Mathematical and computational models of drug transport in tumours.

The ability to predict how far a drug will penetrate into the tumour microenvironment within its pharmacokinetic (PK) lifespan would provide valuable information about therapeutic response. As the PK profile is directly related to the route and schedule of drug administration, an in silico tool that can predict the drug administration schedule that results in optimal drug delivery to tumours would streamline clinical trial design. This paper investigates the application of mathematical and computational modelling techniques to help improve our understanding of the fundamental mechanisms underlying drug delivery, and compares the performance of a simple model with more complex approaches. Three models of drug transport are developed, all based on the same drug binding model and parametrized by bespoke in vitro experiments. Their predictions, compared for a 'tumour cord' geometry, are qualitatively and quantitatively similar. We assess the effect of varying the PK profile of the supplied drug, and the binding affinity of the drug to tumour cells, on the concentration of drug reaching cells and the accumulated exposure of cells to drug at arbitrary distances from a supplying blood vessel. This is a contribution towards developing a useful drug transport modelling tool for informing strategies for the treatment of tumour cells which are 'pharmacokinetically resistant' to chemotherapeutic strategies.

A mathematical model of doxorubicin penetration through multicellular layers.

Inadequate drug delivery to tumours is now recognised as a key factor that limits the efficacy of anticancer drugs. Extravasation and penetration of therapeutic agents through avascular tissue are critically important processes if sufficient drug is to be delivered to be therapeutic. The purpose of this study is to develop an in silico model that will simulate the transport of the clinically used cytotoxic drug doxorubicin across multicell layers (MCLs) in vitro. Three cell lines were employed: DLD1 (human colon carcinoma), MCF7 (human breast carcinoma) and NCI/ADR-Res (doxorubicin resistant and P-glycoprotein [Pgp] overexpressing ovarian cell line). Cells were cultured on transwell culture inserts to various thicknesses and doxorubicin at various concentrations (100 or 50 microM) was added to the top chamber. The concentration of drug appearing in the bottom chamber was determined as a function of time by HPLC-MS/MS. The rate of drug penetration was inversely proportional to the thickness of the MCL. The rate and extent of doxorubicin penetration was no different in the presence of NCI/ADR-Res cells expressing Pgp compared to MCF7 cells. A mathematical model based upon the premise that the transport of doxorubicin across cell membrane bilayers occurs by a passive "flip-flop" mechanism of the drug between two membrane leaflets was constructed. The mathematical model treats the transwell apparatus as a series of compartments and the MCL is treated as a series of cell layers, separated by small intercellular spaces. This model demonstrates good agreement between predicted and actual drug penetration in vitro and may be applied to the prediction of drug transport in vivo, potentially becoming a useful tool in the study of optimal chemotherapy regimes.

Modelling the electrical properties of tissue as a porous medium.

Models of the electrical properties of biological tissue have been the subject of many studies. These models have sought to explain aspects of the dielectric dispersion of tissue. This paper develops a mathematical model of the complex permittivity of tissue as a function of frequency f, in the range 10(4) < f < 10(7) Hz, which is derived from a formulation used to describe the complex permittivity of porous media. The model introduces two parameters, porosity and percolation probability, to the description of the electrical properties of any tissue which comprises a random arrangement of cells. The complex permittivity for a plausible porosity and percolation probability distribution is calculated and compared with the published measured electrical properties of liver tissue. Broad agreement with the experimental data is noted. It is suggested that future detailed experimental measurements should be undertaken to validate the model. The model may be a more convenient method of parameterizing the electrical properties of biological tissue and subsequent measurement of these parameters in a range of tissues may yield information of biological and clinical significance.

On the spread of morphogens.

Pattern organisation of cells and tissue are specified during embryonic development by gradients of morphogens, substances that assign different fates of cells at different concentrations. Morphogen gradients form by transport from a localized site. Whether this occurs by diffusion or some other more elaborate mechanism is controversial. Here we provide an analysis of two models considered by Lander et al. used in support of the diffusive transport hypothesis.

Lattice and non-lattice models of tumour angiogenesis.

In order to progress from the relatively harmless avascular state to the potentially lethal vascular state, solid tumours must induce the growth of new blood vessels from existing ones, a process called angiogenesis. The capillary growth centres around endothelial cells: there are several cell-based models of this process in the literature and these have reproduced some of the key microscopic features of capillary growth. The most common approach is to simulate the movement of leading endothelial cells on a regular lattice. Here, we apply a circular random walk model to the process of angiogenesis, and thus allow the cells to move independently of a lattice; the results display good agreement with empirical observations. We also run simulations of two lattice-based models in order to make a critical comparison of the different modelling approaches. Finally, non-lattice simulations are carried out in the context of a realistic model of tumour angiogenesis, and potential anti-angiogenic strategies are evaluated.

A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins.

