Effect of tumor shape,size, and tissue transport properties on drug delivery to solid tumors

Background

The computational methods provide condition for investigation related to the process of drug delivery, such as convection and diffusion of drug in extracellular matrices, drug extravasation from microvessels or to lymphatic vessels. The information of this process clarifies the mechanisms of drug delivery from the injection site to absorption by a solid tumor. In this study, an advanced numerical method is used to solve fluid flow and solute transport equations simultaneously to investigate the effect of tumor shape and size on drug delivery to solid tumor.

Methods

The advanced mathematical model used in our previous work is further developed by adding solute transport equation to the governing equations. After applying appropriate boundary and initial conditions on tumor and surrounding tissue geometry, the element-based finite volume method is used for solving governing equations of drug delivery in solid tumor. Also, the effects of size and shape of tumor and some of tissue transport parameters such as effective pressure and hydraulic conductivity on interstitial fluid flow and drug delivery are investigated.

Results

Sensitivity analysis shows that drug delivery in prolate shape is significantly better than other tumor shapes. Considering size effect, increasing tumor size decreases drug concentration in interstitial fluid. This study shows that dependency of drug concentration in interstitial fluid to osmotic and intravascular pressure is negligible.

Conclusions

This study shows that among diffusion and convection mechanisms of drug transport, diffusion is dominant in most different tumor shapes and sizes. In tumors in which the convection has considerable effect, the drug concentration is larger than that of other tumors at the same time post injection.

     

Numerical modeling of fluid flow in solid tumors

A mathematical model of interstitial fluid flow is developed, based on the application of the governing equations for fluid flow, i.e., the conservation laws for mass and momentum, to physiological systems containing solid tumors. The discretized form of the governing equations, with appropriate boundary conditions, is developed for a predefined tumor geometry. The interstitial fluid pressure and velocity are calculated using a numerical method, element based finite volume. Simulations of interstitial fluid transport in a homogeneous solid tumor demonstrate that, in a uniformly perfused tumor, i.e., one with no necrotic region, because of the interstitial pressure distribution, the distribution of drug particles is non-uniform. Pressure distribution for different values of necrotic radii is examined and two new parameters, the critical tumor radius and critical necrotic radius, are defined. Simulation results show that: 1) tumor radii have a critical size. Below this size, the maximum interstitial fluid pressure is less than what is generally considered to be effective pressure (a parameter determined by vascular pressure, plasma osmotic pressure, and interstitial osmotic pressure). Above this size, the maximum interstitial fluid pressure is equal to effective pressure. As a consequence, drugs transport to the center of smaller tumors is much easier than transport to the center of a tumor whose radius is greater than the critical tumor radius; 2) there is a critical necrotic radius, below which the interstitial fluid pressure at the tumor center is at its maximum value. If the tumor radius is greater than the critical tumor radius, this maximum pressure is equal to effective pressure. Above this critical necrotic radius, the interstitial fluid pressure at the tumor center is below effective pressure. In specific ranges of these critical sizes, drug amount and therefore therapeutic effects are higher because the opposing force, interstitial fluid pressure, is low in these ranges.     

Numerical Modeling of Interstitial Fluid Flow Coupled with Blood Flow through a Remodeled Solid Tumor Microvascular Network

Modeling of interstitial fluid flow involves processes such as fluid diffusion, convective transport in extracellular matrix, and extravasation from blood vessels. To date, majority of microvascular flow modeling has been done at different levels and scales mostly on simple tumor shapes with their capillaries. However, with our proposed numerical model, more complex and realistic tumor shapes and capillary networks can be studied. Both blood flow through a capillary network, which is induced by a solid tumor, and fluid flow in tumor’s surrounding tissue are formulated. First, governing equations of angiogenesis are implemented to specify the different domains for the network and interstitium. Then, governing equations for flow modeling are introduced for different domains. The conservation laws for mass and momentum (including continuity equation, Darcy’s law for tissue, and simplified Navier–Stokes equation for blood flow through capillaries) are used for simulating interstitial and intravascular flows and Starling’s law is used for closing this system of equations and coupling the intravascular and extravascular flows. This is the first study of flow modeling in solid tumors to naturalistically couple intravascular and extravascular flow through a network. This network is generated by sprouting angiogenesis and consisting of one parent vessel connected to the network while taking into account the non-continuous behavior of blood, adaptability of capillary diameter to hemodynamics and metabolic stimuli, non-Newtonian blood flow, and phase separation of blood flow in capillary bifurcation. The incorporation of the outlined components beyond the previous models provides a more realistic prediction of interstitial fluid flow pattern in solid tumors and surrounding tissues. Results predict higher interstitial pressure, almost two times, for realistic model compared to the simplified model.     

