Alf Gerisch

Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation.

The ability of cancer cells to break out of tissue compartments and invade locally gives solid tumours a defining deadly characteristic. One of the first steps of invasion is the remodelling of the surrounding tissue or extracellular matrix (ECM) and a major part of this process is the over-expression of proteolytic enzymes, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs), by the cancer cells to break down ECM proteins. Degradation of the matrix enables the cancer cells to migrate through the tissue and subsequently to spread to secondary sites in the body, a process known as metastasis. In this paper we undertake an analysis of a mathematical model of cancer cell invasion of tissue, or ECM, which focuses on the role of the urokinase plasminogen activation system. The model consists of a system of five reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, uPA, uPA inhibitors, plasmin and the host tissue. Cancer cells react chemotactically and haptotactically to the spatio-temporal effects of the uPA system. The results obtained from computational simulations carried out on the model equations produce dynamic heterogeneous spatio-temporal solutions and using linear stability analysis we show that this is caused by a taxis-driven instability of a spatially homogeneous steady-state. Finally we consider the biological implications of the model results, draw parallels with clinical samples and laboratory based models of cancer cell invasion using three-dimensional invasion assay, and go on to discuss future development of the model.

Robust numerical methods for taxis-diffusion-reaction systems: Applications to biomedical problems

In this paper we consider the numerical solution of taxis-diffusion-reaction models. Such nonlinear partial differential equation models appear often in mathematical biology and we consider examples from tumour growth and invasion, and the aggregation of amoebae. These examples are characterised by a taxis term which is large in magnitude compared with the diffusion term and this can cause numerical problems. The numerical technique presented here follows the method of lines. Special attention is paid to the discretization of the taxis term in space to avoid oscillations and negative solution values. We employ splitting techniques in the time discretization to deal with the complex structure of the model and to reduce the amount of computational linear algebra. These techniques are based on explicit Runge-Kutta and linearly implicit Runge-Kutta-Rosenbrock methods. A series of numerical experiments demonstrates the good performance of the algorithm and gives rise to some implications for future modelling.

Mathematical modeling of fracture healing in mice: comparison between experimental data and numerical simulation results

The combined use of experimental and mathematical models can lead to a better understanding of fracture healing. In this study, a mathematical model, which was originally established by Bailón-Plaza and van der Meulen (J Theor Biol 212:191–209, 2001), was applied to an experimental model of a semi-stabilized murine tibial fracture. The mathematical model was implemented in a custom finite volumes code, specialized in dealing with the model’s requirements of mass conservation and non-negativity of the variables. A qualitative agreement between the experimentally measured and numerically simulated evolution in the cartilage and bone content was observed. Additionally, an extensive parametric study was conducted to assess the influence of the model parameters on the simulation outcome. Finally, a case of pathological fracture healing and its treatment by administration of growth factors was modeled to demonstrate the potential therapeutic value of this mathematical model.

A hybrid bioregulatory model of angiogenesis during bone fracture healing

Bone fracture healing is a complex process in which angiogenesis or the development of a blood vessel network plays a crucial role. In this paper, a mathematical model is presented that simulates the biological aspects of fracture healing including the formation of individual blood vessels. The model consists of partial differential equations, several of which describe the evolution in density of the most important cell types, growth factors, tissues and nutrients. The other equations determine the growth of blood vessels as a result of the movement of leading endothelial (tip) cells. Branching and anastomoses are accounted for in the model. The model is applied to a normal fracture healing case and subjected to a sensitivity analysis. The spatiotemporal evolution of soft tissues and bone, as well as the development of a blood vessel network are corroborated by comparison with experimental data. Moreover, this study shows that the proposed mathematical framework can be a useful tool in the research of impaired healing and the design of treatment strategies.

Spatial distribution of tissue level properties in a human femoral cortical bone.

