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Mathematical Biosciences & Engineering

2013 , Volume 10 , Issue 3

Special issue on Mathematical Methods in Systems Biology and Population Dynamics

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From the guest editors
Urszula Ledzewicz, Avner Friedman, Jacek Banasiak, Heinz Schättler and Edward M. Lungu
2013, 10(3): i-ii doi: 10.3934/mbe.2013.10.3i +[Abstract](3994) +[PDF](78.0KB)
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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A singularly perturbed SIS model with age structure
Jacek Banasiak, Eddy Kimba Phongi and MirosŁaw Lachowicz
2013, 10(3): 499-521 doi: 10.3934/mbe.2013.10.499 +[Abstract](3628) +[PDF](447.7KB)
We present a preliminary study of an SIS model with a basic age structure and we focus on a disease with quick turnover, such as influenza or common cold. In such a case the difference between the characteristic demographic and epidemiological times naturally introduces two time scales in the model which makes it singularly perturbed. Using the Tikhonov theorem we prove that for certain classes of initial conditions the nonlinear structured SIS model can be approximated with very good accuracy by lower dimensional linear models.
Diffusion rate determines balance between extinction and proliferation in birth-death processes
Hilla Behar, Alexandra Agranovich and Yoram Louzoun
2013, 10(3): 523-550 doi: 10.3934/mbe.2013.10.523 +[Abstract](3001) +[PDF](991.8KB)
We here study spatially extended catalyst induced growth processes. This type of process exists in multiple domains of biology, ranging from ecology (nutrients and growth), through immunology (antigens and lymphocytes) to molecular biology (signaling molecules initiating signaling cascades). Such systems often exhibit an extinction-proliferation transition, where varying some parameters can lead to either extinction or survival of the reactants.
    When the stochasticity of the reactions, the presence of discrete reactants and their spatial distribution is incorporated into the analysis, a non-uniform reactant distribution emerges, even when all parameters are uniform in space.
    Using a combination of Monte Carlo simulation and percolation theory based estimations; the asymptotic behavior of such systems is studied. In all studied cases, it turns out that the overall survival of the reactant population in the long run is based on the size and shape of the reactant aggregates, their distribution in space and the reactant diffusion rate. We here show that for a large class of models, the reactant density is maximal at intermediate diffusion rates and low or zero at either very high or very low diffusion rates. We give multiple examples of such system and provide a generic explanation for this behavior. The set of models presented here provides a new insight on the population dynamics in chemical, biological and ecological systems.
Gompertz model with delays and treatment: Mathematical analysis
Marek Bodnar, Monika Joanna Piotrowska and Urszula Foryś
2013, 10(3): 551-563 doi: 10.3934/mbe.2013.10.551 +[Abstract](2874) +[PDF](630.9KB)
In this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy. We mainly consider two types of delayed models, the one with the delay introduced in the per capita growth rate (we call it the single delayed model) and the other with the delay introduced in the net growth rate (the double delayed model). We focus on stability and possible stability switches with increasing delay for the positive steady state. Moreover, we study a Hopf bifurcation, including stability of arising periodic solutions for a constant treatment. The analytical results are extended by numerical simulations for a pharmacokinetic treatment function.
Mathematical modeling of glioma therapy using oncolytic viruses
Baba Issa Camara, Houda Mokrani and Evans K. Afenya
2013, 10(3): 565-578 doi: 10.3934/mbe.2013.10.565 +[Abstract](3417) +[PDF](872.3KB)
Diffuse infiltrative gliomas are adjudged to be the most common primary brain tumors in adults and they tend to blend in extensively in the brain micro-environment. This makes it difficult for medical practitioners to successfully plan effective treatments. In attempts to prolong the lengths of survival times for patients with malignant brain tumors, novel therapeutic alternatives such as gene therapy with oncolytic viruses are currently being explored. Based on such approaches and existing work, a spatio-temporal model that describes interaction between tumor cells and oncolytic viruses is developed. Conditions that lead to optimal therapy in minimizing cancer cell proliferation and otherwise are analytically demonstrated. Numerical simulations are conducted with the aim of showing the impact of virotherapy on proliferation or invasion of cancer cells and of estimating survival times.