A Mathematical Model of the Enhanced Permeability and Retention Effect for Liposome Transport in Solid Tumors

The discovery of the enhanced permeability and retention (EPR) effect has resulted in the development of nanomedicines, including liposome-based formulations of drugs, as cancer therapies. The use of liposomes has resulted in substantial increases in accumulation of drugs in solid tumors; yet, significant improvements in therapeutic efficacy have yet to be achieved. Imaging of the tumor accumulation of liposomes has revealed that this poor or variable performance is in part due to heterogeneous inter-subject and intra-tumoral liposome accumulation, which occurs as a result of an abnormal transport microenvironment. A mathematical model that relates liposome accumulation to the underlying transport properties in solid tumors could provide insight into inter and intra-tumoral variations in the EPR effect. In this paper, we present a theoretical framework to describe liposome transport in solid tumors. The mathematical model is based on biophysical transport equations that describe pressure driven fluid flow across blood vessels and through the tumor interstitium. The model was validated by direct comparison with computed tomography measurements of tumor accumulation of liposomes in three preclinical tumor models. The mathematical model was fit to liposome accumulation curves producing predictions of transport parameters that reflect the tumor microenvironment. Notably, all fits had a high coefficient of determination and predictions of interstitial fluid pressure agreed with previously published independent measurements made in the same tumor type. Furthermore, it was demonstrated that the model attributed inter-subject heterogeneity in liposome accumulation to variations in peak interstitial fluid pressure. These findings highlight the relationship between transvascular and interstitial flow dynamics and variations in the EPR effect. In conclusion, we have presented a theoretical framework that predicts inter-subject and intra-tumoral variations in the EPR effect based on fundamental properties of the tumor microenvironment and forms the basis for transport modeling of liposome drug delivery.     

Effect of the glycocalyx layer on transmission of interstitial flow shear stress to embedded cells

In this paper, a simple theoretical model is developed to describe the transmission of force from interstitial fluid flow to the surface of a cell covered by a proteoglycan / glycoprotein layer (glycocalyx) and embedded in an extracellular matrix. Brinkman equations are used to describe flow through the extracellular matrix and glycocalyx layers and the solid mechanical stress developed in the glycocalyx by the fluid flow loading is determined. Using reasonable values for the Darcy permeability of extracellular matrix and glycocalyx layers and interstitial flow velocity, we are able to estimate the fluid and solid shear stresses imposed on the surface of embedded vascular, cartilage and tumor cells in vivo and in vitro. The principal finding is that the surface solid stress is typically one to two orders of magnitude larger than the surface fluid stress. This indicates that interstitial flow shear stress can be sensed by the cell surface glycocalyx, supporting numerous recent observations that interstitial flow can induce mechanotransduction in embedded cells. This study may contribute to understanding of interstitial flow-related mechanobiology in embryogenesis, tumorigenesis, tissue physiology and diseases and has implications in tissue engineering.     

Drug delivery to brain tumors

We develop a macroscopic model for delivering drug to brain tumors. The model accounts for bulk convective and diffusive transport across the blood-brain barrier and through the interstitial space. Through mathematical analysis and simulations, we assess the effects of changing parameters (within physiological bounds) on drug delivery. We find that there is an optimal treatment for convective drug delivery to the center of the tumor. We interpret this phenomenon in terms of traffic flow. The implications of our analyses on existing chemotherapeutic protocols are discussed.     