Angiogenesis--the growth of new blood vessels from existing ones--is a prerequisite for the growth of solid tumours beyond a diameter of approximately 2 mm. In recent years, the angiopoietins have emerged as important regulators of angiogenesis. They mediate a delicate balance between vascular quiescence, regression and new growth, but their mechanism of action is not fully understood. This work attempts to provide a mathematical description of the role of the angiopoietins in angiogenesis. The model is formulated within the framework of reinforced random walks, which allows easy transition between the continuum (macroscopic) and discrete (microscopic) forms. Model predictions are in qualitative agreement with experimental observations, and may have implications for anti-cancer therapies based on the prevention of angiogenesis.

The role of the angiopoietins in tumour angiogenesis.

Angiogenesis--the growth of new blood vessels from existing ones--is a prerequisite for the growth of solid tumours beyond a diameter of approximately 2 mm. In recent years, the angiopoietins have emerged as important regulators of angiogenesis. They mediate a delicate balance between vascular quiescence, regression and new growth, but their mechanism of action is not fully understood. This work attempts to provide a mathematical description of the role of the angiopoietins in angiogenesis. The model is formulated within the framework of reinforced random walks, which allows easy transition between the continuum (macroscopic) and discrete (microscopic) forms. Model predictions are in qualitative agreement with experimental observations, and may have implications for antiangiogenic cancer therapies.

A reinforced random walk model of tumour angiogenesis and anti-angiogenic strategies.

It is now well accepted that the growth of a tumour beyond approximately 2 mm in diameter is dependent on its ability to induce the growth of new blood vessels, a process called angiogenesis. This has raised hope that an anti-angiogenic treatment may be effective in the fight against cancer. Here we formulate, using the theory of reinforced random walks, an individual cell-based mathematical model of tumour angiogenesis in response to a diffusible angiogenic factor. The early stages of angiogenesis, in which endothelial cells (EC) escape the parent vessel and invade the extra-cellular matrix, are included in the model, as are the action of a proteolytic enzyme, EC proliferation and capillary branching and anastomosis. The anti-angiogenic potential of angiostatin, a known inhibitor of angiogenesis, is also examined. The capillary networks predicted by the model are in qualitative agreement with experimental observations. Proteolysis and proliferation are shown to be crucial for vascularization, whilst angiostatin is seen to be capable of limiting capillary growth.

Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma.

The purpose of this paper is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 1970s and subsequently established experimentally by him and his coworkers [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 53-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: an experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413-419]. In the simplest version of this model, an avascular tumor secretes a tumor growth factor (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network. In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase. Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of molecular species in porous media. One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 73-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Nat. Cancer Inst., 52, 413-419]. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor [Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64].

Mathematical modeling of the onset of capillary formation initiating angiogenesis.

It is well accepted that neo-vascular formation can be divided into three main stages (which may be overlapping): (1) changes within the existing vessel, (2) formation of a new channel, (3) maturation of the new vessel. In this paper we present a new approach to angiogenesis, based on the theory of reinforced random walks, coupled with a Michaelis-Menten type mechanism which views the endothelial cell receptors as the catalyst for transforming angiogenic factor into proteolytic enzyme in order to model the first stage. In this model, a single layer of endothelial cells is separated by a vascular wall from an extracellular tissue matrix. A coupled system of ordinary and partial differential equations is derived which, in the presence of an angiogenic agent, predicts the aggregation of the endothelial cells and the collapse of the vascular lamina, opening a passage into the extracellular matrix. We refer to this as the onset of vascular sprouting. Some biological evidence for the correctness of our model is indicated by the formation of teats in utero. Further evidence for the correctness of the model is given by its prediction that endothelial cells will line the nascent capillary at the onset of capillary angiogenesis.

A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I. The role of protease inhibitors in preventing angiogenesis.

In this paper, a simple mathematical model developed in H.A. Levine, B.D. Sleeman, M. Nilsen-Hamilton [J. Math. Biol., in press] to describe the initiation of capillary formation in tumor angiogenesis is extended to include the roles of pericytes and macrophages in regulating angiogenesis. The model also allows for the presence of anti-angiogenic (angiostatic) factors. The model is based on the observation that angiostatin can prevent the degradation of fibronectin in the basal lamina by inhibiting the catalytic action of active proteolytic enzyme. That is, it is proposed that the inhibitor 'deactivates' the protease but that it does not reduce the over all concentration of the protease. It consequently explores the possibility of preventing neovascular capillaries from migrating through the extra-cellular matrix toward the tumor by inhibiting protease action. The model is based on the theory of reinforced random walks coupled with Michaelis-Menten mechanisms which view endothelial cell receptors as the catalysts for transforming both tumor and macrophage derived angiogenic factors into proteolytic enzyme which in turn degrade the basal lamina. A simple catalytic reaction is proposed for the degradation of the basal lamina by the active proteases. A mechanism, in which the angiostatin acts as a protease inhibitor is discussed which has been substantiated experimentally. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase.

A mathematical model of tumour angiogenesis incorporating cellular traction and viscoelastic effects.

Angiogenesis is defined as the outgrowth and formation of new vessels from a pre-existing vascular network (Rakusan, In: Cardiac Growth and Regeneration. Annals of the New York Academy of Sciences, 1995), and is of fundamental importance in understanding the processes by which a tumour achieves vascularization. Diffusible substances, collectively called tumour angiogenesis factors are released from the tumour to elicit a variety of responses from the surrounding tissues, most importantly the migration of endothelial cells (lining neighbouring vessels) towards the tumour. To facilitate locomotion, the cells exert appreciable traction forces upon the interstitial extracellular matrix which, in turn, influences the resulting direction of their migration. In this paper, we examine the role played by cellular traction during cell migration and the corresponding viscoelastic effects of the extracellular matrix.

