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Mathematical Biosciences & Engineering

2015 , Volume 12 , Issue 6

Special issue on application of ecological and mathematical theory to cancer: New challenges

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Application of ecological and mathematical theory to cancer: New challenges
Yangjin Kim , Avner Friedman , Eugene Kashdan , Urszula Ledzewicz and  Chae-Ok Yun
2015, 12(6): i-iv doi: 10.3934/mbe.2015.12.6i +[Abstract](32) +[PDF](120.7KB)
According to the World Health Organization, cancer is among the leading causes of morbidity and mortality worldwide. Despite enormous efforts of cancer researchers all around the world, the mechanisms underlying its origin, formation, progression, therapeutic cure or control are still not fully understood. Cancer is a complex, multi-scale process, in which genetic mutations occurring at a sub-cellular level manifest themselves as functional changes at the cellular and tissue scale.

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Hybrid models of cell and tissue dynamics in tumor growth
Yangjin Kim and  Hans G. Othmer
2015, 12(6): 1141-1156 doi: 10.3934/mbe.2015.12.1141 +[Abstract](44) +[PDF](3437.4KB)
Hybrid models of tumor growth, in which some regions are described at the cell level and others at the continuum level, provide a flexible description that allows alterations of cell-level properties and detailed descriptions of the interaction with the tumor environment, yet retain the computational advantages of continuum models where appropriate. We review aspects of the general approach and discuss applications to breast cancer and glioblastoma.
A data-motivated density-dependent diffusion model of in vitro glioblastoma growth
Tracy L. Stepien , Erica M. Rutter and  Yang Kuang
2015, 12(6): 1157-1172 doi: 10.3934/mbe.2015.12.1157 +[Abstract](48) +[PDF](664.7KB)
Glioblastoma multiforme is an aggressive brain cancer that is extremely fatal. It is characterized by both proliferation and large amounts of migration, which contributes to the difficulty of treatment. Previous models of this type of cancer growth often include two separate equations to model proliferation or migration. We propose a single equation which uses density-dependent diffusion to capture the behavior of both proliferation and migration. We analyze the model to determine the existence of traveling wave solutions. To prove the viability of the density-dependent diffusion function chosen, we compare our model with well-known in vitro experimental data.
Mathematical model and its fast numerical method for the tumor growth
Hyun Geun Lee , Yangjin Kim and  Junseok Kim
2015, 12(6): 1173-1187 doi: 10.3934/mbe.2015.12.1173 +[Abstract](37) +[PDF](1149.1KB)
In this paper, we reformulate the diffuse interface model of the tumor growth (S.M. Wise et al., Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524--543). In the new proposed model, we use the conservative second-order Allen--Cahn equation with a space--time dependent Lagrange multiplier instead of using the fourth-order Cahn--Hilliard equation in the original model. To numerically solve the new model, we apply a recently developed hybrid numerical method. We perform various numerical experiments. The computational results demonstrate that the new model is not only fast but also has a good feature such as distributing excess mass from the inside of tumor to its boundary regions.
Synergistic effect of blocking cancer cell invasion revealed by computer simulations
Kazuhisa Ichikawa
2015, 12(6): 1189-1202 doi: 10.3934/mbe.2015.12.1189 +[Abstract](79) +[PDF](590.4KB)
Invasion and metastasis are the main cause of death in cancer patients. The initial step of invasion is the degradation of extracellular matrix (ECM) by primary cancer cells in a tissue. Membranous metalloproteinase MT1-MMP and soluble metalloproteinase MMP-2 are thought to play an important role in the degradation of ECM. In the previous report, we found that the repetitive insertion of MT1-MMP to invadopodia was crucial for the effective degradation of ECM (Hoshino, D., et al., PLoS Comp. Biol., 2012, e1002479). However, the role of MMP-2 and the effect of inhibitors for these ECM-degrading proteases were still obscure. Here we investigated these two problems by using the same model as in the previous report. First we tested the effect of MMP-2 and found that while MT1-MMP played a major role in the degradation of ECM, MMP-2 played only a marginal effect on the degradation of ECM. Based on these findings, we next tested the effect of a putative inhibitor for MT1-MMP and found that such inhibitor was ineffective in blocking ECM degradation. Then we tested combined strategy including inhibitor for MT1-MMP, reduction of its turnover and its content in vesicles. A synergistic effect of combined strategy was observed in the decrease in the efficacy of ECM degradation. Our simulation study suggests the importance of combined strategy in blocking cancer invasion and metastasis.
