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# A Stochastic Model for the Normal Tissue Complication Probability (NTCP) in Radiation Treatment of Cancer

The normal tissue complication probability (NTCP) is a measure for the estimated side effects of a given radiation treatment schedule. Here we use a stochastic logistic birth death process to define an organ specific and patient specific NTCP. We emphasise an asymptotic simplification which relates the NTCP to the solution of a logistic differential equation. This framework allows for a direct use of the NTCP model in clinical practice. We formulate, but do not solve, related optimization problems.

link: http://arxiv.org/abs/1412.3441

# Model for Acid-Mediated Tumour Invasion with Chemotherapy Intervention I: Homogeneous Populations

The acid-mediation hypothesis, that is, the hypothesis that acid produced by tumours, as a result of aerobic glycolysis, provides a mechanism for invasion, has so far been considered as a relatively closed system. The focus has mainly been on the dynamics of the tumour, normal-tissue, acid and possibly some other bodily components, without considering the effect of an external intervention such as a cytotoxic treatment. This article aims to examine the effect that a cytotoxic treatment has on a tumour growing under the acid-mediation hypothesis by using a simple set of ordinary differential equations that consider the interaction between normal-tissue, tumour-tissue, acid and chemotherapy drug.

link: http://arxiv.org/abs/1412.0748

# Dynamics and bifurcations in a simple quasispecies model of tumorigenesis

Submitted on 24 Nov 2014

Cancer is a complex disease and thus is complicated to model. However, simple models that describe the main processes involved in tumoral dynamics, e.g., competition and mutation, can give us clues about cancer behaviour, at least qualitatively, also allowing us to make predictions. Here we analyze a simplified quasispecies mathematical model given by differential equations describing the time behaviour of tumor cells populations with different levels of genomic instability. We find the equilibrium points, also characterizing their stability and bifurcations focusing on replication and mutation rates. We identify a transcritical bifurcation at increasing mutation rates of the tumor cells population. Such a bifurcation involves an scenario with dominance of healthy cells and impairment of tumor populations. Finally, we characterize the transient times for this scenario, showing that a slight increase beyond the critical mutation rate may be enough to have a fast response towards the desired state (i.e., low tumor populations) during directed mutagenic therapies.

# Replicator Dynamics of of Cancer Stem Cell; Selection in the Presence of Differentiation and Plasticity

Stem cells have the potential to produce lineages of non-stem cell populations (differentiated cells) via a ubiquitous hierarchal division scheme. Differentiation of a stem cell into (partially) differentiated cells can happen either symmetrically or asymmetrically. The selection dynamics of a mutant cancer stem cell should be investigated in the light of a stem cell proliferation hierarchy and presence of a non-stem cell population. By constructing a three-compartment Moran-type model composed of normal stem cells, mutant (cancer) stem cells and differentiated cells, we derive the replicator dynamics of stem cell frequencies where asymmetric differentiation and differentiated cell death rates are included in the model. We determine how these new factors change the conditions for a successful mutant invasion and discuss the variation on the steady state fraction of the population as different model parameters are changed. By including the phenotypic plasticity/dedifferentiation, in which a progenitor/differentiated cell can transform back into a cancer stem cell, we show that the effective fitness of mutant stem cells is not only determined by their proliferation and death rates but also according to their dedifferentiation potential. By numerically solving the model we derive the phase diagram of the advantageous and disadvantageous phases of cancer stem cells in the space of proliferation and dedifferentiation potentials. The result shows that at high enough dedifferentiation rates even a previously disadvantageous mutant can take over the population of normal stem cells. This observation has implications in different areas of cancer research including experimental observations that imply metastatic cancer stem cell types might have lower proliferation potential than other stem cell phenotypes while showing much more phenotypic plasticity and can undergo clonal expansion.

