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Modelling Cheyne-Stokes Respiration and other aspects of the control of respiration
by
Alona Ben-Tal
Massey University
Cheyne-Stokes Respiration is a form of periodic breathing where a person experiences cycles of increasing followed by decreasing ventilation, followed by periods of breath-holding. To study this puzzling phenomenon and other aspects of the control of respiration a mathematical model has been developed. The model integrates a reduced representation of the brainstem respiratory neural controller together with peripheral gas exchange and transport. Some features of experimental data are captured by the model and new predictions are made.
Date received: October 29, 2007
Cellular Automata Model of Radiation Therapy in Cervical Cancer
by
Robert Donnelly
Pomona College, Claremont, CA
Coauthors: K. Belsky, H. Ueda, A. Radunskaya, L. dePillis
Spatial interactions and the local chemical environment can play a major role both in the growth of a tumor and its resistance to radiation treatment. We propose a cellular automata (CA) model of radiation therapy in early cervical cancer. This model not only incorporates cellular metabolism and ATP production as functions of glucose, oxygen, and pH levels, but also models diffusion of these nutrients with a modified random walk. In particular, since tissue oxygenation plays a major role in the success of radiation therapy in solid tumors, we have included realistic determination of oxygen levels and the formation of a hypoxic core. Radiation damage is determined using an empirically-supported modified linear-quadratic (LQ) model. Our model can simulate fractionated doses of both external beam radiotherapy and brachytherapy, similar to in vivo treatments described in medical literature. Better understanding the interactions between a tumor and its environment may enhance not only our understanding of tumor growth but also allow us to better predict the effect of radiation therapy on a given tumor. Successful modeling of the effects of radiation therapy on tumor cells and normal cells may prove helpful in optimizing radiation treatment protocols to minimize collateral damage to healthy cells while still effectively treating the cancer.
Date received: November 27, 2007
Mathematical Modeling of GnRH neurons in the Rat Brain
by
Wen Duan
University of Auckland
Mathematical modeling of GnRH neurons in the rat brain. Some biology background and the mathematical model I am using will be introduced.
Date received: October 10, 2007
Optimal sampling for identification of models of cell signaling pathways
by
Krzysztof Fujarewicz
Silesian University of Technology, Poland
Modeling of cell signaling pathways attracted a lot of interest in recent years. Such models let scientists to understand mechanisms governing the cell functioning which plays a crucial role in many areas, for example in new drug development. To obtain a mathematical model that behaves similarly to observed biological process the estimation of model’s parameters is required. In case of cell signaling pathways appropriate measurements, for example DNA microarrays or different blotting techniques, are relatively expensive. Hence it is very important to choose right times of measurements in order to obtain low variances of estimates of parameters. This problem is somehow similar to estimation of parameters in pharmacokinetics. The classical approach is to use the Fisher information matrix (FIM), which inverse, under some assumptions, is a lower bound for the covariance matrix of parameter’s estimates. One of possible approach to sampling schedule optimization is to maximize the determinant of FIM. It is usually performed using any non-gradient method.
We present formulas for calculation of the gradient of FIM in the space of sampling times and we propose the gradient-based optimization approach.
Date received: November 6, 2007
TGF - A Renal Feedback Mechanism
by
Scott Graybill
Mathematics and Statistics Department, University of Canterbury, New Zealand
Coauthors: Alex James (University of Canterbury)
Mike Plank (University of Canterbury)
Tim David (University of Canterbury)
Zoltán Endre (Christchurch School of Medicine)
The tubulo-glomerular feedback (TGF) mechanism is one of two widely recognised feedback mechanisms in the kidney. TGF acts to maintain a constant blood flow to the organ despite fluctuations in blood pressure. Sustained oscillations in flow, pressure and salt concentration, that are attributed to the TGF mechanism, are observed in vivo. A physiologically realistic TGF model that captures these dynamics will be presented.
Date received: October 30, 2007
Complex oscillations in mathematical models of calcium dynamics
by
Emily Harvey
University of Auckland
The dynamics of calcium (Ca2+) is of interest as it is known to play a crucial role in many types of cellular functioning. A common feature of mathematical models of intracellular Ca2+ dynamics are that they have some variables that evolve much slower than others. In this talk I will demonstrate the presence of complicated oscillatory patterns known as mixed-mode oscillations (MMOs) in a few key models of intracellular Ca2+ dynamics. I will then show how these MMOs can arise due to the presence of slower timescales in the models and the existence of special solutions called canards.