Evolution of osmotic pressure in solid tumors

The mechanical microenvironment of solid tumors includes both fluid and solid stresses. These stresses play a crucial role in cancer progression and treatment and have been analyzed rigorously both mathematically and experimentally. The magnitude and spatial distribution of osmotic pressures in tumors, however, cannot be measured experimentally and to our knowledge there is no mathematical model to calculate osmotic pressures in the tumor interstitial space. In this study, we developed a triphasic biomechanical model of tumor growth taking into account not only the solid and fluid phase of a tumor, but also the transport of cations and anions, as well as the fixed charges at the surface of the glycosaminoglycan chains. Our model predicts that the osmotic pressure is negligible compared to the interstitial fluid pressure for values of glycosaminoglycans (GAGs) taken from the literature for sarcomas, melanomas and adenocarcinomas. Furthermore, our results suggest that an increase in the hydraulic conductivity of the tumor, increases considerably the intratumoral concentration of free ions and thus, the osmotic pressure but it does not reach the levels of the interstitial fluid pressure.     

Interstitial Fluid Flow and Drug Delivery in Vascularized Tumors: A Computational Model

Interstitial fluid is a solution that bathes and surrounds the human cells and provides them with nutrients and a way of waste removal. It is generally believed that elevated tumor interstitial fluid pressure (IFP) is partly responsible for the poor penetration and distribution of therapeutic agents in solid tumors, but the complex interplay of extravasation, permeabilities, vascular heterogeneities and diffusive and convective drug transport remains poorly understood. Here we consider–with the help of a theoretical model–the tumor IFP, interstitial fluid flow (IFF) and its impact upon drug delivery within tumor depending on biophysical determinants such as vessel network morphology, permeabilities and diffusive vs. convective transport. We developed a vascular tumor growth model, including vessel co-option, regression, and angiogenesis, that we extend here by the interstitium (represented by a porous medium obeying Darcy''s law) and sources (vessels) and sinks (lymphatics) for IFF. With it we compute the spatial variation of the IFP and IFF and determine its correlation with the vascular network morphology and physiological parameters like vessel wall permeability, tissue conductivity, distribution of lymphatics etc. We find that an increased vascular wall conductivity together with a reduction of lymph function leads to increased tumor IFP, but also that the latter does not necessarily imply a decreased extravasation rate: Generally the IF flow rate is positively correlated with the various conductivities in the system. The IFF field is then used to determine the drug distribution after an injection via a convection diffusion reaction equation for intra- and extracellular concentrations with parameters guided by experimental data for the drug Doxorubicin. We observe that the interplay of convective and diffusive drug transport can lead to quite unexpected effects in the presence of a heterogeneous, compartmentalized vasculature. Finally we discuss various strategies to increase drug exposure time of tumor cells.     

Evaluation of a Voxelized Model Based on DCE-MRI for Tracer Transport in Tumor

Recent advances in the treatment of cancer involving therapeutic agents have shown promising results. However, treatment efficacy can be limited due to inadequate and uneven uptake in solid tumors, thereby making the prediction of drug transport important for developing effective therapeutic strategies. In this study, a patient-specific computational porous media model (voxelized model) was developed for predicting the interstitial flow field and distribution of a systemically delivered magnetic resonance (MR) visible tracer in a tumor. The benefits of a voxel approach include less labor and less computational time (approximately an order of magnitude reduction compared to the traditional computational fluid dynamics (CFD) approach developed earlier by our group). The model results were compared with that obtained from a previous approach based on unstructured meshes along with MR-measured tracer concentration data within tumors, using statistical analysis and qualitative representations. The statistical analysis indicated the similarity between the structured and unstructured models' results with a low root mean square error (RMS) and a high correlation coefficient. The voxelized model captured features of the flow field and tracer distribution such as high interstitial fluid pressure inside the tumor and the heterogeneous distribution of the tracer. Predictions of tracer distribution by the voxelized approach also resulted in low RMS error when compared with MR-measured data over a 1?h time course. The similarity in the voxelized model results with experiment and the nonvoxelized model predictions were maintained across three different tumors. Overall, the voxelized model serves as a reliable and swift alternative to approaches using unstructured meshes in predicting extracellular transport within tumors.     