Aggregation and collapse in a mechanical model of fungal tip growth.

Interconnected hyphal tubes form the mycelia of a fungal colony. The growth of the colony results from the elongation and branching of these single hyphae. The material being incorporated into the extending hyphal wall is supplied by vesicles which are formed further back in the hyphal tip. Such wall-destined vesicles appear conspicuously concentrated in the interior of the hypha, just before the hyphal apex, in the form of an apical body or Spitzenkörper. The cytoskeleton of the hyphal tube has been implicated in the organisation of the Spitzenkörper and the transport of vesicles, but as yet there is no postulated mechanism for this. We propose a mechanism by which forces generated by the cytoskeleton are responsible for biasing the movement of vesicles. A mathematical model is derived where the cytoskeleton is described as a viscoelastic fluid. Viscoelastic forces are coupled to the conservation equation governing the vesicle dynamics, by weighting the diffusion of vesicles via the strain tensor. The model displays collapse and aggregation patterns in one and two dimensions. These are interpreted in terms of the formation of the Spitzenkörper and the initiation of apical branching.

Velocity and stability of solitary planar travelling wave solutions of intracellular [Ca]2+.

Normal cardiac muscle contraction occurs in response to a rapid rise followed by a slower decay in intracellular calcium concentration. When cardiac muscle cells are loaded with calcium, an intracellular store releases calcium into the cytosol by the process of calcium-induced calcium release (CICR). This release contributes to the rise in intracellular calcium which in turn triggers contraction. We use two qualitative piecewise linear reaction-diffusion models of this behaviour to investigate the speed, stability and waveform of plane waves using singular perturbation techniques.

Fluid transport in vascularized tumours and metastasis.

In this paper we propose a modification to a model of R. K. Jain and L. T. Baxter used to describe fluid transport in vascularized tumours. The model predicts high interstitial pressure in the interior of the tumour together with a rapid decrease in pressure to its periphery in agreement with experiment and the predictions of the Jain-Baxter model. By the addition of a pressure-curvature (Gibbs-Thomson) condition on the periphery of the tumour we are able to show by a perturbation analysis that a lowering of the interstitial pressure can retard tumour growth. Lowering interstitial pressure may also be beneficial to the targeting of therapeutic drugs.

A mathematical analysis of a model for tumour angiogenesis.

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Neighbouring endothelial cells respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. A mathematical model is presented which takes into account two of the most important events associated with the endothelial cells as they form capillary sprouts and make their way towards the tumour i.e. cell migration and proliferation. The numerical simulations of the model compare very well with the actual experimental observations. We subsequently investigate the model analytically by making some relevant biological simplifications. The mathematical analysis helps to clarify the particular contributions to the model of the two independent processes of endothelial cell migration and proliferation.

Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory.

Medically, tumours are classified into two important classes--benign and malignant. Generally speaking, the two classes display different behaviour with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumour growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid tumour using membrane and thick-shell theory. A central feature of the model is the characterisation of the material composition of the model through the use of a strain-energy function, thus permitting a mathematical description of the degree of differentiation of the tumour explicitly in the model. Conditions are given in terms of the strain-energy function for the processes of invasion and metastasis occurring in a tumour, being interpreted as the bifurcation modes of the spherical shell which the tumour is essentially modelled as. Our results are compared with actual experimental results and with the general behaviour shown by benign and malignant tumours. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumours (in particular, the Gaussian and mean curvatures of the surface of a solid tumour) in an attempt to produce a mathematical formulation and description of the important medical processes of staging and grading cancers. We hope that this approach may form the basis of a practical application.

A mathematical model for the growth and classification of a solid tumor: a new approach via nonlinear elasticity theory using strain-energy functions.

Medically, tumors are classified into two important classes--benign and malignant. Generally speaking, the two classes display different behaviour with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumor growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid tumor using membrane and thick-shell theory. A central feature of the model is the characterization of the material composition of the tumor through the use of a strain energy function, thus permitting a mathematical description of the degree of differentiation of the tumor explicitly in the model. Conditions are given in terms of the strain energy function for the processes of invasion and metastasis occurring in a tumor, being interpreted as the bifurcation modes of the spherical shell, which the tumor is essentially modeled as. Our results are compared with actual medical experimental results and with the general behavior shown by benign and malignant tumors. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumors (in particular, the Gaussian and mean curvatures of the surface of a solid tumor) in an attempt to produce a mathematical formulation and description of the important medical processes of staging and grading cancers. We hope that this approach may form the basis of a practical application.

A note on the reconstruction of ellipsoids from the X-ray transform.

This paper is concerned with the problem of locating a solid tumour in X-ray tomography. Given that the unknown tumour is ellipsoidal and of uniform density, it is shown that the location, orientation, principal axes of the tumour are uniquely determined from six radiographs.