Date received: October 30, 2007
A Mathematical Model Quantifies Proliferation and Motility Effects of TGF-b on Cancer Cells
by
Peter Hinow
Institute for Mathematics and its Applications, University of Minnesota
Coauthors: Shizhen Emily Wang, Nicole Bryce (Department of Cancer Biology, Vanderbilt University), Glenn F. Webb (Department of Mathematics, Vanderbilt University)
Transforming growth factor (TGF) b is a signaling molecule involved in a variety of cellular processes including growth, differentiation, apoptosis and cell motility. While TGF-b slows proliferation of certain cell types it also increases their motility and may decrease cell-cell adhesion. Thus, it has properties of both a tumor suppressor and a tumor promoting factor. We have carried out experiments to quantify cell motility and growth in presence of TGF-b and use a version of the classical Fisher-Kolmogorov equation to interpret the experimental findings. We find that TGF-b increases the tendency of individual cells and cell clusters to move randomly, while simultaneously diminishing overall population growth. Our model, which can also be adopted to simulate other growth-regulating signals, will provide a unique insight into the TGF-b function in both normal and cancer cells, and further understanding on targeted therapeutic strategies that aim at interfering with TGF-b signaling.
Date received: September 26, 2007
Chronological calculus and nonlinear feedback loops
by
Matthias Kawski
Arizona State University
Many models in biomedicine involve feedback loops and nonlinearly interacting dynamics. Often it can be advantageous to consider these as systems made up of collections of interacting sub-systems. Such splitting may be based on physical characteristics, or they may be abstract mathematical factorizations.
Commonly, the individual subsystems are comparatively straightforward to analyze, but the nonlinear, generally noncommuting effects of the subsystems on each other present challenges for the analysis of the combined system.
We present tools from the chronological calculus and recent combinatorial simplifications that facilitate the analysis and design and control of such composite systems that involve generally noncommuting nonlinear interactions.
Date received: October 30, 2007
Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities
by
Carlo Laing
Massey University, Auckland
Coauthors: Markus Owen and Steve Coombes, University of Nottingham, UK
We consider instabilities of localised solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilise spatially localised solutions. For a scalar model with Heaviside firing rate function we calculate symmetric one-bump and ring solutions explicitly and use an Evans function approach to predict the point of instability, and the shape of the dominant growing modes. Our predictions are in excellent agreement with direct numerical simulations.
With the addition of spike-frequency adaptation, numerical simulations of the resulting vector model show that it is possible for structures without rotational symmetry, and in particular multi-bumps, to undergo an instability to a rotating wave. We use a general argument, valid for smooth firing rate functions, to establish the conditions necessary to generate such a rotational instability. Numerical continuation of the rotating wave is used to quantify the emergent angular velocity as a bifurcation parameter is varied. Wave stability is found via the numerical evaluation of an associated eigenvalue problem.
Date received: October 8, 2007
Optimal and Suboptimal Protocols for a Class of Mathematical Models of Tumor Growth under Angiogenic Inhibitors
by
Urszula Ledzewicz
Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, USA
Coauthors: Heinz Schättler, Department of Electrical and Systems Engineering, Washington University, St. Louis, USA
Tumor anti-angiogenesis is a novel medical approach to cancer treatment that aims at preventing the development of the blood vessel network a tumor needs for growth. In this talk we shall show how tools from optimal control theory can be used to analyze a class of mathematical models for tumor anti-angiogenesis based on a paper by Hahnfeldt et al. (Cancer Research, 59, 1999). In these models the state of the system represents the primary tumor volume and the carrying capacity of the vasculature related to the endothelial cells. The nonlinear dynamics models how control functions representing angiogenic inhibitors effect the growths of these variables. The objective is to minimize the tumor volume with a fixed total amount of inhibitors.
In the talk we shall present a full theoretical solution to the problem in terms of a synthesis of optimal controls and trajectories. Using tools of geometric control theory (e.g., Lie bracket computations), analytic formulas for the theoretically optimal solutions will be given. Optimal controls are concatenations of bang-bang controls (representing therapies of full dose with rest periods) and singular controls (therapies with specific time-varying partial doses). Singular controls, however, are of feedback type and as such do not lead to implementable therapy protocols. Properties of the dynamics and knowledge of the theoretically optimal solution are used to formulate practically realizable suboptimal protocols and evaluate their efficiency. Specifically, for the original model by Hahnfeldt et al., it is shown that a constant dose protocol with the dose given by the averaged values of the theoretically optimal control is an excellent suboptimal protocol that achieves tumor volumes that lie within 1theoretically optimal values.
Date received: October 29, 2007
A Hybrid CA-PDE Model of Chlamydia Trachomatis Infection in the Female Genital Tract
by
Dann Mallet
School of Mathematical Sciences, Queensland University of Technology
Coauthors: Kelly-Jean Heymer, David P. Wilson
Chlamydia trachomatis is the most common sexually transmitted pathogen of humans, with the World Health Organisation (WHO) estimating 91.98 million new cases in adults occurring world wide each year. It typically infects the genitals and sometimes the eyes, throat and internal organs.
In this talk I will present the first spatio-temporal model of Chlamydial infection in the genital tract, along with some initial results and directions for future work.