Effect of wall compliance and permeability on blood-flow rate in counter-current microvessels formed from anastomosis during tumor-induced angiogenesis

Tumor blood-flow is inhomogeneous because of heterogeneity in tumor vasculature, vessel-wall leakiness, and compliance. Experimental studies have shown that normalization of tumor vasculature by antiangiogenic therapy can improve tumor microcirculation and enhance the delivery of therapeutic agents to tumors. To elucidate the quantitative relationship between the vessel-wall compliance and permeability and the blood-flow rate in the microvessels of the tumor tissue, the tumor tissue with the normalized vasculature, and the normal tissue, we developed a transport model to simultaneously predict the interstitial fluid pressure (IFP), interstitial fluid velocity (IFV) and the blood-flow rate in a counter-current microvessel loop, which occurs from anastomosis in tumor-induced angiogenesis during tumor growth. Our model predicts that although the vessel-wall leakiness greatly affects the IFP and IFV, it has a negligible effect on the intravascular driving force (pressure gradient) for both rigid and compliant vessels, and thus a negligible effect on the blood-flow rate if the vessel wall is rigid. In contrast, the wall compliance contributes moderately to the IFP and IFV, but significantly to the vessel radius and to the blood-flow rate. However, the combined effects of vessel leakiness and compliance can increase IFP, which leads to a partial collapse in the blood vessels and an increase in the flow resistance. Furthermore, our model predictions speculate a new approach for enhancing drug delivery to tumor by modulating the vessel-wall compliance in addition to reducing the vessel-wall leakiness and normalizing the vessel density.     

MHD Free Convective Boundary Layer Flow of a Nanofluid past a Flat Vertical Plate with Newtonian Heating Boundary Condition

Steady two dimensional MHD laminar free convective boundary layer flows of an electrically conducting Newtonian nanofluid over a solid stationary vertical plate in a quiescent fluid taking into account the Newtonian heating boundary condition is investigated numerically. A magnetic field can be used to control the motion of an electrically conducting fluid in micro/nano scale systems used for transportation of fluid. The transport equations along with the boundary conditions are first converted into dimensionless form and then using linear group of transformations, the similarity governing equations are developed. The transformed equations are solved numerically using the Runge-Kutta-Fehlberg fourth-fifth order method with shooting technique. The effects of different controlling parameters, namely, Lewis number, Prandtl number, buoyancy ratio, thermophoresis, Brownian motion, magnetic field and Newtonian heating on the flow and heat transfer are investigated. The numerical results for the dimensionless axial velocity, temperature and nanoparticle volume fraction as well as the reduced Nusselt and Sherwood number have been presented graphically and discussed. It is found that the rate of heat and mass transfer increase as Newtonian heating parameter increases. The dimensionless velocity and temperature distributions increase with the increase of Newtonian heating parameter. The results of the reduced heat transfer rate is compared for convective heating boundary condition and found an excellent agreement.     

Tumor microenvironment abnormalities: causes, consequences, and strategies to normalize

A solid tumor is an organ-like entity comprised of neoplastic cells and non-transformed host stromal cells embedded in an extracellular matrix. The expression of various genes is influenced by interactions among these cells, surrounding matrix, and their local physical and biochemical microenvironment. The products encoded by these genes, in turn, control the pathophysiological characteristics of the tumor, and give rise to the abnormal organization, structure, and function of tumor blood vessels. These abnormalities contribute to heterogeneous blood flow, vascular permeability, and microenvironment. Proliferating tumor cells produce solid stress which compresses blood and lymphatic vessels. As a result of vessel leakiness and lack of functional lymphatics, interstitial fluid pressure is significantly elevated in solid tumors. Each of these abnormalities forms a physiological barrier to the delivery of therapeutic agents to tumors. Furthermore, the metabolic microenvironment in tumors such as hypoxia and acidosis hinder the efficacy of anti-tumor treatments such as radiation therapy and chemotherapy. A judicious application of anti-angiogenic therapy has the potential to overcome these problems by normalizing the tumor vessels and making them more efficient for delivery of oxygen and drugs. Combined anti-angiogenic and conventional therapies have shown promise in the clinic.     