link: http://arxiv.org/abs/1411.1399

# Modeling and simulation of a low grade urinary bladder carcinoma

In this work, we present a mathematical model of the initiation and progression of a low-grade urinary bladder carcinoma. We simulate the crucial processes involved in tumor growth, such as oxygen diffusion, carcinogen penetration, and angiogenesis, within the framework of the urothelial cell dynamics. The cell dynamics are modeled using the discrete technique of Cellular Automata, while the continuous processes of carcinogen penetration and oxygen diffusion are described by nonlinear diffusion-absorption equations. As the availability of oxygen is necessary for tumor progression, processes of oxygen transport to the tumor growth site seem most important. Our model yields a theoretical insight into the main stages of development and growth of urinary bladder carcinoma with emphasis on two most common types: bladder polyps and carcinoma {\it in situ}. Analysis of histological structure of bladder tumor is important to avoid misdiagnosis and wrong treatment and we expect our model to be a valuable tool in the prediction of tumor grade and progression patterns, based on the exposure to carcinogens and an oxygen dependent expression of genes promoting tumor growth. Our numerical simulations have good qualitative agreement with {\it in vivo} results reported in the corresponding medical literature.
 Comments: The paper has been withdrawn due to the disagreement with the journal Subjects: Quantitative Methods (q-bio.QM); Tissues and Organs (q-bio.TO) Cite as: arXiv:1310.3301 [q-bio.QM] (or arXiv:1310.3301v3 [q-bio.QM] for this version)link: http://arxiv.org/abs/1310.3301CAN THE AUTHORS ELABORATE ON THE WITHDRAWAL FROM THE JOURNAL?

# A pedagogical walkthrough of computational modeling and simulation of Wnt signaling pathway using static causal models in Matlab

A tutorial introduction to computational modeling of Wnt signaling pathway in a human colorectal cancer dataset using static Bayesian network models is provided. This work endeavours to expound in detail the simulation study in Matlab along with the code while explaining the concepts related to Bayesian networks. This is done in order to ease the understanding of beginner students and researchers in transition to computational signaling biology, who intend to work in the field of modeling of signaling pathways. The case study is based on the contents of the advance article by Sinha (2014) and takes the reader in a step by step process of how (1) the collection and the transformation of the available biological information from literature is done, (2) the integration of the heterogeneous data and prior biological knowledge in the network is achieved, (3) the simulation study is designed, (4) the hypothesis regarding a biological phenomena is transformed into computational framework, and (5) results and inferences drawn using d-connectivity/separability are reported. It is hoped that the walkthrough will aid biologists understand the design of the computational experiments using causal models. The manuscript finally ends with a programming assignment to help the readers get hands on experience of a perturbation project. Matlab code with dataset is made available under GNU GPL v3 license at google code project on https://code.google.com/p/static-bn-for-wnt-signaling-pathway

# Stochastic model for computer simulation of the number of cancer cells and lymphocytes in homogeneous sections of cancer tumors

We deal with a small enough tumor section to consider it homogeneous, such that populations of lymphocytes and cancer cells are independent of spatial coordinates. A stochastic model based in one step processes is developed to take into account natural birth and death rates. Other rates are also introduced to consider medical treatment: natural birth rate of lymphocytes and cancer cells; induced death rate of cancer cells due to self-competition, and other ones caused by the activated lymphocytes acting on cancer cells. Additionally, a death rate of cancer cells due to induced apoptosis is considered. Weakness due to the advance of sickness is considered by introducing a lymphocytes death rate proportional to proliferation of cancer cells.
Simulation is developed considering different combinations of the parameters and its values, so that several strategies are taken into account to study the effect of anti-angiogenic drugs as well the self-competition between cancer cells. Immune response, with the presence of a kind of specialized lymphocytes, is introduced such that they appear once cancer cells are detected. Induced apoptosis of cancer cells is introduced to model the action of several drugs under development right now. Besides, the model predicts the cancer relapse even from a very small number of cells. Simulation is done by using Gillespie algorithm.
link

# Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach

We consider quasi-stationary (travelling wave type) solutions to a nonlinear reaction-diffusion equation with arbitrary, autonomous coefficients, describing the evolution of glioblastomas, aggressive primary brain tumors that are characterized by extensive infiltration into the brain and are highly resistant to treatment. The second order nonlinear equation describing the glioblastoma growth through travelling waves can be reduced to a first order Abel type equation. By using the integrability conditions for the Abel equation several classes of exact travelling wave solutions of the general reaction-diffusion equation that describes glioblastoma growth are obtained, corresponding to different forms of the product of the diffusion and reaction functions. The solutions are obtained by using the Chiellini lemma and the Lemke transformation, respectively, and the corresponding equations represent generalizations of the classical Fisher--Kolmogorov equation. The biological implications of two classes of solutions are also investigated by using both numerical and semi-analytical methods for realistic values of the biological parameters.
link: http://arxiv.org/abs/1409.0605

# Computational Screening of Angiogenesis Model Variants Predicts that Differential Chemotaxis Helps Tip Cells Move to the Sprout Tip and Accelerates Sprouting

Angiogenesis involves the formation of new blood vessels by sprouting or splitting of existing blood vessels. During sprouting, a highly motile type of endothelial cell, called the tip cell, migrates from the blood vessels followed by stalk cells, an endothelial cell type that forms the body of the sprout. In vitro models and computational models can recapitulate much of the phenomenology of angiogenesis in absence of tip and stalk cell differentiation. Therefore it is unclear how the presence of tip cells contributes to angiogenesis. To get more insight into how tip cells contribute to angiogenesis, we extended an existing computational model of vascular network formation based on the cellular Potts model with tip and stalk differentiation, without making a priori assumptions about the specific rules that tip cells follow. We then screened a range of model variants, looking for rules that make tip cells (a) move to the sprout tip, and (b) change the morphology of the angiogenic networks. The screening predicted that if tip cells respond less effectively to an endothelial chemoattractant than stalk cells, they move to the tips of the sprouts, which impacts the morphology of the networks. A comparison of this model prediction with genes expressed differentially in tip and stalk cells revealed that the endothelial chemoattractant Apelin and its receptor APJ may match the model prediction. To test the model prediction we inhibited Apelin signaling in our model and in an in vitro model of angiogenic sprouting, and found that in both cases inhibition of Apelin or of its receptor APJ reduces sprouting. Based on the prediction of the computational model, we propose that the differential expression of Apelin and APJ yields a "self-generated" gradient mechanisms that accelerates the extension of the sprout.
link: http://arxiv.org/abs/1409.5895

# The degenerative evolution from multicellularity to unicellularity during cancer

Theoretical reasoning suggests that human cancer may result from knocking down the genetic constraints evolved for maintenance of the metazoan multicellularity, which, however, requires a critical test. Using xenograft-based experimental evolution we characterized for the first time the full life history from initiation to metastasis of a tumor at the genomic and transcriptomic levels, and observed metastasis-driving positive selection for generally loss-of-function mutations on a set of multicellularity-related genes, which is further supported by large-scale exome data of clinical tumor samples. Subsequent expression analysis revealed mainly expression down-regulation of multicellularity-related genes, which form an evolving expression profile approaching that of embryonic stem cells, the cell type with the most characteristics of unicellular life. The theoretical conjecture predicts that genes born at the emergence of metazoan multicellularity tend to be cancer drivers, which we validated using a rigorous phylostratigraphy analysis on the birth rate of genes annotated by Cancer Gene Census. Also, the number of loss-of-function tumor suppressors often predominates over activated oncogenes in a typical tumor of human patients. These data collectively suggest that, different from typical organismal evolution in which gain of new genes is the mainstream, cancer represents a loss-of-function-driven degenerative evolution back to the unicellular ground state. This cancer evolution model may explain the enormous tumoral genetic heterogeneity in the clinic, underlie how distant-organ metastases originate in primary tumors despite distinct environmental requirements, and hold implications for designing effective cancer therapy.

http://arxiv.org/abs/1408.3236

# Stress-Induced Mutagenesis and Complex Adaptation

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## Wednesday, July 16, 2014

### The Role of Migration in the Evolution of Phenotypic Switching

Not strictly cancer but the topic is very relevant to many of us trying to understand the emergence of invasiveness.

