PDE problems arising in mathematical biology
Avner Friedman
This article reviews biological processes that can be modeled by PDEs, it describes mathematical results, and suggests open problems. The first topic deals with tumor growth. This is modeled as a free boundary problem for a coupled system of elliptic, hyperbolic and parabolic equations. Existence theorems, stability of radially symmetric stationary solutions, and symmetry-breaking bifurcation results are stated. Next, a free boundary problem for wound healing is described, again involving a coupled system of PDEs. Other topics include movement of molecules in a neuron, modeled as a system of reaction-hyperbolic equations, and competition for resources, modeled as a system of reaction-diffusion equations.
keywords: reaction-hyperbolic equations reaction-diffusion equations Free boundary problems tumor growth wound healing.
A hierarchy of cancer models and their mathematical challenges
Avner Friedman
A variety of PDE models for tumor growth have been developed in the last three decades. These models are based on mass conservation laws and on reaction-diffusion processes within the tumor.
keywords: Cancer free boundary problems proliferating cells bifurcation.
Conservation laws in mathematical biology
Avner Friedman
Many mathematical models in biology can be described by conservation laws of the form \begin{equation}\tag{0.1} \frac{\partial{\bf{u}}}{\partial t} + \rm{div}(V{\bf{u}})=F(t,{\bf{x}}, {\bf{u}})\quad ({\bf{x}}=(x_1,\dots, x_n)) \end{equation} where ${\bf{u}}={\bf{u}}(t,{\bf{x}})$ is a vector $(u_1,\dots,u_k)$, ${\bf{F}}$ is a vector $(F_1,\dots,F_k)$, $V$ is a matrix with elements $V_{ij}(t,{\bf{x}},{\bf{u}})$, and $F_i(t,{\bf{x}}, {\bf{u}})$, $V_{ij}(t,{\bf{x}}, {\bf{u}})$ are nonlinear and/or non-local functions of ${\bf{u}}$. From a mathematical point of view one would like to establish, first of all, the existence and uniqueness of solutions under some prescribed initial (and possibly also boundary) conditions. However, the more interesting questions relate to establishing properties of the solutions that are of biological interest.
    In this article we give examples of biological processes whose mathematical models are represented in the form (0.1). We describe results and present open problems.
keywords: free boundary problems tumor growth. Hyperbolic equations reaction-diffusion equations cell differentiation parabolic equations drug resistance wound healing
Energy Considerations in a Model of Nematode Sperm Crawling
Borys V. Bazaliy Ya. B. Bazaliy Avner Friedman Bei Hu
In this paper we propose a mathematical model for nematode sperm cell crawling. The model takes into account both force and energy balance in the process of lamellipodium protrusion and cell nucleus drag. It is shown that by specifying the (possibly variable) efficiency of the major sperm protein biomotor one completely determines a self-consistent problem of the lamellipodium-nucleus motion. The model thus obtained properly accounts for the feedback of the load on the lamellipodium protrusion, which in general should not be neglected. We study and analyze the steady crawling state for a particular efficiency function and find that all nonzero modes, up to a large magnitude, are linearly asymptotically stable, thus reproducing the experimental observations of the long periods of steady crawling exhibited by the nematode sperm cells.
keywords: crawling stability nematode sperm cell travelling wave solution. biomotor thermodynamics free bound- ary problem
Tumor cells proliferation and migration under the influence of their microenvironment
Avner Friedman Yangjin Kim
It is well known that tumor microenvironment affects tumor growth and metastasis: Tumor cells may proliferate at different rates and migrate in different patterns depending on the microenvironment in which they are embedded. There is a huge literature that deals with mathematical models of tumor growth and proliferation, in both the avascular and vascular phases. In particular, a review of the literature of avascular tumor growth (up to 2006) can be found in Lolas [8] (G. Lolas, Lecture Notes in Mathematics, Springer Berlin / Heidelberg, 1872, 77 (2006)). In this article we report on some of our recent work. We consider two aspects, proliferation and of migration, and describe mathematical models based on in vitro experiments. Simulations of the models are in agreement with experimental results. The models can be used to generate hypotheses regarding the development of drugs which will confine tumor growth.
keywords: microenvironment. Cancer tumor growth
Can malaria parasite pathogenesis be prevented by treatment with tumor necrosis factor-alpha?
