## Friday, June 28, 2013

### Quantitative Analysis of the Tumor/Metastasis System and its Optimal Therapeutic Control

This preprint presents a model which looks like a merge of two previous models on different biological scales: one is a tumour growth model by Hahnfeldt et al. the other is a model of metastatic spread in terms of a transport equation originally formulated by Iwata et al. With this new model the authors investigate the efficacy of different schedules for chemotherapy.

# Quantitative Analysis of the Tumor/Metastasis System and its Optimal Therapeutic Control

Sebastien Benzekry (CCSB, INRIA Bordeaux - Sud-Ouest), Dominique Barbolosi (CRO2), Assia Benabdallah (LATP), Florence Hubert (LATP), Philip Hahnfeldt (CCSB)
A mathematical model for time development of metastases and their distribution in size and carrying capacity is presented. The model is used to theoretically investigate anti-cancer therapies such as surgery and chemical treatments (cytotoxic or anti-angiogenic), in monotherapy or in combination. Quantification of the effect of surgery on the size distribution of metastatic colonies is derived. For systemic therapies, emphasis is placed on the differences between the treatment of an isolated lesion and a population of metastases. Combination therapy is addressed, in particular the problem of the drugs administration sequence. Theoretical optimal schedules are derived that show the superiority of a metronomic administration scheme (defined as a continuous administration of a given amount of drug spread during the whole therapeutic cycle) on a classical Maximum Tolerated Dose scheme (where the dose is given as a few concentrated administrations at the beginning of the cycle), for the total metastatic burden in the organism.

## Tuesday, June 25, 2013

### Maximum Tolerated Dose Versus Metronomic Scheduling in the Treatment of Metastatic Cancers

This paper was posted to the arxiv along with another paper, titled

The authors are clearly interested in the implications of optimal control in the metastatic setting. The paper below leaves me wondering from an evolutionary selection point of view, if metronomic therapy is actually increasing the probability of metastasis.

## Maximum Tolerated Dose Versus Metronomic Scheduling in the Treatment of Metastatic Cancers

Authors
Sébastien Benzekry, Philip Hahnfeldt

Abstract

Although optimal control theory has been used for the theoretical study of anticancerous drugs scheduling optimization,with the aim of reducing the primary tumor volume, the effect on metastases is often ignored. Here, we use a previously published model for metastatic development to define an optimal control problem at the scale of the entire organism of the patient. In silico study of the impact of different scheduling strategies for anti-angiogenic and cytotoxic agents (either in monotherapy or in combination) is performed to compare a low-dose, continuous, metronomic administration scheme with a more classical maximum tolerated dose schedule. Simulation results reveal differences between primary tumor reduction and control of metastases but overall suggest use of the metronomic protocol.

# Moderate stem cell telomere shortening rate postpones cancer onset in stochastic model

Mammalian cells are restricted from proliferating indefinitely. Telomeres at the end of each chromosome are shortened at cell division and, when they reach a critical length, the cell will enter permanent cell cycle arrest - a state known as senescence. This mechanism is thought to be tumor suppressing, as it helps prevent precancerous cells from dividing uncontrollably.
Stem cells express the enzyme telomerase, which elongates the telomeres, thereby postponing senescence. However, unlike germ cells and most types of cancer cells, stem cells only express telomerase at levels insufficient to fully maintain the length of their telomeres leading to a slow decline in proliferation potential. It is not yet fully understood how this decline influences the risk of cancer and the longevity of the organism. We here develop a stochastic model to explore the role of telomere dynamics in relation to both senescence and cancer. The model describes the accumulation of cancerous mutations in a multicellular organism and creates a coherent theoretical framework for interpreting the results of several recent experiments on telomerase regulation. We demonstrate that the longest average cancer free life span before cancer onset is obtained when stem cells start with relatively long telomeres that are shortened at a steady rate at cell division. Furthermore, the risk of cancer early in life can be reduced by having a short initial telomere length. Finally, our model suggests that evolution will favour a shorter than optimal average cancer free life span in order to postpone cancer onset until late in life.