# The Role of Migration in the Evolution of Phenotypic Switching

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## Investigating the development of chemotherapeutic drug resistance in cancer: A multiscale computational study

By: Gibin G Powathil, Mark AJ Chaplain, Maciej Swat

Chemotherapy is one of the most important therapeutic options used to treat human cancers, either alone or in combination with radiation therapy and surgery. Recent studies have indicated that intra-tumoural heterogeneity has a significant role in driving resistance to chemotherapy in many human malignancies. Multiple factors including the internal cell-cycle dynamics and the external microenvironement contribute to the intra-tumoural heterogeneity. In this paper we present a hybrid, multiscale, individual-based mathematical model, incorporating internal cell-cycle dynamics and changes in oxygen concentration, to study the effects of delivery of several different chemotherapeutic drugs on the heterogeneous subpopulations of cancer cells with varying cell-cycle dynamics. The computational simulation results from the multiscale model are in good agreement with available experimental data and support the hypothesis that slow-cycling sub-populations of tumour cells within a growing tumour mass can induce drug resistance to chemotherapy and thus the use of conventional chemotherapy may actually result in the emergence of dominant, therapy-resistant, slow-cycling subpopulations of tumour cells. Our results indicate that the appearance of this chemotherapeutic resistance is mainly due to the inability of the administered drug to target all cancer cells irrespective of the stage in the cell-cycle they are in i.e. most chemotherapeutic drugs target cells in a particular phase/phases of the cell-cycle, and hence always spare some cancer cells that are not in the targeted cell-cycle phase/phases. The results also suggest that this cell-cycle-mediated drug resistance may be overcome by using multiple doses of cell-cycle, phase-specific chemotherapy that targets cells in all phases and its appropriate sequencing and scheduling.

## Tuesday, June 10, 2014

### Spatial evolutionary games with small selection coeﬃcients

Spatial evolutionary games with small selection coefficients

Rick Durrett May 5, 2014

Abstract
Here we will use results of Cox, Durrett, and Perkins [56] for voter model perturba- tions to study spatial evolutionary games on Zd, d 3 when the interaction kernel is finite range, symmetric, and has covariance matrix σ2I. The games we consider have payoff matrices of the form 1 + wG where 1 is matrix of all 1’s and w is small and positive. Since our population size N = , we call our selection small rather than weak which usually means w = O(1/N). We prove that the effect of space is equiv- alent to replacing the replicator ODE by a related PDE where the reaction term is the replicator equation for a game matrix with some of the entries changed. The first idea is well known in the theory of stochastic spatial processes [58, 16, 62, 63]. The second is inspired by work of Ohtsuki and Nowak [28] (for the pair approximation). A remarkable aspect of our result is that the modifications of the game matrix depend on the interaction kernel only through the values of two simple probabilities for an associated coalescing random walk.

link: http://www.math.duke.edu/~rtd/evog/spaceg.pdf

# Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth

Despite internal complexity, tumor growth kinetics follow relatively simple macroscopic laws that have been quantified by mathematical models. To resolve this further, quantitative and discriminant analyses were performed for the purpose of comparing alternative models for their abilities to describe and predict tumor growth. For this we used two in vivo experimental systems, an ectopic syngeneic tumor (Lewis lung carcinoma) and an orthotopically xenografted human breast carcinoma. The goals were threefold: to 1) determine a statistical model for description of the volume measurement error, 2) establish the descriptive power of each model, using several goodness-of-fit metrics and a study of parametric identifiability, and 3) assess the models ability to forecast future tumor growth.
Nine models were compared that included the exponential, power law, Gompertz and (generalized) logistic formalisms. The Gompertz and power law provided the most parsimonious and parametrically identifiable description of the lung data, whereas the breast data were best captured by the Gompertz and exponential-linear models. The latter also exhibited the highest predictive power for the breast tumor growth curves, with excellent prediction scores (greater than 80%) extending out as far as 12 days. In contrast, for the lung data, none of the models were able to achieve substantial prediction rates (greater than 70%) further than the next day data point. In this context, adjunction of a priori information on the parameter distribution led to considerable improvement of predictions.
These results not only have important implications for biological theories of tumor growth and the use of mathematical modeling in preclinical anti-cancer drug investigations, but also may assist in defining how mathematical models could serve as potential prognostic tools in the clinical setting.

