April 22, 2018

Workshop on Mathematical Oncology
April 23-24, 2004


John A. Adam, Old Dominion University
Developing Mathematical Models in Cancer Biology: Some Personal Reflections
As more becomes known about the nature of cancer and other diseases in the light of recent advances in molecular immunology and genomics, the potential for complex mathematical models increases. There are several questions that come to mind when recognizing that, at a basic level, "Biology = Information". Ultimately, will such models be as complex as the phenomena they seek to describe? What is the relationship between "description" and "explanation" in these models? Does there come a point when another approach to viewing the problem is necessary? If so, how can this be accomplished? Are simplified "toy" models (or mathematical metaphors?) ever useful? What about their predictive capabilities? I am better at posing questions than answering them…

In this talk I will recount some of the pitfalls I have encountered, together with some of the frustrations and pleasures I have experienced in attempting to build some simple mathematical models of prevascular tumor growth, chemotherapy, immune system response, and (time permitting), wound healing in bone. In so doing I will attempt to address aspects of the nature, philosophy and methodology of mathematical modeling, at least as it applies to the above topics. I will also discuss the kind of limitations (other than basic stupidity) and attitudes that have prevented me from taking these models to the "next level", in the hope that this will help others to avoid the kinds of trap I fell into while (radically) changing research direction twelve years or so after completing my Ph.D in theoretical astrophysics.

David Hogg, University of Toronto
Developmental second hits and the concept of a mutation field
Knudson's two hit paradigm has driven research in the field of cancer genetics for over thirty years. This model explains many of the molecular and clinical observations in sporadic and familial cancers. However, it fails to predict the phenotypic consequences of second hits that occur during the development of an organism. In familial glomuvenous malformations (GVM), loss of the wild type FAP68 allele may produce variable numbers of discrete or clustered lesions. Herein, we develop a mathematical model in which stochastic second hits during different stages of development predict differing spatial distributions of vascular malformations. More generally, this model should help describe the phenotypic effects of mutations in cancer stem cells.

Trachette Jackson, University of Michigan
Mathematical Models of Traditional and Targeted Chemotherapeutic Approaches.
In the first part of this talk I will present a mathematical model of a two-step approach to cancer chemotherapy involving the use of targeted monoclonal antibody-enzyme conjugates for the selective activation of anti-cancer prodrugs. This mathematical formulation is used to isolate critical parameters for improving the therapeutic index of this treatment strategy. In the second part of the talk, I will present a mathematical model that addresses the issue of tumor response to traditional and targeted chemotherapeutic approaches. An important feature of this model formulation is the characterization of the different mechanisms of action by which anti-cancer agents can initiate the cell death cascade. This model is used to predict the long-term response of tumors to repeated rounds of therapy.

Igor Jurisica, Ontario Cancer Institute, PMH
Integrative approach to molecular medicine
Our goal is to understand disease such as cancer at molecular level to develop early detection methods, accurate prognosis and effective therapies. Addressing these important clinical questions requires systematic knowledge management and integrative analysis of large volumes of information.

We can increase our understanding of the disease origin and tumorigenesis by integrating existing large scale genomic and proteomic data sets. This requires new analysis methods to combine, consolidate and interpret heterogeneous data. No single database or algorithm will be successful at solving these complex analytical problems. Thus, we need to integrate different tools and approaches, multiple sources of single type of data, and diverse data types.

We have developed several algorithms for supervised and unsupervised analysis of gene/protein expression profiles; e.g., Binary Tree- Structured Vector Quantization (BTSVQ), case-based reasoning, probabilistic iterative clustering, and association mining. We have also introduced a comprehensive approach for the analysis of large protein interaction networks to support functional analysis and hypothesis generation. By combining results from gene/protein expression data in the context of protein-protein interactions we can visualize, integrate and comprehend these complex datasets in a functional context. This approach will allow us to identify markers reflective of altered pathways.

Philip Maini, University of Oxford
Modelling aspects of vascular cancer
The modelling of cancer provides an enormous mathematical challenge because of its inherent multi-scale nature. For example, in vascular tumours, nutrient is transported by the vascular system, which operates on a tissue level. However, it effects processes occuring on a molecular level. Molecular and intra-cellular events in turn effect the vascular network and therefore the nutrient dynamics. Our modelling approach is to model, using partial differential equations, processes on the tissue level and couple these to the intercellular events (modelled by ordinary differential equations) via cells modelled as automaton units. Thusfar, within this framework we have modelled structural adaptation at the vessel level and we have modelled the cell cycle in order to account for the effects of p27 during hypoxia. These preliminary results will be presented.