# Phylogenetic quantification of intra-tumour heterogeneity

Background: Intra-tumour heterogeneity (ITH) is the result of ongoing evolutionary change within each cancer. The expansion of genetically distinct sub-clonal populations may explain the emergence of drug resistance and if so would have prognostic and predictive utility. However, methods for objectively quantifying ITH have been missing and are particularly difficult to establish in cancers where predominant copy number variation prevents accurate phylogenetic reconstruction owing to horizontal dependencies caused by long and cascading genomic rearrangements.
Results: To address these challenges we present MEDICC, a method for phylogenetic reconstruction and ITH quantification based on a Minimum Event Distance for Intra-tumour Copynumber Comparisons. Using a transducer-based pairwise comparison function we determine optimal phasing of major and minor alleles, as well as evolutionary distances between samples, and are able to reconstruct ancestral genomes. Rigorous simulations and an extensive clinical study show the power of our method, which outperforms state-of-the-art competitors in reconstruction accuracy and additionally allows unbiased numerical quantification of ITH.
Conclusions: Accurate quantification and evolutionary inference are essential to understand the functional consequences of ITH. The MEDICC algorithms are independent of the experimental techniques used and are applicable to both next-generation sequencing and array CGH data.

# Adaptation and learning of molecular networks as a description of cancer development at the systems-level: Potential use in anti-cancer therapies

There is a widening recognition that cancer cells are products of complex developmental processes. Carcinogenesis and metastasis formation are increasingly described as systems-level, network phenomena. Here we propose that malignant transformation is a two-phase process, where an initial increase of system plasticity is followed by a decrease of plasticity at late stages of carcinogenesis as a model of cellular learning. We describe the hallmarks of increased system plasticity of early, tumor initiating cells, such as increased noise, entropy, conformational and phenotypic plasticity, physical deformability, cell heterogeneity and network rearrangements. Finally, we argue that the large structural changes of molecular networks during cancer development necessitate a rather different targeting strategy in early and late phase of carcinogenesis. Plastic networks of early phase cancer development need a central hit, while rigid networks of late stage primary tumors or established metastases should be attacked by the network influence strategy, such as by edgetic, multi-target, or allo-network drugs. Cancer stem cells need special diagnosis and targeting, since their dormant and rapidly proliferating forms may have more rigid, or more plastic networks, respectively. The extremely high ability to change their rigidity/plasticity may be a key differentiating hallmark of cancer stem cells. The application of early stage-optimized anti-cancer drugs to late-stage patients may be a reason of many failures in anti-cancer therapies. Our hypotheses presented here underlie the need for patient-specific multi-target therapies applying the correct ratio of central hits and network influences -- in an optimized sequence.

## Thursday, June 13, 2013

### The time-evolution of DCIS size distributions with applications to breast cancer growth and progression

This paper looks at the growth dynamics of ductal carcinoma in situ (DCIS) in the breast, but instead of focusing on the dynamics of a single tumour the authors derive an equation for the size distribution of DCIS across an entire population. The equation for the size distribution has a stationary solution, and by comparing the analytical expression with data from mammographic screening the parameters of the growth model were estimated.

# The time-evolution of DCIS size distributions with applications to breast cancer growth and progression

Ductal carcinoma {\em in situ} (DCIS) lesions are non-invasive tumours of the breast which are thought to precede most invasive breast cancers (IBC). As individual DCIS lesions are initiated, grow and invade (i.e. become IBC) the size distribution of the DCIS lesions present in a given human population will evolve. We derive a differential equation governing this evolution and show, for given assumptions about growth and invasion, that there is a unique distribution which does not vary with time. Further, we show that any initial distribution converges to this stationary distribution exponentially quickly. It is therefore reasonable to assume that the stationary distribution governs the size of DCIS lesions in human populations which are relatively stable with respect to the determinants of breast cancer. Based on this assumption and the size data of 110 DCIS lesions detected in a mammographic screening program between 1993 and 2000, we produce maximum likelihood estimates for certain growth and invasion parameters. Assuming that DCIS size is proportional to a positive power $p$ of the time since tumour initiation, we estimate $p$ to be 0.50 with a 95% confidence interval of $(0.35, 0.71)$. Therefore we estimate that DCIS lesions follow a square-root growth law and hence that they grow rapidly when small and relatively slowly when large. Our approach and results should be useful for other mathematical studies of cancer, especially those investigating biological mechanisms of invasion.