link: http://arxiv.org/abs/1406.1446

# Bridging scales in cancer progression: Mapping genotype to phenotype using neural networks

In this review we summarize our recent efforts in trying to understand the role of heterogeneity in cancer progression by using neural networks to characterise different aspects of the mapping from a cancer cells genotype and environment to its phenotype. Our central premise is that cancer is an evolving system subject to mutation and selection, and the primary conduit for these processes to occur is the cancer cell whose behaviour is regulated on multiple biological scales. The selection pressure is mainly driven by the microenvironment that the tumour is growing in and this acts directly upon the cell phenotype. In turn, the phenotype is driven by the intracellular pathways that are regulated by the genotype. Integrating all of these processes is a massive undertaking and requires bridging many biological scales (i.e. genotype, pathway, phenotype and environment) that we will only scratch the surface of in this review. We will focus on models that use neural networks as a means of connecting these different biological scales, since they allow us to easily create heterogeneity for selection to act upon and importantly this heterogeneity can be implemented at different biological scales. More specifically, we consider three different neural networks that bridge different aspects of these scales and the dialogue with the micro-environment, (i) the impact of the micro-environment on evolutionary dynamics, (ii) the mapping from genotype to phenotype under drug-induced perturbations and (iii) pathway activity in both normal and cancer cells under different micro-environmental conditions.

link: http://arxiv.org/abs/1404.7108

# Niche inheritance: a cooperative pathway to enhance cancer cell fitness though ecosystem engineering

Cancer cells can be described as an invasive species that is able to establish itself in a new environment. The concept of niche construction can be utilized to describe the process by which cancer cells terraform their environment, thereby engineering an ecosystem that promotes the genetic fitness of the species. Ecological dispersion theory can then be utilized to describe and model the steps and barriers involved in a successful diaspora as the cancer cells leave the original host organ and migrate to new host organs to successfully establish a new metastatic community. These ecological concepts can be further utilized to define new diagnostic and therapeutic areas for lethal cancers.

link: http://arxiv.org/abs/1403.7413

## Thursday, March 13, 2014

### A continuous model of osteocyte generation in bone matrix

A continuous model of osteocyte generation in bone matrix

Author: P R Buenzli

Abstract:

The formation of new bone involves both the deposition of bone matrix by cells called osteoblasts, and the formation of a network of cells embedded within the bone matrix, called osteocytes. Osteocytes derive from osteoblasts that become buried in bone matrix during bone deposition. There has been a growing interest in osteocytes in recent years with the realisation that these cells are essential to the detection of micro-damage in bone, and that they participate in the orchestration of local bone renewal. However, the generation of osteocytes is a complex process that remains incompletely understood. Whilst osteoblast burial determines the density of osteocytes, the expanding network of osteocytes regulates in turn osteoblast activity and osteoblast burial through their interconnected cell processes. In this contribution, a spatiotemporal continuous model is proposed to investigate the osteoblast-to- osteocyte transition. The model elucidates the interplays between matrix secretory rate, rate of entrapment, and curvature of the bone substrate in determining the density of osteocytes in the new bone matrix. We find that the density of osteocytes generated at the moving deposition front depends solely on the ratio of the instantaneous burial rate and matrix secretory rate. It is remarkably independent of osteoblast density and substrate curvature. This mathematical result is used with experimental measurements of osteocyte lacuna distributions in a human cortical bone sample to determine for the first time the rate of burial of osteoblasts in bone matrix. Our results suggest that in the bone specimen analysed: (i) burial rate decreases during osteonal infilling, and (ii) the control of osteoblast burial by osteocytes is likely to emanate as a collective signal from a large group of osteocytes, rather than from the osteocytes closest to the bone deposition front.

