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.

     

Exploring the Competition between Proliferative and Invasive Cancer Phenotypes in a Continuous Spatial Model

Tumor is characterized by extensive heterogeneity with respect to its microenvironment and its genetic composition. We extend a previously developed monoclonal continuous spatial model of tumor growth to account for polyclonal cell populations and investigate the interplay between a more proliferative and a more invasive phenotype under different conditions. The model simulations demonstrate a transition from the dominance of the proliferative to the dominance of the invasive phenotype resembling malignant tumor progression and show a time period where both subpopulations are abundant. As the dominant phenotype switches from proliferative to invasive, the geometry of tumor changes from a compact and almost spherical shape to a more diffusive and fingered morphology with the proliferative phenotype to be restricted in the tumor bulk and the invasive to dominate at tumor edges. Different micro-environmental conditions and different phenotypic properties can promote or inhibit invasion demonstrating their mutual importance. The model provides a computational framework to investigate tumor heterogeneity and the constant interplay between the environment and the specific characteristics of phenotypes that should be taken into account for the prediction of tumor evolution, morphology and effective treatment.     

Elastic Free Energy Drives the Shape of Prevascular Solid Tumors

It is well established that the mechanical environment influences cell functions in health and disease. Here, we address how the mechanical environment influences tumor growth, in particular, the shape of solid tumors. In an in vitro tumor model, which isolates mechanical interactions between cancer tumor cells and a hydrogel, we find that tumors grow as ellipsoids, resembling the same, oft-reported observation of in vivo tumors. Specifically, an oblate ellipsoidal tumor shape robustly occurs when the tumors grow in hydrogels that are stiffer than the tumors, but when they grow in more compliant hydrogels they remain closer to spherical in shape. Using large scale, nonlinear elasticity computations we show that the oblate ellipsoidal shape minimizes the elastic free energy of the tumor-hydrogel system. Having eliminated a number of other candidate explanations, we hypothesize that minimization of the elastic free energy is the reason for predominance of the experimentally observed ellipsoidal shape. This result may hold significance for explaining the shape progression of early solid tumors in vivo and is an important step in understanding the processes underlying solid tumor growth.     

血管内皮标记物及血管内皮生长因子在胸水细胞中的表达和意义

探讨CD31、CD34、CD105及VEGF在胸水中的表达.应用免疫细胞化学染色,免疫荧光染色,Western-blot技术检测200例非小细胞肺癌患者胸水、30例增生胸水和20例炎性胸水中CD31、CD34、CD105及VEGF的表达.CD31、CD34、CD105及VEGF在非小细胞肺癌胸水中的表达量明显高于增生和炎性胸水表达(P＜0.05).非小细胞肺癌胸水患者CD31、CD34、CD105及VEGF高表达,并且在存在着肿瘤细胞血管样管型结构中表达量明显高于未发现肿瘤细胞血管样管型的胸腔积液.检测CD31、CD34、CD105及VEGF在胸腔积液中的表达情况可能对判断患者的预后有一定价值.     

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.     

Effect of tumor shape and size on drug delivery to solid tumors

ABSTRACT: Tumor shape and size effect on drug delivery to solid tumors are studied, 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 predefined tumor geometries. The governing equations are solved using a numerical method, the element-based finite volume method. Interstitial fluid pressure and velocity are used to show the details of drug delivery in a solid tumor, under an assumption that drug particles flow with the interstitial fluid. Drug delivery problems have been most extensively researched in spherical tumors, which have been the simplest to examine with the analytical methods. With our numerical method, however, more complex shapes of the tumor can be studied. The numerical model of fluid flow in solid tumors previously introduced by our group is further developed to incorporate and investigate non-spherical tumors such as prolate and oblate ones. Also the effects of the surface area per unit volume of the tissue, vascular and interstitial hydraulic conductivity on drug delivery are investigated.     