Identifying preseizure state in intracranial EEG data using diffusion kernels
Dominique Duncan, Ronen Talmon, Hitten P. Zaveri and Ronald R. Coifman
2013, 10(3): 579-590 doi: 10.3934/mbe.2013.10.579 +[Abstract](2576) +[PDF](830.5KB)
The goal of this study is to identify preseizure changes in intracranial EEG (icEEG). A novel approach based on the recently developed diffusion map framework, which is considered to be one of the leading manifold learning methods, is proposed. Diffusion mapping provides dimensionality reduction of the data as well as pattern recognition that can be used to distinguish different states of the patient, for example, interictal and preseizure. A new algorithm, which is an extension of diffusion maps, is developed to construct coordinates that generate efficient geometric representations of the complex structures in the icEEG data. In addition, this method is adapted to the icEEG data and enables the extraction of the underlying brain activity.
    The algorithm is tested on icEEG data recorded from several electrode contacts from a patient being evaluated for possible epilepsy surgery at the Yale-New Haven Hospital. Numerical results show that the proposed approach provides a distinction between interictal and preseizure states.
A partial differential equation model of metastasized prostatic cancer
Avner Friedman and Harsh Vardhan Jain
2013, 10(3): 591-608 doi: 10.3934/mbe.2013.10.591 +[Abstract](3665) +[PDF](7367.7KB)
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.
Can malaria parasite pathogenesis be prevented by treatment with tumor necrosis factor-alpha?
Avner Friedman and Edward M. Lungu
2013, 10(3): 609-624 doi: 10.3934/mbe.2013.10.609 +[Abstract](2302) +[PDF](406.9KB)
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.
Identifying requirements for the invasion of a tick species and tick-borne pathogen through TICKSIM
Holly Gaff and Robyn Nadolny
2013, 10(3): 625-635 doi: 10.3934/mbe.2013.10.625 +[Abstract](2650) +[PDF](306.6KB)
Ticks and tick-borne diseases have been on the move throughout the United State over the past twenty years. We use an agent-based model, TICKSIM, to identify the key parameters that determine the success of invasion of the tick and if that is successful, the succees of the tick-borne pathogen. We find that if an area has competent hosts, an initial population of ten ticks is predicted to always establish a new population. The establishment of the tick-borne pathogen depends on three parameters: the initial prevalence in the ten founding ticks, the probability that a tick infects the longer-lived hosts and the probability that a tick infects the shorter-lived hosts. These results indicate that the transmission rates to hosts in the newly established area can be used to predict the potential risk of disease to humans.
Finite element approximation of a population spatial adaptation model
Gonzalo Galiano and Julián Velasco
2013, 10(3): 637-647 doi: 10.3934/mbe.2013.10.637 +[Abstract](2991) +[PDF](201.0KB)
In [18], Sighesada, Kawasaki and Teramoto presented a system of partial differential equations for modeling spatial segregation of interacting species. Apart from competitive Lotka-Volterra (reaction) and population pressure (cross-diffusion) terms, a convective term modeling the populations attraction to more favorable environmental regions was included. In this article, we study numerically a modification of their convective term to take account for the notion of spatial adaptation of populations. After describing the model, in which a time non-local drift term is considered, we propose a numerical discretization in terms of a mass-preserving time semi-implicit finite element method. Finally, we provied the results of some biologically inspired numerical experiments showing qualitative differences between the original model of [18] and the model proposed in this article.
Shear-thinning effects of hemodynamics in patient-specific cerebral aneurysms
Alberto Gambaruto, João Janela, Alexandra Moura and Adélia Sequeira
2013, 10(3): 649-665 doi: 10.3934/mbe.2013.10.649 +[Abstract](3137) +[PDF](7063.0KB)
Two different generalized Newtonian mathematical models for blood flow, derived for the same experimental data, are compared, together with the Newtonian model, in three different anatomically realistic geometries of saccular cerebral aneurysms obtained from rotational CTA. The geometries differ in size of the aneurysm and the existence or not of side branches within the aneurysm. Results show that the differences between the two generalized Newtonian mathematical models are smaller than the differences between these and the Newtonian solution, in both steady and unsteady simulations.
On the sensitivity of feature ranked lists for large-scale biological data
Danuta Gaweł and Krzysztof Fujarewicz
2013, 10(3): 667-690 doi: 10.3934/mbe.2013.10.667 +[Abstract](2925) +[PDF](641.5KB)
The problem of feature selection for large-scale genomic data, for example from DNA microarray experiments, is one of the fundamental and well-investigated problems in modern computational biology. From the computational point of view, a selected gene list should be characterized by good predictive power and should be understood and well explained from the biological point of view. Recently, another feature of selected gene lists is increasingly investigated, namely their stability which measures how the content and/or the gene order change when the data are perturbed. In this paper we propose a new approach to analysis of gene list stability, termed the sensitivity index, that does not require any data perturbation and allows the gene list that is most reliable in a biological sense to be chosen.