# A tug-of-war between driver and passenger mutations in cancer and other adaptive processes

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## Abstract

Cancer progression is an example of a rapid adaptive process where evolving new traits is essential for survival and requires a high mutation rate. Precancerous cells acquire a few key mutations that drive rapid population growth and carcinogenesis. Cancer genomics demonstrates that these few ‘driver’ mutations occur alongside thousands of random ‘passenger’ mutations––a natural consequence of cancer's elevated mutation rate. Some passengers can be deleterious to cancer cells, yet have been largely ignored in cancer research. In population genetics, however, the accumulation of mildly deleterious mutations has been shown to cause population meltdown. Here we develop a stochastic population model where beneficial drivers engage in a tug-of-war with frequent mildly deleterious passengers. These passengers present a barrier to cancer progression that is described by a critical population size, below which most lesions fail to progress, and a critical mutation rate, above which cancers meltdown. We find support for the model in cancer age-incidence and cancer genomics data that also allow us to estimate the fitness advantage of drivers and fitness costs of passengers. We identify two regimes of adaptive evolutionary dynamics and use these regimes to rationalize successes and failures of different treatment strategies. We find that a tumor’s load of deleterious passengers can explain previously paradoxical treatment outcomes and suggest that it could potentially serve as a biomarker of response to mutagenic therapies. The collective deleterious effect of passengers is currently an unexploited therapeutic target. We discuss how their effects might be exacerbated by both current and future therapies.

# Cancer-driven dynamics of immune cells in a microfluidic environment

Scope of the present work is to frame into a rigorous, quantitative scaffold - stemmed from stochastic process theory - two sets of experiments designed to infer the spontaneous organization of leukocytes against cancer cells, namely mice splenocytes vs. B16 mouse tumor cells, and embedded in an "ad hoc" microfluidic environment developed on a LabOnChip technology. In the former, splenocytes from knocked out (KO) mice engineered to silence the transcription factor IRF-8, crucial for the development and function of several immune populations, were used. In this case lymphocytes and cancer cells exhibited a poor reciprocal exchange, resulting in the inability of coordinating or mounting an effective immune response against melanoma. In the second class of tests, wild type (WT) splenocytes were able to interact with and to coordinate a response against the tumor cells through physical interaction. The environment where cells moved was built of by two different chambers, containing respectively melanoma cells and splenocytes, connected by capillary migration channels allowing leucocytes to migrate from their chamber toward the melanoma one. We collected and analyzed data on the motility of the cells and found that the first ensemble of IRF-8 KO cells performed pure uncorrelated random walks, while WT splenocytes were able to make singular drifted random walks, that, averaged over the ensemble of cells, collapsed on a straight ballistic motion for the system as a whole. At a finer level of investigation, we found that IRF-8 KO splenocytes moved rather uniformly since their step lengths were exponentially distributed, while WT counterpart displayed a qualitatively broader motion as their step lengths along the direction of the melanoma were log-normally distributed.
http://arxiv.org/abs/1402.0451

# Evolutionary dynamics of shared niche construction

Many species engage in niche construction that ultimately leads to an increase in the carrying capacity of the population. We have investigated how the specificity of this behaviour affects evolutionary dynamics using a set of coupled logistic equations, where the carrying capacity of each genotype consists of two components: an intrinsic part and a contribution from all genotypes present in the population. The relative contribution of the two components is controlled by a specificity parameter $\gamma$, and we show that the ability of a mutant to invade a resident population depends strongly on this parameter. When the carrying capacity is intrinsic, selection is almost exclusively for mutants with higher carrying capacity, while a shared carrying capacity yields selection purely on growth rate. This result has important implications for our understanding of niche construction, in particular the evolutionary dynamics of tumor growth.
on the arXiv:  http://arxiv.org/abs/1402.0757
or the bioRxiv: http://www.biorxiv.org/content/early/2014/02/05/002378