Mutations in foregut SOX2+ cells induce efficient proliferation via CXCR2 pathway

Identification of the precise molecular pathways involved in oncogene-induced transformation may help us gain a better understanding of tumor initiation and promotion. Here, we demonstrate that SOX2⁺ foregut epithelial cells are prone to oncogenic transformation upon mutagenic insults, such as Kras^G12D and p53 deletion. GFP-based lineage-tracing experiments indicate that SOX2⁺ cells are the cells-of-origin of esophagus and stomach hyperplasia. Our observations indicate distinct roles for oncogenic KRAS mutation and P53 deletion. p53 homozygous deletion is required for the acquisition of an invasive potential, and Kras^G12D expression, but not p53 deletion, suffices for tumor formation. Global gene expression analysis reveals secreting factors upregulated in the hyperplasia induced by oncogenic KRAS and highlights a crucial role for the CXCR2 pathway in driving hyperplasia. Collectively, the array of genetic models presented here demonstrate that stratified epithelial cells are susceptible to oncogenic insults, which may lead to a better understanding of tumor initiation and aid in the design of new cancer therapeutics.     

Short dysfunctional telomeres impair tumorigenesis in the INK4a(delta2/3) cancer-prone mouse.

Maintenance of telomere length is predicted to be essential for bypass of senescence and crisis checkpoints in cancer cells. The impact of telomere dysfunction on tumorigenesis was assessed in successive generations of mice doubly null for the telomerase RNA (mTR) and the INK4a tumor suppressor genes. Significant reductions in tumor formation in vivo and oncogenic potential in vitro were observed in late generations of telomerase deficiency, coincident with severe telomere shortening and associated dysfunction. Reintroduction of mTR into cells significantly restored the oncogenic potential, indicating telomerase activation is a cooperating event in the malignant transformation of cells containing critically short telomeres. The results described here demonstrate that loss of telomere function in a cancer-prone mouse model possessing intact DNA damage responses impairs, but does not prevent, tumor formation.     

On the infusion of a therapeutic agent into a solid tumor modeled as a poroelastic medium

The direct infusion of an agent into a solid tumor, modeled as a spherical poroelastic material with anisotropic dependence of the tumor hydraulic conductivity upon the tissue deformation, is treated both by solving the coupled fluid/elastic equations, and by expressing the solution as an asymptotic expansion in terms of a small parameter, ratio between the driving pressure force in the fluid system, and the elastic properties of the solid. Results at order one match almost perfectly the solutions of the full system over a large range of infusion pressures. Comparison with experimental results is acceptable after the hydraulic conductivity of the medium is properly calibrated. Given the uncertain estimates of some model constants, the order zero solution of the expansion, for which fluid and porous matrix are decoupled, yields acceptable values and trends for all the physical fields of interest, rendering the coupled analysis (in the limit of small displacements) of little use. When the deformation of the tissue becomes large nonlinear elasticity theory must be resorted to.     

Selective evolution of stromal mesenchyme with p53 loss in response to epithelial tumorigenesis

Our understanding of cancer has largely come from the analysis of aberrations within the tumor cell population. Yet it is increasingly clear that the tumor microenvironment can significantly influence tumorigenesis. For example, the mesenchyme can support the growth of tumorigenic epithelium. However, whether fibroblasts are subject to genetic/epigenetic changes as a result of selective pressures conferred by oncogenic stress in the epithelium has not been experimentally assessed. Recent analyses of some human carcinomas have shown tumor-suppressor gene mutations within the stroma, suggesting that the interplay among multiple cell types can select for aberrations nonautonomously during tumor progression. We demonstrate that this indeed occurs in a mouse model of prostate cancer where epithelial cell cycle disruption via cell-specific inhibition of pRb function induces a paracrine p53 response that suppresses fibroblast proliferation in associated stroma. This interaction imposes strong selective pressure yielding a highly proliferative mesenchyme that has undergone p53 loss.     

