same reason, the currently available mathematical models can estimate incorrectly the effects of treatment modalities (local heating, ultrasound, radiofrequency ablation, promoter drugs, etc., that affect the local microenvironment in the different parts of the tumor tissue) on the intratumoral drug/DDS disposition and pharmacological effects. Limited prediction capabilities of the developed mathematical models

8.4

Prediction of intratumoral drug/DDS disposition and of the resulting pharmacological effects is one of the major goals of mathematical modeling. Ideally, such models would predict the drug/DDS disposition and pharmacological effects for different treatment scenarios, such as change of the drug, DDS properties, administration schedule, type of tumor (tumor cell type, stage and location) and so on. Indeed, mathematical models can be used for estimating the contribution of the individual pathways to the intratumoral disposition of a specific drug/DDS. For instance, sensitivity analysis can be used to estimate the changes in intratumoral drug levels due to increased expression of P-gp or BCRP efflux pumps or CYP450 enzymes, increased convection due to the effect of promoter drugs and so on. However, mathematical modeling of stochastic events that shape the intratumoral drug/DDS disposition (e.g., mutations, expression levels of the proteins that mediate drug disposition, responsiveness of the individual cells to the drug) is more problematic. Due to the above-described limitations of the mathematical models (e.g., oversimplification of the tumor properties and lack of account for dynamic changes in these properties; see

Sections 8.2 and 8.3), it is expected that the currently available mathematical models have limited extrapolation capabilities and provide unreliable predictions. Indeed, majority of the mathematical models that are summarized in this review served to describe the experimental data on drug/ DDS disposition and pharmacological effects obtained in a selected study or set of studies. Only in a few cases, the developed mathematical model was partially validated and modeling-based predictions were compared to external experimental data. This lack of clarity regarding the robustness of the individual mathematical models of intratumoral drug/ DDS disposition and prediction abilities for different treatment scenarios is a major bottleneck for application of these models in preclinical and clinical studies. During the recent years, individualization of drug treatment has been the major trend in the field of management of neoplastic diseases. To increase the efficiency of anticancer treatment in the individual patient, we need to choose the most appropriate drugs/DDSs, to select their optimal dosing schedule, and consider use of specific treatment modalities to enhance the magnitude of the desired effects and reduce the magnitude of the adverse effects. Mathematical models of drug/DDS disposition and pharmacological effects can greatly assist in the design of optimal anticancer treatments tailored for individual cancer patients. However, the currently available mathematical models appear to neglect some of the major factors that govern the intratumoral drug/DDS disposition. More sophisticated and detailed mathematical models and their extensive validation are needed for reliable modelbased prediction of different treatment scenarios, and for efficient application of these models for design of optimal anticancer treatment therapy with specific drug/DDS in a specific patient.

Declaration of interest This research project of analysis and mathematical modeling of intratumoral drug/drug delivery system disposition was funded by the US--Israel Binational Science Foundation grant No 2013/131. D Stepensky serves as a consultant of LipoCure Ltd, Jerusalem, Israel, that develops proprietary liposome-based nanodrugs. Except of this, the authors have no relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript. This includes employment, consultancies, honoraria, stock ownership or options, expert testimony, grants or patents, received or pending, or royalties.


15


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Affiliation

Hen Popilski & David Stepensky† † Author for correspondence Ben-Gurion University of the Negev, Department of Clinical Biochemistry and Pharmacology, The Faculty of Health Sciences, P.O.Box 653, Beer Sheva 84105, Israel Tel: +972 8 6477381; E-mail: [email protected]

Mathematical models in drug delivery: how modeling has shaped the way we design new drug delivery systems.

Classification of stimuli-responsive polymers as anticancer drug delivery systems.

Molecular Communication Modeling of Antibody-Mediated Drug Delivery Systems.

Drug delivery systems 5A. Oral drug delivery.

Drug delivery systems. 6. Transdermal drug delivery.

Targeted delivery of anticancer agents via a dual function nanocarrier with an interfacial drug-interactive motif.

Molecular modeling of potential anticancer agents from African medicinal plants.

Mathematical and Computational Modeling in Complex Biological Systems.

Implantable drug-delivery systems.

Multimodality imaging and mathematical modelling of drug delivery to glioblastomas.

Nanomicellar carriers for targeted delivery of anticancer agents.

Modeling and analysis of drug-eluting stents with biodegradable PLGA coating: consequences on intravascular drug delivery.

Fluorescence optical imaging in anticancer drug delivery.

Mathematical modeling for novel cancer drug discovery and development.

Lipid-Based Drug Delivery Systems.

ATP-Responsive Drug Delivery Systems.

MEMS: Enabled Drug Delivery Systems.

Kidney-targeted drug delivery systems.

pH-responsive drug-delivery systems.

Implantable neurosurgical drug delivery systems.

Radiation sterilization of new drug delivery systems.

Raman imaging of drug delivery systems.

Improved delivery of the natural anticancer drug tetrandrine.

Development of polycationic amphiphilic cyclodextrin nanoparticles for anticancer drug delivery.