# Symmetric vs asymmetric stem cell divisions: an adaptation against cancer?

Traditionally, it has been held that a central characteristic of stem cells is their ability to divide asymmetrically. Recent advances in inducible genetic labeling provided ample evidence that symmetric stem cell divisions play an important role in adult mammalian homeostasis. It is well understood that the two types of cell divisions differ in terms of the stem cells' flexibility to expand when needed. On the contrary, the implications of symmetric and asymmetric divisions for mutation accumulation are still poorly understood. In this paper we study a stochastic model of a renewing tissue, and address the optimization problem of tissue architecture in the context of mutant production. Specifically, we study the process of tumor suppressor gene inactivation which usually takes place as a sequence of two consecutive "hits", and which is one of the most common patterns in carcinogenesis. We compare and contrast symmetric and asymmetric (and mixed) stem cell divisions, and focus on the rate at which double-hit mutants are generated. It turns out that symmetrically-dividing cells generate such mutants at a rate which is significantly lower than that of asymmetrically-dividing cells. This result holds whether single-hit (intermediate) mutants are disadvantageous, neutral, or advantageous. It is also independent on whether the carcinogenic double-hit mutants are produced only among the stem cells or also among more differentiated cells. We argue that symmetric stem cell divisions in mammals could be an adaptation which helps delay the onset of cancers. We further investigate the question of the optimal fraction of stem cells in the tissue, and quantify the contribution of non-stem cells in mutant production. Our work provides a hypothesis to explain the observation that in mammalian cells, symmetric patterns of stem cell division seem to be very common.
Preprint: http://arxiv.org/abs/1305.0100

## Monday, June 10, 2013

### Evolution of intratumoral phenotypic heterogeneity: the role of trait inheritance

A new paper from +Jill Gallaher and +Alexander Anderson is out on the arXiv.  I asked Jill for a PLoS style 'author summary' in non-technical language, and here it is:

Author Summary:

A tumor can be thought of as an ecosystem, which critically means that we cannot just consider it as a collection of mutated cells. A tumor is more of a complex system of many interacting cellular and microenvironmental elements. There is variation among cells within the tumor, and with an increased proliferation capacity, there is competition for space, so evolution and selection occurs.  Because our current understanding at the genetic scale gives little information on translating to actual changes in cell behavior, we bypass the translation of genetics to behavior by focussing on the functional end result of the cell’s traits (phenotype) combined with the environmental influence of limited space, which will ultimately dictate tumor aggressiveness and treatability.

The evolution of the population depends on the way in which traits are passed on as cells divide. We investigate trait inheritance by building a cell based simulation in which individual cells with varied trait combinations compete for space over time. Specifically, we characterize cell behavior in terms of two traits: proliferation rate and migration speed. The mode in which these traits are inherited significantly affects the evolution, composition, and fitness of a tumor population. To investigate competition for space, we initiate the population as a tight cluster, representing a growing tumor mass, and as a dispersed population, representing a cell culture experiment. We find that the dispersed population has more space, less competition, and reduced selection.  With a growing cluster of cells, there is more competition and selection. But constraining the allowable trait combinations so that several phenotypes are equally fit reduces competition and leads to the coexistence of several phenotypes. In this case, local heterogeneity may be advantageous to maximize growth.

# Evolution of intratumoral phenotypic heterogeneity: the role of trait inheritance

A tumor can be thought of as an ecosystem, which critically means that we cannot just consider it as a collection of mutated cells but more as a complex system of many interacting cellular and microenvironmental elements. At its simplest, a growing tumor with increased proliferation capacity must compete for space as a limited resource. Hypercellularity leads to a contact-inhibited core with a competitive proliferating rim. Evolution and selection occurs, and an individual cell's capacity to survive and propagate is determined by its combination of traits and interaction with the environment. With heterogeneity in phenotypes, the clone that will dominate is not always obvious as there are both local interactions and global pressures. Several combinations of phenotypes can coexist, changing the fitness of the whole.
To understand some aspects of heterogeneity in a growing tumor we build an off-lattice agent based model consisting of individual cells with assigned trait values for proliferation and migration rates. We represent heterogeneity in these traits with frequency distributions and combinations of traits with density maps. How the distributions change over time is dependent on how traits are passed on to progeny cells, which is our main inquiry. We bypass the translation of genetics to behavior by focussing on the functional end result of inheritance of the phenotype combined with the environmental influence of limited space.

## Friday, June 7, 2013

### Cooperation and competition in the dynamics of tissue architecture during homeostasis and tumorigenesis

Interesting review about how game theory (including game theory on networks) can be used to study the interactions between cells in a homeostatic tissue (made of cooperators) and how defectors (cancer cells) can disrupt that homeostasis.