In vivo mimicking model for solid tumor towards hydromechanics of tissue deformation and creation of necrosis

The present work addresses transvascular and interstitial fluid transport inside a solid tumor surrounded by normal tissue (close to an in vivo mimicking setup). In general, biological tissues behave like a soft porous material and show mechanical behavior towards the fluid motion through the interstitial space. In general, forces like viscous drag that are associated with the fluid flow may compress the tissue material. On the macroscopic level, we try to model the motion of fluids and macromolecules through the interstitial space of solid tumor and the normal tissue layer. The transvascular fluid transport is assumed to be governed by modified Starling’s law. The poroelastohydrodynamics (interstitial hydrodynamics and the deformation of tissue material) inside the tumor and normal tissue regions is modeled using linearized biphasic mixture theory. Correspondingly, the velocity distribution of fluid is coupled to the displacement field of the solid phase (mainly cellular phase and extracellular matrix) in both the normal and tumor tissue regions. The corresponding velocity field is used within the transport reaction equation for fluids and macromolecules through interstitial space to get the overall solute (e.g., nutrients, drug, and other macromolecules) distribution. This study justifies that the presence of the normal tissue layer plays a significant role in delaying/assisting necrosis inside the tumor tissue. It is observed that the exchange process of fluids and macromolecules across the interface of the tumor and normal tissue affects the effectiveness factor corresponding to the tumor tissue.     

Theory of deformable substrates for cell motility studies.

Linear theory is used to relate the tractions F applied by a cell to the resulting deformation of fluid, viscoelastic, or solid substrates. The theory is used to fit data in which the motion of a fluid surface in the neighborhood of a motile keratocyte is visualized with the aid of embedded beads. The data are best fit by modeling the surface layer as a two-dimensional, nearly incompressible fluid. The data favor this model over another plausible model, the planar free boundary of a three-dimensional fluid. In the resulting diagrams for the distribution of F, it is found that both curl F and div F are concentrated in the lateral extrema of the lamellipodium. In a second investigation, a nonlinear theory of weak wrinkles in a solid substrate is proposed. The in-plane stress tensor plays the role of a metric. Compression wrinkles are found in regions where this metric is negative definite. Tension wrinkles arise, in linear approximation, at points on the boundary between positive definite and indefinite regions, and are conjectured to be stabilized by nonlinear effects. Data for the wrinkles that would be produced by keratocyte traction are computed, and these agree qualitatively with observed keratocyte wrinkles.     

The control of human WI-38 cell proliferation by extracellular calcium and its elimination by SV-40 virus-induced proliferative transformation.

The proliferative activity of diploid human WI-38 cells in sparse cultures depends on the extracellular concentration of free (or physiologically available) calcium, and cultivation in a medium having a calcium concentration of 0.1 mM or less gradually, but reversibly, arrest their proliferative development in the prereplicative (G1) phase of the cell cycle. Calcium's proliferative control of this cell type is eliminated by proliferative and morphological transformation by the oncogenic SV-40 virus, and the proliferative activity of SV-WI-38 cells in sparse cultures is unaffected by variation of the extracellular free calcium concentration between 0.00 and 1.25 mM.     

A Simple,Flexible Failure Model

A new failure model is introduced in the form of a four-parameter nonlinear differential equation, with failure probability as the dependent variable and failure time as the independent variable. The first parameter characterizes the location, the second the scale, and the other two the shape of the model. The type of the accompanying hazard function is immediately read off the shape parameters. The new model approximates the classical failure models with rather high precision, but also models cases where the failure density is skewed to the left. It can be used to analyze survival data objectively, based on the shape of the failure distribution. The computation of quantiles and moments is easy and fast. Nonlinear regression methods are used to estimate parameter values.     