# Mesoscopic and continuum modelling of angiogenesis

Angiogenesis is the formation of new blood vessels from pre-existing ones in response to chemical signals secreted by, for example, a wound or a tumour. In this paper, we propose a mesoscopic lattice-based model of angiogenesis, in which processes that include proliferation and cell movement are considered as stochastic events. By studying the dependence of the model on the lattice spacing and the number of cells involved, we are able to derive the deterministic continuum limit of our equations and compare it to similar existing models of angiogenesis. We further identify conditions under which the use of continuum models is justified, and others for which stochastic or discrete effects dominate. We also compare different stochastic models for the movement of endothelial tip cells which have the same macroscopic, deterministic behaviour, but lead to markedly different behaviour in terms of production of new vessel cells.

http://arxiv.org/abs/1401.5701

# Delay effects in the response of low grade gliomas to radiotherapy: A mathematical model and its therapeutical implications

Low grade gliomas (LGGs) are a group of primary brain tumors usually encountered in young patient populations. These tumors represent a difficult challenge because many patients survive a decade or more and may be at a higher risk for treatment-related complications. Specifically, radiation therapy is known to have a relevant effect on survival but in many cases it can be deferred to avoid side effects while maintaining its beneficial effect. However, a subset of low-grade gliomas manifests more aggressive clinical behavior and requires earlier intervention. Moreover, the effectiveness of radiotherapy depends on the tumor characteristics. Recently Pallud et al., [Neuro-oncology, 14(4):1-10, 2012], studied patients with LGGs treated with radiation therapy as a first line therapy. and found the counterintuitive result that tumors with a fast response to the therapy had a worse prognosis than those responding late. In this paper we construct a mathematical model describing the basic facts of glioma progression and response to radiotherapy. The model provides also an explanation to the observations of Pallud et al. Using the model we propose radiation fractionation schemes that might be therapeutically useful by helping to evaluate the tumor malignancy while at the same time reducing the toxicity associated to the treatment.

http://arxiv.org/abs/1401.2603

# Combined therapies of antithrombotics and antioxidants delay in silico brain tumor progression

Glioblastoma multiforme, the most frequent type of primary brain tumor, is a rapidly evolving and spatially heterogeneous high-grade astrocytoma that presents areas of necrosis, hypercellularity and microvascular hyperplasia. The aberrant vasculature leads to hypoxic areas and results in an increase of the oxidative stress selecting for more invasive tumor cell phenotypes. In our study we assay in silico different therapeutic approaches which combine antithrombotics, antioxidants and standard radiotherapy. To do so, we have developed a biocomputational model of glioblastoma multiforme that incorporates the spatio-temporal interplay among two glioma cell phenotypes corresponding to oxygenated and hypoxic cells, a necrotic core and the local vasculature whose response evolves with tumor progression. Our numerical simulations predict that suitable combinations of antithrombotics and antioxidants may diminish, in a synergetic way, oxidative stress and the subsequent hypoxic response. This novel therapeutical strategy, with potentially low or no toxicity, might reduce tumor invasion and further sensitize glioblastoma multiforme to conventional radiotherapy or other cytotoxic agents, hopefully increasing median patient overall survival time.

http://arxiv.org/abs/1401.2397

# Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors

Histopathological evidence supports the idea that the emergence of phenotypic heterogeneity and resistance to cytotoxic drugs can be considered as a process of adaptation, or evolution, in tumor cell populations. In this framework, can we explain intra-tumor heterogeneity in terms of cell adaptation to local conditions? How do anti-cancer therapies affect the outcome of cell competition for nutrients within solid tumors? Can we overcome the emergence of resistance and favor the eradication of cancer cells by using combination therapies? Bearing these questions in mind, we develop a model describing cell dynamics inside a tumor spheroid under the effects of cytotoxic and cytostatic drugs. Cancer cells are assumed to be structured as a population by two real variables standing for space position and the expression level of a cytotoxic resistant phenotype. The model takes explicitly into account the dynamics of resources and anti-cancer drugs as well as their interactions with the cell population under treatment. We analyze the effects of space structure and combination therapies on phenotypic heterogeneity and chemotherapeutic resistance. Furthermore, we study the efficacy of combined therapy protocols based on constant infusion and/or bang-bang delivery of cytotoxic and cytostatic drugs.

http://arxiv.org/abs/1312.6237