An example from the world of tsetse flies
John Hargrove
2013, 10(3): 691-704 doi: 10.3934/mbe.2013.10.691 +[Abstract](2183) +[PDF](716.8KB)
In biomathematics, communication between mathematicians and biologists is crucial. This matter is illustrated using studies aimed at estimating mortality rates of tsetse flies (Glossina spp.). Examples are provided of apparently sound pieces of mathematics which, when applied to real data, provide obviously erroneous results. More serious objections arise when mathematical models make no attempt to address the real world in such a way that they can be tested. Unless models account for the known biology of the problem under investigation, and are challenged with data, the existence and nature of imperfections in the models will likely not be detected.
Mathematical modeling of citrus groves infected by huanglongbing
Karly Jacobsen, Jillian Stupiansky and Sergei S. Pilyugin
2013, 10(3): 705-728 doi: 10.3934/mbe.2013.10.705 +[Abstract](3478) +[PDF](486.3KB)
Huanglongbing (citrus greening) is a bacterial disease that is significantly impacting the citrus industry in Florida and poses a risk to the remaining citrus-producing regions of the United States. A mathematical model of a grove infected by citrus greening is developed. An equilibrium stability analysis is presented. The basic reproductive number and its relation to the persistence of the disease is discussed. A numerical study is performed to illustrate the theoretical findings.
Hybrid discrete-continuous model of invasive bladder cancer
Eugene Kashdan and Svetlana Bunimovich-Mendrazitsky
2013, 10(3): 729-742 doi: 10.3934/mbe.2013.10.729 +[Abstract](2828) +[PDF](448.8KB)
Bladder cancer is the seventh most common cancer worldwide. Epidemiological studies and experiments implicated chemical penetration into urothelium (epithelial tissue surrounding bladder) in the etiology of bladder cancer. In this work we model invasive bladder cancer. This type of cancer starts in the urothelium and progresses towards surrounding muscles and tissues, causing metastatic disease. Our mathematical model of invasive BC consists of two coupled sub-models: (i) living cycle of the urothelial cells (normal and mutated) simulated using discrete technique of Cellular Automata and (ii) mechanism of tumor invasion described by the system of reaction-diffusion equations. Numerical simulations presented here are in good qualitative agreement with the experimental results and reproduce in vitro observations described in medical literature.
Calcium waves with mechano-chemical couplings
Bogdan Kazmierczak and Zbigniew Peradzynski
2013, 10(3): 743-759 doi: 10.3934/mbe.2013.10.743 +[Abstract](2686) +[PDF](435.4KB)
As follows from experiments, waves of calcium concentration in biological tissues can be easily excited by a local mechanical stimulation. Therefore the complete theory of calcium waves should also take into account coupling between mechanical and chemical processes. In this paper we consider the existence of travelling waves for buffered systems, as in [22], completed, however, by an equation for mechanical equilibrium and respective mechanochemical coupling terms. Thus the considered, coupled system consists of reaction-diffusion equations (for the calcium and buffers concentrations) and equations for the balance of mechanical forces.
Spatial stochastic models of cancer: Fitness, migration, invasion
Natalia L. Komarova
2013, 10(3): 761-775 doi: 10.3934/mbe.2013.10.761 +[Abstract](3456) +[PDF](437.0KB)
Cancer progression is driven by genetic and epigenetic events giving rise to heterogeneity of cell phenotypes, and by selection forces that shape the changing composition of tumors. The selection forces are dynamic and depend on many factors. The cells favored by selection are said to be more ``fit'' than others. They tend to leave more viable offspring and spread through the population. What cellular characteristics make certain cells more fit than others? What combinations of the mutant characteristics and ``background'' characteristics make the mutant cells win the evolutionary competition? In this review we concentrate on two phenotypic characteristics of cells: their reproductive potential and their motility. We show that migration has a direct positive impact on the ability of a single mutant cell to invade a pre-existing colony. Thus, a decrease in the reproductive potential can be compensated by an increase in cell migration. We further demonstrate that the neutral ridges (the set of all types with the invasion probability equal to that of the host cells) remain invariant under the increase of system size (for large system sizes), thus making the invasion probability a universal characteristic of the cells' selection status. We list very general conditions under which the optimal phenotype is just one single strategy (thus leading to a nearly-homogeneous type invading the colony), or a large set of strategies that differ by their reproductive potentials and migration characteristics, but have a nearly-equal fitness. In the latter case the evolutionary competition will result in a highly heterogeneous population.