Regulation of tumor invasion by interstitial fluid flow

The importance of the tumor microenvironment in cancer progression is undisputed, yet the significance of biophysical forces in the microenvironment remains poorly understood. Interstitial fluid flow is a nearly ubiquitous and physiologically relevant biophysical force that is elevated in tumors because of tumor-associated angiogenesis and lymphangiogenesis, as well as changes in the tumor stroma. Not only does it apply physical forces to cells directly, but interstitial flow also creates gradients of soluble signals in the tumor microenvironment, thus influencing cell behavior and modulating cell-cell interactions. In this paper, we highlight our current understanding of interstitial fluid flow in the context of the tumor, focusing on the physical changes that lead to elevated interstitial flow, how cells sense flow and how they respond to changes in interstitial flow. In particular, we emphasize that interstitial flow can directly promote tumor cell invasion through a mechanism known as autologous chemotaxis, and indirectly support tumor invasion via both biophysical and biochemical cues generated by stromal cells. Thus, interstitial fluid flow demonstrates how important biophysical factors are in cancer, both by modulating cell behavior and coupling biophysical and biochemical signals.     

Edemagenic gain and interstitial fluid volume regulation

Under physiological conditions, interstitial fluid volume is tightly regulated by balancing microvascular filtration and lymphatic return to the central venous circulation. Even though microvascular filtration and lymphatic return are governed by conservation of mass, their interaction can result in exceedingly complex behavior. Without making simplifying assumptions, investigators must solve the fluid balance equations numerically, which limits the generality of the results. We thus made critical simplifying assumptions to develop a simple solution to the standard fluid balance equations that is expressed as an algebraic formula. Using a classical approach to describe systems with negative feedback, we formulated our solution as a "gain" relating the change in interstitial fluid volume to a change in effective microvascular driving pressure. The resulting "edemagenic gain" is a function of microvascular filtration coefficient (K(f)), effective lymphatic resistance (R(L)), and interstitial compliance (C). This formulation suggests two types of gain: "multivariate" dependent on C, R(L), and K(f), and "compliance-dominated" approximately equal to C. The latter forms a basis of a novel method to estimate C without measuring interstitial fluid pressure. Data from ovine experiments illustrate how edemagenic gain is altered with pulmonary edema induced by venous hypertension, histamine, and endotoxin. Reformulation of the classical equations governing fluid balance in terms of edemagenic gain thus yields new insight into the factors affecting an organ's susceptibility to edema.     

Multiscale Modelling of Fluid and Drug Transport in Vascular Tumours

A model for fluid and drug transport through the leaky neovasculature and porous interstitium of a solid tumour is developed. The transport problems are posed on a micro-scale characterized by the inter-capillary distance, and the method of multiple scales is used to derive the continuum equations describing fluid and drug transport on the length scale of the tumour (under the assumption of a spatially periodic microstructure). The fluid equations comprise a double porous medium, with coupled Darcy flow through the interstitium and vasculature, whereas the drug equations comprise advection–reaction equations; in each case the dependence of the transport coefficients on the vascular geometry is determined by solving micro-scale cell problems.     

血小板源生长因子受体与肿瘤

血小板源生长因子(platelet-derived growth factor,PDGF)经由其受体(platelet-derived growth fac tor receptor,PDGFR)表现细胞效应。PDGF和PDGFR涉及多种肿瘤的发病机制并在血管生成中起重要作用。PDGF在肿瘤中的自分泌刺激、PDGFR的过表达或过度活化或者刺激肿瘤内血管生成都会促进肿瘤生长;PDGFR的阻断可以降低实体瘤中组织间质液压而增强药物传送。这些机制可能提示在肿瘤治疗中PDGFR抑制剂单用、与化疗药物或者和其他靶点药物联合用药的可能性和可行性。随着PDGFR拮抗剂,如imatinib的上市,PDGFR作为抗肿瘤药物的靶点备受瞩目。     

Three-dimensional Cell Culture Model for Measuring the Effects of Interstitial Fluid Flow on Tumor Cell Invasion

