### Abstracts

**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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