Equilibrium solutions for microscopic stochastic systems in population dynamics
MirosŁaw Lachowicz and Tatiana Ryabukha
2013, 10(3): 777-786 doi: 10.3934/mbe.2013.10.777 +[Abstract](3184) +[PDF](368.4KB)
The present paper deals with the problem of existence of equilibrium solutions of equations describing the general population dynamics at the microscopic level of modified Liouville equation (individually--based model) corresponding to a Markov jump process. We show the existence of factorized equilibrium solutions and discuss uniqueness. The conditions guaranteeing uniqueness or non-uniqueness are proposed under the assumption of periodic structures.
On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth
Urszula Ledzewicz, Omeiza Olumoye and Heinz Schättler
2013, 10(3): 787-802 doi: 10.3934/mbe.2013.10.787 +[Abstract](3143) +[PDF](449.3KB)
In this paper, a mathematical model for chemotherapy that takes tumor immune-system interactions into account is considered for a strongly targeted agent. We use a classical model originally formulated by Stepanova, but replace exponential tumor growth with a generalised logistic growth model function depending on a parameter $\nu$. This growth function interpolates between a Gompertzian model (in the limit $\nu\rightarrow0$) and an exponential model (in the limit $\nu\rightarrow\infty$). The dynamics is multi-stable and equilibria and their stability will be investigated depending on the parameter $\nu$. Except for small values of $\nu$, the system has both an asymptotically stable microscopic (benign) equilibrium point and an asymptotically stable macroscopic (malignant) equilibrium point. The corresponding regions of attraction are separated by the stable manifold of a saddle. The optimal control problem of moving an initial condition that lies in the malignant region into the benign region is formulated and the structure of optimal singular controls is determined.
On the MTD paradigm and optimal control for multi-drug cancer chemotherapy
Urszula Ledzewicz, Heinz Schättler, Mostafa Reisi Gahrooi and Siamak Mahmoudian Dehkordi
2013, 10(3): 803-819 doi: 10.3934/mbe.2013.10.803 +[Abstract](3244) +[PDF](483.9KB)
In standard chemotherapy protocols, drugs are given at maximum tolerated doses (MTD) with rest periods in between. In this paper, we briefly discuss the rationale behind this therapy approach and, using as example multi-drug cancer chemotherapy with a cytotoxic and cytostatic agent, show that these types of protocols are optimal in the sense of minimizing a weighted average of the number of tumor cells (taken both at the end of therapy and at intermediate times) and the total dose given if it is assumed that the tumor consists of a homogeneous population of chemotherapeutically sensitive cells. A $2$-compartment linear model is used to model the pharmacokinetic equations for the drugs.
Modeling bacterial attachment to surfaces as an early stage of biofilm development
Fadoua El Moustaid, Amina Eladdadi and Lafras Uys
2013, 10(3): 821-842 doi: 10.3934/mbe.2013.10.821 +[Abstract](3206) +[PDF](900.7KB)
Biofilms are present in all natural, medical and industrial surroundings where bacteria live. Biofilm formation is a key factor in the growth and transport of both beneficial and harmful bacteria. While much is known about the later stages of biofilm formation, less is known about its initiation which is an important first step in the biofilm formation. In this paper, we develop a non-linear system of partial differential equations of Keller-Segel type model in one-dimensional space, which couples the dynamics of bacterial movement to that of the sensing molecules. In this case, bacteria perform a biased random walk towards the sensing molecules. We derive the boundary conditions of the adhesion of bacteria to a surface using zero-Dirichlet boundary conditions, while the equation describing sensing molecules at the interface needed particular conditions to be set. The numerical results show the profile of bacteria within the space and the time evolution of the density within the free-space and on the surface. Testing different parameter values indicate that significant amount of sensing molecules present on the surface leads to a faster bacterial movement toward the surface which is the first step of biofilm initiation. Our work gives rise to results that agree with the biological description of the early stages of biofilm formation.
Modelling the role of drug barons on the prevalence of drug epidemics
John Boscoh H. Njagarah and Farai Nyabadza
2013, 10(3): 843-860 doi: 10.3934/mbe.2013.10.843 +[Abstract](2486) +[PDF](551.9KB)
Substance abuse is a global menace with immeasurable consequences to the health of users, the quality of life and the economy of countries affected. Although the prominently known routes of initiation into drug use are; by contact between potential users and individuals already using the drugs and self initiation, the role played by a special class of individuals referred to as drug lords can not be ignored. We consider a simple but useful compartmental model of drug use that accounts for the contribution of contagion and drug lords to initiation into drug use and drug epidemics. We show that the model has a drug free equilibrium when the threshold parameter $R_{0}$ is less that unity and a drug persistent equilibrium when $R_{0}$ is greater than one. In our effort to ascertain the effect of policing in the control of drug epidemics, we include a term accounting for law enforcement. Our results indicate that increased law enforcement greatly reduces the prevalence of substance abuse. In addition, initiation resulting from presence of drugs in circulation can be as high as seven times higher that initiation due to contagion alone.