The growth and progression of most solid tumors depend on the initial transformation of the cancer cells and their response to stroma-associated signaling in the tumor microenvironment ¹. Previously, research on the tumor microenvironment has focused primarily on tumor-stromal interactions ^1-2. However, the tumor microenvironment also includes a variety of biophysical forces, whose effects remain poorly understood. These forces are biomechanical consequences of tumor growth that lead to changes in gene expression, cell division, differentiation and invasion³. Matrix density ⁴, stiffness ^5-6, and structure ^6-7, interstitial fluid pressure ⁸, and interstitial fluid flow ⁸ are all altered during cancer progression.Interstitial fluid flow in particular is higher in tumors compared to normal tissues ^8-10. The estimated interstitial fluid flow velocities were measured and found to be in the range of 0.1-3 μm s^-1, depending on tumor size and differentiation ^{9, 11}. This is due to elevated interstitial fluid pressure caused by tumor-induced angiogenesis and increased vascular permeability ¹². Interstitial fluid flow has been shown to increase invasion of cancer cells ^13-14, vascular fibroblasts and smooth muscle cells ¹⁵. This invasion may be due to autologous chemotactic gradients created around cells in 3-D ¹⁶ or increased matrix metalloproteinase (MMP) expression ¹⁵, chemokine secretion and cell adhesion molecule expression ¹⁷. However, the mechanism by which cells sense fluid flow is not well understood. In addition to altering tumor cell behavior, interstitial fluid flow modulates the activity of other cells in the tumor microenvironment. It is associated with (a) driving differentiation of fibroblasts into tumor-promoting myofibroblasts ¹⁸, (b) transporting of antigens and other soluble factors to lymph nodes ¹⁹, and (c) modulating lymphatic endothelial cell morphogenesis ²⁰.The technique presented here imposes interstitial fluid flow on cells in vitro and quantifies its effects on invasion (Figure 1). This method has been published in multiple studies to measure the effects of fluid flow on stromal and cancer cell invasion ^{13-15, 17}. By changing the matrix composition, cell type, and cell concentration, this method can be applied to other diseases and physiological systems to study the effects of interstitial flow on cellular processes such as invasion, differentiation, proliferation, and gene expression.     

Numerical Simulation for the Unsteady MHD Flow and Heat Transfer of Couple Stress Fluid over a Rotating Disk

The present work is devoted to study the numerical simulation for unsteady MHD flow and heat transfer of a couple stress fluid over a rotating disk. A similarity transformation is employed to reduce the time dependent system of nonlinear partial differential equations (PDEs) to ordinary differential equations (ODEs). The Runge-Kutta method and shooting technique are employed for finding the numerical solution of the governing system. The influences of governing parameters viz. unsteadiness parameter, couple stress and various physical parameters on velocity, temperature and pressure profiles are analyzed graphically and discussed in detail.     

Multiscale Measurements Distinguish Cellular and Interstitial Hindrances to Diffusion In Vivo

Molecular cancer therapy relies on interstitial diffusion for drug distribution in solid tumors. A mechanistic understanding of how tumor components affect diffusion is necessary to advance cancer drug development. Yet, because of limitations in current techniques, it is unclear how individual tissue components hinder diffusion. We developed multiscale fluorescence recovery after photobleaching (MS-FRAP) to address this deficiency. Diffusion measurements facilitated by MS-FRAP distinguish the diffusive hindrance of the interstitial versus cellular constituents in living tissue. Using multiscale diffusion measurements in vivo, we resolved the contributions of these two major tissue components toward impeding diffusive transport in solid tumors and subcutaneous tissue in mice. We further used MS-FRAP in interstitial matrix-mimetic gels and in vivo to show the influence of physical interactions between collagen and hyaluronan on diffusive hindrance through the interstitium. Through these studies, we show that interstitial hyaluronan paradoxically improves diffusion and that reducing cellularity enhances diffusive macromolecular transport in solid tumors.     

A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy

A mathematical model is developed that describes the reduction in volume of a vascular tumor in response to specific chemotherapeutic administration strategies. The model consists of a system of partial differential equations governing intratumoral drug concentration and cancer cell density. In the model the tumor is treated as a continuum of two types of cells which differ in their proliferation rates and their responses to the chemotherapeutic agent. The balance between cell proliferation and death within the tumor generates a velocity field which drives expansion or regression of the spheroid. Insight into the tumor's response to therapy is gained by applying a combination of analytical and numerical techniques to the model equations.