Thomas Hillen

A user’s guide to PDE models for chemotaxis

Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225–234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “auto-aggregation”, has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes

In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropic diffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak--Keller--Segel--Alt...

THE DIFFUSION LIMIT OF TRANSPORT EQUATIONS II: CHEMOTAXIS EQUATIONS ∗

In this paper, we use the diffusion-limit expansion of transport equations developed earlier [T. Hillen and H. G. Othmer, SIAM J. Appl. Math., 61 (2000), pp. 751--775] to study the limiting equation under a variety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modification of the turning rate, the movement speed, or the preferred direction of movement. Depending on the strength of the bias, it leads to anisotropic diffusion, to a drift term in the flux, or to both, in the parabolic limit. We show that the classical chemotaxis equation---which we call the Patlak--Keller--Segel--Alt (PKSA) equation---arises only when the bias is sufficiently small. Using this general framework, we derive phenomenological models for chemotaxis of flagellated bacteria, of slime molds, and of myxobacteria. We also show that certain results derived earlier for one-dimensional motion can easily be generalized to two- or three-dimensional motion as well.

HYPERBOLIC MODELS FOR CHEMOSENSITIVE MOVEMENT

Chemosensitive movement describes the active orientation of individuals on chemical signals. In cases of cellular slime molds or flagellated bacteria, chemosensitive movement leads to aggregation and pattern formation. The classical mathematical model to describe chemosensitive movement is the diffusion based Patlak–Keller–Segel model. It suffers from the drawback of infinite propagation speeds. The relevant model parameters (motility and chemosensitivity) are related to population statistics. Hyperbolic models respect finite propagation speeds and the relevant model parameters (turning rate, distribution of new chosen velocities) are based on the individual movement patterns of the species at hand. In this paper hyperbolic models (in 1-D) and a transport model (in n-D) for chemosensitive movement are discussed and compared to the classical model. For the hyperbolic and transport models the following topics are reviewed: parabolic limit (which in some cases leads to the Patlak–Keller–Segel model), local and global existence, asymptotic behavior and moment closure. The moment closure approach leads to models based on Cattaneos law of heat conduction (telegraph equation).

Classical solutions and pattern formation for a volume filling chemotaxis model.

We establish the global existence of classical solutions to a generalized chemotaxis model, which includes the volume filling effect expressed through a nonlinear squeezing probability. This novel choice of squeezing probability reflects the elastic properties of cells. Necessary and sufficient conditions for spatial pattern formation are given and the underlying bifurcations are analyzed. In numerical simulations, the complex dynamics of merging and emerging patterns are shown for zero cell kinetics and nonzero cell kinetics, respectively. We conclude that the emerging process of pattern formation is due to cell growth.

Linear quadratic and tumour control probability modelling in external beam radiotherapy

The standard linear-quadratic (LQ) survival model for external beam radiotherapy is reviewed with particular emphasis on studying how different schedules of radiation treatment planning may be affected by different tumour repopulation kinetics. The LQ model is further examined in the context of tumour control probability (TCP) models. The application of the Zaider and Minerbo non-Poissonian TCP model incorporating the effect of cellular repopulation is reviewed. In particular the recent development of a cell cycle model within the original Zaider and Minerbo TCP formalism is highlighted. Application of this TCP cell-cycle model in clinical treatment plans is explored and analysed.

Mathematical modelling of glioma growth: the use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion.

The nonuniform growth of certain forms of cancer can present significant complications for their treatment, a particularly acute problem in gliomas. A number of experimental results have suggested that invasion is facilitated by the directed movement of cells along the aligned neural fibre tracts that form a large component of the white matter. Diffusion tensor imaging (DTI) provides a window for visualising this anisotropy and gaining insight on the potential invasive pathways. In this paper we develop a mesoscopic model for glioma invasion based on the individual migration pathways of invading cells along the fibre tracts. Via scaling we obtain a macroscopic model that allows us to explore the overall growth of a tumour. To connect DTI data to parameters in the macroscopic model we assume that directional guidance along fibre tracts is described by a bimodal von Mises-Fisher distribution (a normal distribution on a unit sphere) and parametrised according to the directionality and degree of anisotropy in the diffusion tensors. We demonstrate the results in a simple model for glioma growth, exploiting both synthetic and genuine DTI datasets to reveal the potentially crucial role of anisotropic structure on invasion.

The Tumor Growth Paradox and Immune System-Mediated Selection for Cancer Stem Cells

Cancer stem cells (CSCs) drive tumor progression, metastases, treatment resistance, and recurrence. Understanding CSC kinetics and interaction with their nonstem counterparts (called tumor cells, TCs) is still sparse, and theoretical models may help elucidate their role in cancer progression. Here, we develop a mathematical model of a heterogeneous population of CSCs and TCs to investigate the proposed “tumor growth paradox”—accelerated tumor growth with increased cell death as, for example, can result from the immune response or from cytotoxic treatments. We show that if TCs compete with CSCs for space and resources they can prevent CSC division and drive tumors into dormancy. Conversely, if this competition is reduced by death of TCs, the result is a liberation of CSCs and their renewed proliferation, which ultimately results in larger tumor growth. Here, we present an analytical proof for this tumor growth paradox. We show how numerical results from the model also further our understanding of how the fraction of cancer stem cells in a solid tumor evolves. Using the immune system as an example, we show that induction of cell death can lead to selection of cancer stem cells from a minor subpopulation to become the dominant and asymptotically the entire cell type in tumors.

CONVERGENCE OF A CANCER INVASION MODEL TO A LOGISTIC CHEMOTAXIS MODEL

A characteristic feature of tumor invasion is the destruction of the healthy tissue surrounding it. Open space is generated, which invasive tumor cells can move into. One such mechanism is the urokinase plasminogen system (uPS), which is found in many processes of tissue reorganization. Lolas, Chaplain and collaborators have developed a series of mathematical models for the uPS and tumor invasion. These models are based upon degradation of the extracellular material through plasmid plus chemotaxis and haptotaxis. In this paper we consider the uPS invasion models in one-space dimension and we identify a condition under which this cancer invasion model converges to a chemotaxis model with logistic growth. This condition assumes that the density of the extracellular material is not too large. Our result shows that the complicated spatio-temporal patterns, which were observed by Lolas and Chaplain et al. are organized by the chaotic attractor of the logistic chemotaxis system. Our methods are based on energy estimates, where, for convergence, we needed to find lower estimates in Lγ for 0 < γ < 1. This is a new method for these types of PDE.

Derivation of the tumour control probability (TCP) from a cell cycle model

In this paper, a model for the radiation treatment of cancer which includes the effects of the cell cycle is derived from first principles. A malignant cell population is divided into two compartments based on radiation sensitivities. The active compartment includes the four phases of the cell cycle, while the quiescent compartment consists of the G0 state. Analysis of this active-quiescent radiation model confirms the classical interpretation of the linear quadratic (LQ) model, which is that a larger a/b ratio corresponds to a fast cell cycle, while a smaller ratio corresponds to a slow cell cycle. Additionally, we find that a large a/b ratio indicates the existence of a significant quiescent phase. The active-quiescent model is extended as a nonlinear birth‐death process in order to derive an explicit time dependent expression for the tumour control probability (TCP). This work extends the TCP formula from Zaider and Minerbo and it enables the TCP to be calculated for general time dependent treatment schedules.