Mitigation of destructive fluid force on buildings due to trapping of floating debris by coastal forest during the Great East Japan tsunami

The effectiveness of coastal or inland forests in trapping floating debris produced by destructive tsunamis has long been discussed but never clarified quantitatively. To estimate the effectiveness, especially in reducing the damage to wooden buildings (houses) behind the forest, a nonlinear long wave equation model that includes not only destruction of the coastal forest but also a trap mechanism has been developed, and a simulation was conducted where trapping was observed during the 2011 Great East Japan tsunami. A post-tsunami survey of Sendai City showed that some houses just behind the coastal forest remained, although the inland-side houses were washed out. Based on the damage to houses, fragility curves as a function of moment by fluid force were analyzed. Although only the fringe of a coastal forest increases the fluid force due to a tsunami current concentration where a coastal forest is damaged, analyses using the model with and without the trapping function suggest that the debris-trapping function of a coastal forest greatly mitigates the damage observed in the post-tsunami survey, not only just behind the forest, but further inland, at around 1300 m from the coast. By including the trapping effect, the inundation height can be well reproduced, especially at around 2000–3000 m. The fluid force reduction was increased inland because some houses there remained standing in the tsunami and maintained resistance to the current.     

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.     

Nonlinear simulation of tumor growth

 We study solid tumor (carcinoma) growth in the nonlinear regime using boundary-integral simulations. The tumor core is nonnecrotic and no inhibitor chemical species are present. A new formulation of the classical models [18,24,8,3] is developed and it is demonstrated that tumor evolution is described by a reduced set of two dimensionless parameters and is qualitatively unaffected by the number of spatial dimensions. One parameter describes the relative rate of mitosis to the relaxation mechanisms (cell mobility and cell-to-cell adhesion). The other describes the balance between apoptosis (programmed cell-death) and mitosis. Both parameters also include the effect of vascularization. Our analysis and nonlinear simulations reveal that the two new dimensionless groups uniquely subdivide tumor growth into three regimes associated with increasing degrees of vascularization: low (diffusion dominated, e.g., in vitro), moderate and high vascularization, that correspond to the regimes observed in vivo. We demonstrate that critical conditions exist for which the tumor evolves to nontrivial dormant states or grows self-similarly (i.e., shape invariant) in the first two regimes. This leads to the possibility of shape control and of controlling the release of tumor angiogenic factors by restricting the tumor volume-to-surface-area ratio. Away from these critical conditions, evolution may be unstable leading to invasive fingering into the external tissues and to topological transitions such as tumor breakup and reconnection. Interestingly we find that for highly vascularized tumors, while they grow unbounded, their shape always stays compact and invasive fingering does not occur. This is in agreement with recent experimental observations [30] of in vivo tumor growth, and suggests that the invasive growth of highly-vascularized tumors is associated to vascular and elastic anisotropies, which are not included in the model studied here. Received: 1 May 2002 / Revised version: 26 August 2002 / Published online: 18 December 2002 Current address: Department of Biomedical Engineering, and Department of Mathematics, University of California at Irvine, Irvine CA 92697. e-mail: cristini@math.uci.edu; lowengrb@math.umn.edu. Key words or phrases: Tumor growth – Linear stability analysis – Self-similarity – Boundary-integral simulations     

A nonlinear structured population model of tumor growth with quiescence

A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators. Supported in part by the National Science Foundation under Grant No. DMS-8722947     

Self-association of the cardiac fatty acid binding protein. Influence on membrane-bound, fatty acid dependent enzymes

The present study on the fatty acid binding protein, purified from pig heart and studied by three independent techniques (electron spin resonance, circular dichroism, and polyacrylamide gel electrophoresis), suggests that the protein self-aggregates and exists in at least four distinct molecular species. This plurality is demonstrated by the presence of four bands after electrophoretic migration at pH 7.2 and by three transitions of molar ellipticity theta 225 that depend on protein concentration. A mathematical model is formulated to simulate the three transitions and to calculate the concentrations of the four species. The multistates manifest themselves in a complex binding capacity for fatty acid, with two sigmoidal components in the binding curve. A general equation for the curve is formulated, and the characteristic constants are evaluated by a nonlinear least-squares fit. The experimental results and their interpretation in quantitative terms lead to a theoretical evaluation of the importance of this new property of self-aggregation of the protein on the activity of membrane-bound model enzymes which are fatty acid or acyl coenzyme A dependent.