The mechanical properties of cortical bone are determined by a combination bone tissue composition, and structure at several hierarchical length scales. In this study the spatial distribution of tissue level properties within a human femoral shaft has been investigated. Cylindrically shaped samples (diameter: 4.4mm, N=56) were prepared from cortical regions along the entire length (20-85% of the total femur length), and around the periphery (anterior, medial, posterior and lateral quadrants). The samples were analyzed using scanning acoustic microscopy (SAM) at 50MHz and synchrotron radiation micro computed tomography (SRμCT). For all samples the average cortical porosity (Ct.Po), tissue elastic coefficients (c(ij)) and the average tissue degree of mineralization (DMB) were determined. The smallest coefficient of variation was observed for DMB (1.8%), followed by BV/TV (5.4%), c(ij) (8.2-45.5%), and Ct.Po (47.5%). Different variations with respect to the anatomical position were found for DMB, Ct.Po and c(ij). These data address the anatomical variations in anisotropic elastic properties and link them to tissue mineralization and porosity, which are important input parameters for numerical multi-scale bone models.

Operator splitting and approximate factorization for taxis-diffusion-reaction models

In this paper we consider the numerical solution of 2D systems of certain types of taxis-diffusion-reaction equations from mathematical biology. By spatial discretization these PDE systems are approximated by systems of positive, nonlinear ODEs (Method of Lines). The aim of this paper is to examine the numerical integration of these ODE systems for low to moderate accuracy by means of splitting techniques. An important consideration is maintenance of positivity. We apply operator splitting and approximate matrix factorization using low order explicit Runge-Kutta methods and linearly implicit Runge-Kutta-Rosenbrock methods. As a reference method the general purpose solver VODPK is applied.

Mathematical modelling of cancer invasion : implications of cell adhesion variability for tumour infiltrative growth patterns

Cancer invasion, recognised as one of the hallmarks of cancer, is a complex, multiscale phenomenon involving many inter-related genetic, biochemical, cellular and tissue processes at different spatial and temporal scales. Central to invasion is the ability of cancer cells to alter and degrade an extracellular matrix. Combined with abnormal excessive proliferation and migration which is enabled and enhanced by altered cell-cell and cell-matrix adhesion, the cancerous mass can invade the neighbouring tissue. Along with tumour-induced angiogenesis, invasion is a key component of metastatic spread, ultimately leading to the formation of secondary tumours in other parts of the host body. In this paper we explore the spatio-temporal dynamics of a model of cancer invasion, where cell-cell and cell-matrix adhesion is accounted for through non-local interaction terms in a system of partial integro-differential equations. The change of adhesion properties during cancer growth and development is investigated here through time-dependent adhesion characteristics within the cell population as well as those between the cells and the components of the extracellular matrix. Our computational simulation results demonstrate a range of heterogeneous dynamics which are qualitatively similar to the invasive growth patterns observed in a number of different types of cancer, such as tumour infiltrative growth patterns (INF).

Mathematical modeling in wound healing, bone regeneration and tissue engineering.

The processes of wound healing and bone regeneration and problems in tissue engineering have been an active area for mathematical modeling in the last decade. Here we review a selection of recent models which aim at deriving strategies for improved healing. In wound healing, the models have particularly focused on the inflammatory response in order to improve the healing of chronic wound. For bone regeneration, the mathematical models have been applied to design optimal and new treatment strategies for normal and specific cases of impaired fracture healing. For the field of tissue engineering, we focus on mathematical models that analyze the interplay between cells and their biochemical cues within the scaffold to ensure optimal nutrient transport and maximal tissue production. Finally, we briefly comment on numerical issues arising from simulations of these mathematical models.

A Positive Splitting Method for Mixed Hyperbolic-Parabolic Systems

In this article we present a method of lines approach to the numerical solution of a system of coupled hyperbolic—parabolic partial differential equations (PDEs). Special attention is paid to preserving the positivity of the solution of the PDEs when this solution is approximated numerically. This is achieved by using a flux-limited spatial discretization for the hyperbolic equation. We use splitting techniques for the solution of the resulting large system of stiff ordinary differential equations. The performance of the approach applied to a biomathematical model is compared with the performance of standard methods.

Building blocks of a simulation environment of the OSI network layer of packet-switching networks

This paper describes the main building blocks of a simulation environment of the OSI network layer of packet-switching networks. The need for such a tool is presented and pitfalls of previous solutions are described. Remedies provided by the most recent solution are discussed. Architecture, organisation, and architectural decisions are explained.