# Cooperation and competition in the dynamics of tissue architecture during homeostasis and tumorigenesis

Attila Csikász-Nagy, Luis M. Escudero, Martial Guillaud, Sean Sedwards, Buzz Baum, Matteo Cavaliere

Abstract: The construction of a network of cell-to-cell contacts makes it possible to characterize the patterns and spatial organisation of tissues. Such networks are highly dynamic, depending on the changes of the tissue architecture caused by cell division, death and migration. Local competitive and cooperative cell-to-cell interactions influence the choices cells make. We review the literature on quantitative data of epithelial tissue topology and present a dynamical network model that can be used to explore the evolutionary dynamics of a two dimensional tissue architecture with arbitrary cell-to-cell interactions. In particular, we show that various forms of experimentally observed types of interactions can be modelled using game theory. We discuss a model of cooperative and non-cooperative cell-to-cell communication that can capture the interplay between cellular competition and tissue dynamics. We conclude with an outlook on the possible uses of this approach in modelling tumorigenesis and tissue homeostasis.

## Thursday, June 6, 2013

### An apology

It came to our attention yesterday that we had inadvertently used an image that belongs to Kasia Rejniak in the background of our logo without her permission.

We apologize for this.  We are in the process of making a new logo now which will maintain the spirit of the message with some other art.  We would happily accept suggestions!

With apologies,

The Warburg's Lens team

## Wednesday, June 5, 2013

### A recent physical approach to angiogenesis modeling: mixture models

Angiogenesis modeling has generally tended towards (1) discrete models of individual sprout tip cells (cellular automata, Plank-Sleeman free-swimming models, etc.), or (2) continuum models of the evolving blood vessel density. At the Mathways into Cancer II workshop last week, I learned of a new approach using mixture (or phase field) models: computational voxels are treated as a conserved mixture of water, matrix, and endothelial cells. Applying conservation laws to each of these phases, plus an energy formulation of how the phases mix, leads to fourth-order Cahn-Hilliard equations for the mixture components. These require some fairly tricky numerics to solve correctly, but the results can be beautiful. And notably, it allows the inclusion of sophisticated cell and tissue mechanics.

Without further ado, I present two papers (by two different groups) on this new approach to angiogenesis modeling, combined as hybrid models with "traditional" sprout tip agents:
Capillary networks in tumor angiogenesis: From discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis
Guillermo Vilanova, Ignasi Colominas, Hector Gomez

Abstract: Tumor angiogenesis, the growth of new capillaries from preexisting ones promoted by the starvation and hypoxia of cancerous cell, creates complex biological patterns. These patterns are captured by a hybrid model that involves high-order partial differential equations coupled with mobile, agent-based components. The continuous equations of the model rely on the phase-field method to describe the intricate interfaces between the vasculature and the host tissue. The discrete equations are posed on a cellular scale and treat tip endothelial cells as mobile agents. Here, we put the model into a coherent mathematical and algorithmic framework and introduce a numerical method based on isogeometric analysis that couples the discrete and continuous descriptions of the theory. Using our algorithms, we perform numerical simulations that show the development of the vasculature around a tumor. The new method permitted us to perform a parametric study of the model. Furthermore, we investigate different initial configurations to study the growth of the new capillaries. The simulations illustrate the accuracy and efficiency of our numerical method and provide insight into the dynamics of the governing equations as well as into the underlying physical phenomenon.

(AFAIK) The first work using this approach was by Rui Travasso and colleagues in 2011, with a focus on the role of mechanistic modeling in driving biological hypotheses:
Tumor Angiogenesis and Vascular Patterning: A Mathematical Model
Rui D. M. Travasso et al.

Abstract: Understanding tumor induced angiogenesis is a challenging problem with important consequences for diagnosis and treatment of cancer. Recently, strong evidences suggest the dual role of endothelial cells on the migrating tips and on the proliferating body of blood vessels, in consonance with further events behind lumen formation and vascular patterning. In this paper we present a multi-scale phase-field model that combines the benefits of continuum physics description and the capability of tracking individual cells. The model allows us to discuss the role of the endothelial cells' chemotactic response and proliferation rate as key factors that tailor the neovascular network. Importantly, we also test the predictions of our theoretical model against relevant experimental approaches in mice that displayed distinctive vascular patterns. The model reproduces the in vivo patterns of newly formed vascular networks, providing quantitative and qualitative results for branch density and vessel diameter on the order of the ones measured experimentally in mouse retinas. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of different parameters in this process, hence underlining the necessary collaboration between mathematical modeling, in vivo imaging and molecular biology techniques to improve current diagnostic and therapeutic tools.

Open access: http://dx.doi.org/10.1371/journal.pone.0019989

# A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues

We model a general, hierarchically organized tissue by a multi compartment approach, allowing any number of mutations within a cell. We derive closed solutions for the deterministic clonal dynamics and the reproductive capacity of single clones. Our results hold for the average dynamics in a hierarchical tissue characterized by an arbitrary combination of proliferation parameters.