Lance L. Munn, E.L. Steele Laboratory for Tumor Biology, Department of Radiation Oncology, Harvard Medical School
The interaction of circulating cells with tumors: pharmacokinetic and fluid dynamic models
Blood-borne cells are involved in both tumor angiogenesis and immune responses. Circulating cells with the potential to differentiate into mature endothelial cells, or endothelial progenitor cells, have recently been shown to contribute to tumor angiogenesis. This modifies the traditional view of angiogenesis in which new vessels form from proliferation and migration of endothelial cells already residing in and around the tumor. Unfortunately, little is known about the relative contributions of EPCs and resident endothelial cells. To address this question, we developed a pharmacokinetic mathematical model that quantifies the incorporation of both resident endothelial cells and circulating EPCs in new vessels. These two mechanisms interact through various growth factors, inhibitors, and their receptors, and ultimately increase vascular density and permit tumor growth. The model predicts that endothelial progenitor cells have a significant impact on tumor growth and angiogenesis, and that this contribution is mediated primarily by their localization in the tumor, not by their proliferation, during the early stages of tumor growth.
To accurately predict overall incorporation rates of circulating cells, we need a better understanding of the fluid dynamics and adhesion mechanisms. To this end, we have developed another model, based on a lattice-Boltzmann approach, to analyze the mechanical interactions that result in cell rolling and adhesion. Although specific adhesion mechanisms are involved in interactions with the endothelium, adhesion patterns in vivo suggest other rheological mechanisms also play a role. To demonstrate this, we analyzed the interactions of red and white blood cells as they flow from a capillary into a postcapillary venule using the lattice Boltzmann model. Our results show that capillary:postcapillary venule diameter ratio, RBC configuration and RBC shape are critical determinants of the initiation of cell rolling in postcapillary venules. This may be important for immune evasion by tumors, which have unique and abnormal vasculature.

Kristin R. Swanson, University of Washington School of Medicine
Insights into the Behavior of Gliomas Provided by Quantitative Modeling
Gliomas account for over half of all primary brain tumors and have been studied extensively for decades. Even with increasingly sophisticated medical imaging technologies, gliomas remain uniformly fatal lesions. A significant gap remains between the goal of designing effective therapy and the present understanding of the dynamics of glioma progression. It has become increasingly clear that, along with the proliferative potential of these neoplasms, it is the subclinically diffuse invasion of gliomas that most contributes to their resistance to treatment. That is, the inevitable recurrence of these tumors is the result of diffusely invaded but practically invisible tumor cells peripheral to the abnormal signal on medical imaging and to the limits of surgical, radiological and chemical treatments.

In this presentation, I will demonstrate how quantitative modeling can not only shed light on the spatio-temporal growth of gliomas but also can have specific clinical application in real patients. Integration of our quantitative model with the T1-weighted and T2-weighted MR imaging characteristics of gliomas can provide estimates of the extent of invasion of glioma cells peripheral to the imaging abnormality. Further model analysis reveals remarkable concordance with patient survival rates. In summary, although current imaging techniques remain woefully inadequate in accurately resolving the true extent of gliomas, quantitative modeling provides a new approach for the dynamic assessment of real patients and helps direct the way to novel therapeutic approaches.

Howard Thames, University of Texas
Cluster Models of Dose-Volume Effects in Radiotherapy
Joint work with Ming Zhang, Susan L. Tucker, H. Helen Liu, Lei Dong, and Radhe Mohan. Radiotherapy treatment optimization is based on algorithms that maximize tumor cure probability under constraints determined by normal-tissue complication probability (NTCP). "Cluster models" comprise a class of NTCP models where both the number and the spatial location of radiation-sterilized functional subunits (FSUs) play a role in defining complication probability. These appear in mathematics under the guise "percolation models", which arose in the 1950s to describe complex interacting random systems, including such widely dissimilar disordered media as ferromagnets, epidemics, and fires in orchards. We focus on the distribution of maximum-sized cluster of sterilized FSUs, and since percolation theory provides only asymptotic estimates of cluster size, our estimates are based on computer simulation. NTCP is defined as the probability that the maximum cluster size exceeds a threshold. For fitting purposes the simplest version has four parameters, n (size of the lattice), t (threshold), and two parameters describing the dose-density relationship and which include dose-rate and fractionation effects. Cluster models show a volume effect, as increasing effect for constant dose when the volume increases or constant effect when the dose is reduced with increasing volume. In one dimension the mean largest cluster size and the variance of largest cluster size increase approximately exponentially with increasing density of sterilized FSUs, while in two dimensions a similar exponential increase gives way to "saturation" phenomena whose behavior depends on the local connectivity imposed on clusters. The density at which saturation occurs is known in percolation theory (where connectivity is limited to what we call 1-connectivity) as the "percolation" probability, and is used to model e.g. a phase transition in statistical physics. Cluster models give similar results to existing models when tissues are irradiated uniformly. With inhomogeneous dose distributions on the other hand, a higher NTCP results when hot spots are contiguous (clustered) than when they are dispersed. These findings have implications for treatment optimization with the newest technologies, such as intensity-modulated radiotherapy.

Marco Zaider, Department of Medical Physics, Memorial Sloan-Kettering Cancer Center
On the question of the distribution of the number of clonogens surviving radiation therapy
The probability of tumor cure (tumor control probability, TCP) is commonly defined as the probability of no clonogenic tumor cells surviving by the end of the treatment. I shall describe a formula for the exact distribution of the number of cells surviving radiotherapy and argue that it provides mechanistic motivation for ab initio parametric estimation of survival models designed to analyze data on the efficacy of radiation therapy. As well, I shall describe its application to the calculation of TCP for prostate cancer based on a study of 1100 patients with clinically localized prostate cancer who were treated with 3D conformal radiation therapy at our institution.

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