The role of the cytokines IL-27 and IL-35 in cancer
Avner Friedman and  Kang-Ling Liao
2015, 12(6): 1203-1217 doi: 10.3934/mbe.2015.12.1203 +[Abstract](33) +[PDF](1021.9KB)
The cancer-immune interaction is a fast growing field of research in biology, where the goal is to harness the immune system to fight cancer more effectively. In the present paper we review recent work of the interaction between T cells and cancer. CD8$^+$ T cells are activated by IL-27 cytokine and they kill tumor cells. Regulatory T cells produce IL-35 which promotes cancer cells by enhancing angiogenesis, and inhibit CD8$^+$ T cells via TGF-$\beta$ production. Hence injections of IL-27 and anti-IL-35 are both potentially anti-tumor drugs. The models presented here are based on experimental mouse experiments, and their simulations agree with these experiments. The models are used to suggest effective schedules for drug treatment.
An integrated cellular and sub-cellular model of cancer chemotherapy and therapies that target cell survival
Alexis B. Cook , Daniel R. Ziazadeh , Jianfeng Lu and  Trachette L. Jackson
2015, 12(6): 1219-1235 doi: 10.3934/mbe.2015.12.1219 +[Abstract](47) +[PDF](1051.7KB)
Apoptosis resistance is a hallmark of human cancer, and tumor cells often become resistant due to defects in the programmed cell death machinery. Targeting key apoptosis regulators to overcome apoptotic resistance and promote rapid death of tumor cells is an exciting new strategy for cancer treatment, either alone or in combination with traditionally used anti-cancer drugs that target cell division. Here we present a multiscale modeling framework for investigating the synergism between traditional chemotherapy and targeted therapies aimed at critical regulators of apoptosis.
Treatment strategies for combining immunostimulatory oncolytic virus therapeutics with dendritic cell injections
Joanna R. Wares , Joseph J. Crivelli , Chae-Ok Yun , Il-Kyu Choi , Jana L. Gevertz and  Peter S. Kim
2015, 12(6): 1237-1256 doi: 10.3934/mbe.2015.12.1237 +[Abstract](185) +[PDF](456.3KB)
Oncolytic viruses (OVs) are used to treat cancer, as they selectively replicate inside of and lyse tumor cells. The efficacy of this process is limited and new OVs are being designed to mediate tumor cell release of cytokines and co-stimulatory molecules, which attract cytotoxic T cells to target tumor cells, thus increasing the tumor-killing effects of OVs. To further promote treatment efficacy, OVs can be combined with other treatments, such as was done by Huang et al., who showed that combining OV injections with dendritic cell (DC) injections was a more effective treatment than either treatment alone. To further investigate this combination, we built a mathematical model consisting of a system of ordinary differential equations and fit the model to the hierarchical data provided from Huang et al. We used the model to determine the effect of varying doses of OV and DC injections and to test alternative treatment strategies. We found that the DC dose given in Huang et al. was near a bifurcation point and that a slightly larger dose could cause complete eradication of the tumor. Further, the model results suggest that it is more effective to treat a tumor with immunostimulatory oncolytic viruses first and then follow-up with a sequence of DCs than to alternate OV and DC injections. This protocol, which was not considered in the experiments of Huang et al., allows the infection to initially thrive before the immune response is enhanced. Taken together, our work shows how the ordering, temporal spacing, and dosage of OV and DC can be chosen to maximize efficacy and to potentially eliminate tumors altogether.
Dynamics and control of a mathematical model for metronomic chemotherapy
Urszula Ledzewicz , Behrooz Amini and  Heinz Schättler
2015, 12(6): 1257-1275 doi: 10.3934/mbe.2015.12.1257 +[Abstract](41) +[PDF](2150.3KB)
