Urszula Ledzewicz Drábek Pavel Avner Friedman Marek Galewski Maria do Rosário de Pinho Bogdan Przeradzki Ewa Schmeidel
Discrete & Continuous Dynamical Systems - B 2018, 23(1): i-ii doi: 10.3934/dcdsb.201801i
A mathematical model for chronic wounds
Avner Friedman Chuan Xue
Mathematical Biosciences & Engineering 2011, 8(2): 253-261 doi: 10.3934/mbe.2011.8.253
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
Mathematical modeling of liver fibrosis
Avner Friedman Wenrui Hao
Mathematical Biosciences & Engineering 2017, 14(1): 143-164 doi: 10.3934/mbe.2017010

Fibrosis is the formation of excessive fibrous connective tissue in an organ or tissue, which occurs in reparative process or in response to inflammation. Fibrotic diseases are characterized by abnormal excessive deposition of fibrous proteins, such as collagen, and the disease is most commonly progressive, leading to organ disfunction and failure. Although fibrotic diseases evolve in a similar way in all organs, differences may occur as a result of structure and function of the specific organ. In liver fibrosis, the gold standard for diagnosis and monitoring the progression of the disease is biopsy, which is invasive and cannot be repeated frequently. For this reason there is currently a great interest in identifying non-invasive biomarkers for liver fibrosis. In this paper, we develop for the first time a mathematical model of liver fibrosis by a system of partial differential equations. We use the model to explore the efficacy of potential and currently used drugs aimed at blocking the progression of liver fibrosis. We also use the model to develop a diagnostic tool based on a combination of two biomarkers.

keywords: Network of liver fibrosis partial differential equations modeling hepatocytes stellate cells Kupffer cells hyalluronic acid
Avner Friedman Mirosław Lachowicz Urszula Ledzewicz Monika Joanna Piotrowska Zuzanna Szymanska
Mathematical Biosciences & Engineering 2017, 14(1): i-i doi: 10.3934/mbe.201701i
PDE problems arising in mathematical biology
Avner Friedman
Networks & Heterogeneous Media 2012, 7(4): 691-703 doi: 10.3934/nhm.2012.7.691
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
Discrete & Continuous Dynamical Systems - B 2004, 4(1): 147-159 doi: 10.3934/dcdsb.2004.4.147
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
Discrete & Continuous Dynamical Systems - A 2012, 32(9): 3081-3097 doi: 10.3934/dcds.2012.32.3081
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
The role of TNF-α inhibitor in glioma virotherapy: A mathematical model
Elzbieta Ratajczyk Urszula Ledzewicz Maciej Leszczynski Avner Friedman
Mathematical Biosciences & Engineering 2017, 14(1): 305-319 doi: 10.3934/mbe.2017020

Virotherapy, using herpes simplex virus, represents a promising therapy of glioma. But the innate immune response, which includes TNF-α produced by macrophages, reduces the effectiveness of the treatment. Hence treatment with TNF-α inhibitor may increase the effectiveness of the virotherapy. In the present paper we develop a mathematical model that includes continuous infusion of the virus in combination with TNF-α inhibitor. We study the efficacy of the treatment under different combinations of the two drugs for different scenarios of the burst size of newly formed virus emerging from dying infected cancer cells. The model may serve as a first step toward developing an optimal strategy for the treatment of glioma by the combination of TNF-α inhibitor and oncolytic virus injection.

keywords: Dynamical system virotherapy TNF-α inhibitors efficacy combination therapy glioma
Energy Considerations in a Model of Nematode Sperm Crawling
Borys V. Bazaliy Ya. B. Bazaliy Avner Friedman Bei Hu
Mathematical Biosciences & Engineering 2006, 3(2): 347-370 doi: 10.3934/mbe.2006.3.347
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
Mathematical Biosciences & Engineering 2011, 8(2): 371-383 doi: 10.3934/mbe.2011.8.371
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

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