Avner Friedman Edward M. Lungu
We consider a model incorporating the influence of innate and adaptive immune responses on malaria pathogenesis. By calculating the model reproduction number for a special representation of cytokine interaction, we have shown that the cytokine tumour necrosis factor-$\alpha$ can be administered to inhibit malaria infection. We have also found that if the cytokine $F^*$ and a generic drug of efficacy $\epsilon$ are administered as dual therapy then clearance of the parasite can be achieved even for a generic drug of low efficacy. Our study is recommending administration of dual therapy as a strategy to prevent parasites from developing resistance to malaria treatment drugs.
keywords: cytokines. Pathogenesis adaptive immune response innate immune response
From the guest editors
Urszula Ledzewicz Avner Friedman Jacek Banasiak Heinz Schättler Edward M. Lungu
This special issue of Mathematical Biosciences and Engineering contains selected papers which were presented at the US-SA Workshop on ``Mathematical Methods in Systems Biology and Population Dynamics'' held at the African Institute for Mathematical Sciences (AIMS) in Muizenberg, South Africa, January 4-7, 2012. The workshop was originally planned as a small US-SA meeting, but with the growing interest of participants from other countries, we ended up with about 60 participants representing 16 countries from Europe, Africa and even Asia and Australia. Topics addressed at the workshop included the spread of infectious diseases and the growing need for robust and reliable models in ecology, both of special importance in the host country of South Africa where research naturally has been focused on fighting disease and epidemics like HIV/AIDS, malaria and others. In the US, on the other hand, a strong emphasis exists on systems biology and on its aspects related to cancer. Therefore, a second focus area of the workshop was on improved and more realistic models for the dynamic progression and treatment of various types of cancer, a truly globally challenging problem. We would also like to take the opportunity to thank all the sponsors: the National Science Foundation and the Society for Mathematical Biology from the US side, the National Research Foundation of South Africa with institutional support of AIMS, the University of KwaZulu-Natal, Durban and Southern Illinois University Edwardsville for making this event possible.

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Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy
Harsh Vardhan Jain Avner Friedman
Due to its dependence on androgens, metastatic prostate cancer is typically treated with continuous androgen ablation. However, such therapy eventually fails due to the emergence of castration-resistance cells. It has been hypothesized that intermittent androgen ablation can delay the onset of this resistance. In this paper, we present a biochemically-motivated ordinary differential equation model of prostate cancer response to anti-androgen therapy, with the aim of predicting optimal treatment protocols based on individual patient characteristics. Conditions under which intermittent scheduling is preferable over continuous therapy are derived analytically for a variety of castration-resistant cell phenotypes. The model predicts that while a cure is not possible for androgen-independent castration-resistant cells, continuous therapy results in longer disease-free survival periods. However, for androgen-repressed castration-resistant cells, intermittent therapy can significantly delay the emergence of resistance, and in some cases induce tumor regression. Numerical simulations of the model lead to two interesting cases, where even though continuous therapy may be non-viable, an optimally chosen intermittent schedule leads to tumor regression, and where a sub-optimally chosen intermittent schedule can initially appear to result in a cure, it eventually leads to resistance emergence. These results demonstrate the model's potential impact in a clinical setting.
keywords: mathematical model PSA. Androgen ablation androgen independent prostate cancer castration resistance androgen repressed
A partial differential equation model of metastasized prostatic cancer
Avner Friedman Harsh Vardhan Jain
Biochemically failing metastatic prostate cancer is typically treated with androgen ablation. However, due to the emergence of castration-resistant cells that can survive in low androgen concentrations, such therapy eventually fails. Here, we develop a partial differential equation model of the growth and response to treatment of prostate cancer that has metastasized to the bone. Existence and uniqueness results are derived for the resulting free boundary problem. In particular, existence and uniqueness of solutions for all time are proven for the radially symmetric case. Finally, numerical simulations of a tumor growing in 2-dimensions with radial symmetry are carried in order to evaluate the therapeutic potential of different treatment strategies. These simulations are able to reproduce a variety of clinically observed responses to treatment, and suggest treatment strategies that may result in tumor remission, underscoring our model's potential to make a significant contribution in the field of prostate cancer therapeutics.
keywords: prostate cancer Androgen ablation mathematical model PSA. free boundary problem
A mathematical model for chronic wounds
Avner Friedman Chuan Xue
Chronic wounds are often associated with ischemic conditions whereby the blood vascular system is damaged. A mathematical model which accounts for these conditions is developed and computational results are described in the two-dimensional radially symmetric case. Preliminary results for the three-dimensional axially symmetric case are also included.
keywords: free boundary problem viscoelasticity. Chronic wound healing

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