A $3$-compartment model for metronomic chemotherapy that takes into account cancerous cells, the tumor vasculature and tumor immune-system interactions is considered as an optimal control problem. Metronomic chemo-therapy is the regular, almost continuous administration of chemotherapeutic agents at low dose, possibly with small interruptions to increase the efficacy of the drugs. There exists medical evidence that such administrations of specific cytotoxic agents (e.g., cyclophosphamide) have both antiangiogenic and immune stimulatory effects. A mathematical model for angiogenic signaling formulated by Hahnfeldt et al. is combined with the classical equations for tumor immune system interactions by Stepanova to form a minimally parameterized model to capture these effects of low dose chemotherapy. The model exhibits bistable behavior with the existence of both benign and malignant locally asymptotically stable equilibrium points. In this paper, the transfer of states from the malignant into the benign regions is used as a motivation for the construction of an objective functional that induces this process and the analysis of the corresponding optimal control problem is initiated.
Oncogene-tumor suppressor gene feedback interactions and their control
Baltazar D. Aguda , Ricardo C.H. del Rosario and  Michael W.Y. Chan
2015, 12(6): 1277-1288 doi: 10.3934/mbe.2015.12.1277 +[Abstract](55) +[PDF](590.5KB)
We propose the hypothesis that for a particular type of cancer there exists a key pair of oncogene (OCG) and tumor suppressor gene (TSG) that is normally involved in strong stabilizing negative feedback loops (nFBLs) of molecular interactions, and it is these interactions that are sufficiently perturbed during cancer development. These nFBLs are thought to regulate oncogenic positive feedback loops (pFBLs) that are often required for the normal cellular functions of oncogenes. Examples given in this paper are the pairs of MYC and p53, KRAS and INK4A, and E2F1 and miR-17-92. We propose dynamical models of the aforementioned OCG-TSG interactions and derive stability conditions of the steady states in terms of strengths of cycles in the qualitative interaction network. Although these conditions are restricted to predictions of local stability, their simple linear expressions in terms of competing nFBLs and pFBLs make them intuitive and practical guides for experimentalists aiming to discover drug targets and stabilize cancer networks.
Algebraic and topological indices of molecular pathway networks in human cancers
Peter Hinow , Edward A. Rietman , Sara Ibrahim Omar and  Jack A. Tuszyński
2015, 12(6): 1289-1302 doi: 10.3934/mbe.2015.12.1289 +[Abstract](30) +[PDF](478.1KB)
Protein-protein interaction networks associated with diseases have gained prominence as an area of research. We investigate algebraic and topological indices for protein-protein interaction networks of 11 human cancers derived from the Kyoto Encyclopedia of Genes and Genomes (KEGG) database. We find a strong correlation between relative automorphism group sizes and topological network complexities on the one hand and five year survival probabilities on the other hand. Moreover, we identify several protein families (e.g. PIK, ITG, AKT families) that are repeated motifs in many of the cancer pathways. Interestingly, these sources of symmetry are often central rather than peripheral. Our results can aide in identification of promising targets for anti-cancer drugs. Beyond that, we provide a unifying framework to study protein-protein interaction networks of families of related diseases (e.g. neurodegenerative diseases, viral diseases, substance abuse disorders).
The performance of discrete models of low reynolds number swimmers
Qixuan Wang and  Hans G. Othmer
2015, 12(6): 1303-1320 doi: 10.3934/mbe.2015.12.1303 +[Abstract](106) +[PDF](1363.9KB)
Swimming by shape changes at low Reynolds number is widely used in biology and understanding how the performance of movement depends on the geometric pattern of shape changes is important to understand swimming of microorganisms and in designing low Reynolds number swimming models. The simplest models of shape changes are those that comprise a series of linked spheres that can change their separation and/or their size. Herein we compare the performance of three models in which these modes are used in different ways.
Models, measurement and inference in epithelial tissue dynamics
Oliver J. Maclaren , Helen M. Byrne , Alexander G. Fletcher and  Philip K. Maini
2015, 12(6): 1321-1340 doi: 10.3934/mbe.2015.12.1321 +[Abstract](98) +[PDF](515.2KB)
The majority of solid tumours arise in epithelia and therefore much research effort has gone into investigating the growth, renewal and regulation of these tissues. Here we review different mathematical and computational approaches that have been used to model epithelia. We compare different models and describe future challenges that need to be overcome in order to fully exploit new data which present, for the first time, the real possibility for detailed model validation and comparison.

2016  Impact Factor: 1.035




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