A simple model of carcinogenic mutations with time delay and diffusion
Monika Joanna Piotrowska, Urszula Foryś, Marek Bodnar and Jan Poleszczuk
2013, 10(3): 861-872 doi: 10.3934/mbe.2013.10.861 +[Abstract](3222) +[PDF](952.5KB)
In the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely favorable, competitive and unfavorable, were considered. We focus on the unfavorable conditions that can result from a given treatment, e.g. chemotherapy. Included delay stands for the interactions between benign and other cells. We compare the dynamics of ODEs system, the system with delay and the system with delay and diffusion. We mainly focus on the dynamics when a positive steady state exists. The system which is globally stable in the case without the delay and diffusion is destabilized by increasing delay, and therefore the underlying kinetic dynamics becomes oscillatory due to a Hopf bifurcation for appropriate values of the delay. This suggests the occurrence of spatially non-homogeneous periodic solutions for the system with the delay and diffusion.
Application of evolutionary games to modeling carcinogenesis
Andrzej Swierniak and Michal Krzeslak
2013, 10(3): 873-911 doi: 10.3934/mbe.2013.10.873 +[Abstract](3178) +[PDF](2530.5KB)
We review a quite large volume of literature concerning mathematical modelling of processes related to carcinogenesis and the growth of cancer cell populations based on the theory of evolutionary games. This review, although partly idiosyncratic, covers such major areas of cancer-related phenomena as production of cytotoxins, avoidance of apoptosis, production of growth factors, motility and invasion, and intra- and extracellular signaling. We discuss the results of other authors and append to them some additional results of our own simulations dealing with the possible dynamics and/or spatial distribution of the processes discussed.
A flexible multivariable model for Phytoplankton growth
Mohammad A. Tabatabai, Wayne M. Eby, Sejong Bae and Karan P. Singh
2013, 10(3): 913-923 doi: 10.3934/mbe.2013.10.913 +[Abstract](2675) +[PDF](438.8KB)
We introduce a new multivariable model to be used to study the growth dynamics of phytoplankton as a function of both time and the concentration of nutrients. This model is applied to a set of experimental data which describes the rate of growth as a function of these two variables. The form of the model allows easy extension to additional variables. Thus, the model can be used to analyze experimental data regarding the effects of various factors on phytoplankton growth rate. Such a model will also be useful in analysis of the role of concentration of various nutrients or trace elements, temperature, and light intensity, or other important explanatory variables, or combinations of such variables, in analyzing phytoplankton growth dynamics.
T model of growth and its application in systems of tumor-immune dynamics
Mohammad A. Tabatabai, Wayne M. Eby, Karan P. Singh and Sejong Bae
2013, 10(3): 925-938 doi: 10.3934/mbe.2013.10.925 +[Abstract](2728) +[PDF](591.0KB)
In this paper we introduce a new growth model called T growth model. This model is capable of representing sigmoidal growth as well as biphasic growth. This dual capability is achieved without introducing additional parameters. The T model is useful in modeling cellular proliferation or regression of cancer cells, stem cells, bacterial growth and drug dose-response relationships. We recommend usage of the T growth model for the growth of tumors as part of any system of differential equations. Use of this model within a system will allow more flexibility in representing the natural rate of tumor growth. For illustration, we examine some systems of tumor-immune interaction in which the T growth rate is applied. We also apply the model to a set of tumor growth data.
Computational modeling approaches to studying the dynamics of oncolytic viruses
Dominik Wodarz
2013, 10(3): 939-957 doi: 10.3934/mbe.2013.10.939 +[Abstract](3088) +[PDF](2822.0KB)
Oncolytic viruses specifically infect cancer cells, replicate in them, kill them, and spread to further tumor cells. They represent a targeted treatment approach that is promising in principle, but consistent success has yet to be observed. Mathematical models can play an important role in analyzing the dynamics between oncolytic viruses and a growing tumor cell population, providing insights that can be useful for the further development of this therapy approach. This article reviews different mathematical modeling approaches ranging from ordinary differential equations to spatially explicit agent-based models. Problems of model robustness are discussed and so are some clinically important insight derived from the models.

2018 Impact Factor: 1.313




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