Date received: October 22, 2007
Optimal multi-drug control of the innate immune response with time delays
by
Helmut Maurer
University of Münster, Germany
Optimal control problems with pure time delays in state or control variables and control-state inequality constraints are considered. We present a Pontryagin type Maximum Principle and numerical solution techniques for computing state, control and adjoint variables. The algorithm proceeds by first discretizing the retarded control problem and then using a large-scale nonlinear programming solver. In this talk, the numerical methods are applied to the optimal control of the immune response; cf. R. Stengel et al., Optimal control of innate immune response, Optimal Control Applications and Methods 23, 91-104 (2002). In that paper, only undelayed equations are considered, therapeutic agents are treated separately, and the objective function is assumed to be of quadratic type. We discuss optimal multi-drug controls in both the unretarded and retarded case as well as for quadratic and linear type objective functions. In the latter case, all controls components are shown to be bang-bang representing therapies that can easily be administered to the patient. Similar results are obained for the optimal control of the chemotherapy of HIV. Parts of the talk are based on joint work with Laurenz Goellmann, Daniela Kern and Lisa Poppe.
Date received: September 14, 2007
Piecewise Constant Estimation Algorithms for Predicting Clinical Outcomes: Applications in Genomic Data
by
Annette Molinaro
Yale University
Coauthors: Karen Lostritto (Yale University)
Clinicians aim toward a more preventative model of attacking cancer by pinpointing and targeting specific early events in disease development. These early events can be measured as genomic, proteomic, epidemiologic, and/or clinical variables. Such measurements are then used to predict clinical outcomes such as primary occurrence, recurrence, metastasis, or mortality. Recursive partitioning seeks to explain the individual contributions of various covariates as well as their interactions for the purposes of predicting outcomes, either continuous or categorical. Potential algorithms such as Classification and Regression Trees (CART) and partDSA aggressively search highly-complex covariate spaces. There are several important considerations when using such algorithms. The first is to not overfit the data. The second consideration is the stability of the resulting predictor. Algorithms such as CART are sensitive to data fluctuations and, thus, given a perturbation will potentially build a different predictor than that built on the original data. A third consideration is variable importance. In this talk, such considerations will be discussed and results comparing both algorithms presented.
Date received: October 31, 2007
A Mathematical Model of B Cell Chronic Lymphocytic Leukemia
by
L.G. de Pillis
Department of Mathematics, Harvey Mudd College, Claremont, California, USA
Coauthors: S. Nanda (Tata Institute, Bangalore, India); A.E. Radunskaya (Department of Mathematics, Pomona College, Claremont, California, USA);
B-cell chronic lymphocytic leukemia (B-CLL) is a disease for which new clinical understanding and treatment strategies continue to emerge. B-CLL is characterized by the existence of large numbers of white blood cells (B cells) in the blood, the bone marrow, the spleen and in the lymph nodes. Until recently it was believed to be a slowly progressing disease of accumulation of abnormal B cells that were immunologically challenged, and not a disease of proliferation of these cells. Over the last decade this view has changed as more is understood about the genetic changes involved in B cell production. Unlike chronic myelogenous leukemia (CML) where the presence of a genetic abnormality in hematopoeitic cells is understood to be the cause of the disease, there is no obvious genetic explanation for B-CLL. It is however understood now that B-CLL cells derive from mature antigen-stimulated cells that are immunologically competent. As a result, questions arise as to how best to treat a patient in light of new information about the disease, and clinical treatment strategies have been evolving. One method for addressing many questions about disease progression and possible treatment approaches is to develop mathematical models that reflect particular disease dynamics. B-CLL is one form of cancer for which very few mathematical models have been developed to date. The goal of the work we will present is to develop a model of B-CLL that is sufficiently complex to reflect key features of disease development, yet sufficiently streamlined to allow for reasonable parameter estimates and to admit computational and mathematical analysis. The biological literature reveals that NK cells, helper T cells and cytotoxic T cells may all play a role in stemming the growth of B-CLL. Therefore, the model we present tracks the progression of diseased B-cells through time together with these three immune cell populations. Such a model can then be used as a test-bed for exploring various treatment options. We will discuss some of these options as well as plans for further model development.
Date received: October 18, 2007
Levy random walks in ecology: fact or fiction?
by
Michael Plank
University of Canterbury
Coauthors: Alex James
A Levy random walk is one where the lengths of the steps have a distribution that is heavy tailed, i.e. does not have a finite variance. All sorts of ecological data sets have been claimed to support the idea that Levy walks are prevalent in nature, for example in the foraging movements of seals, albatrosses and spider monkeys to name a few. Furthermore, it has been suggested that a power law with an exponent of 2 provides an optimal walk for maximising foraging efficiency. In this talk, the evidence supporting this widely accepted theory will be examined. An alterative, non-Levy model for foraging will also be discussed, based on a stochastic differential equation. This model can provide higher foraging efficiency than a Levy walk, whilst producing distributions consistent with field data that supposedly support the Levy hypothesis. In conclusion, it is important to remember that a Levy walk is not the only random walk, and caution should be used when using data to infer information about an underlying process.
Date received: October 30, 2007
A delayed-differential model of the immune response: optimization and analysis.
by
Ami Radunskaya
Pomona College, Claremont, California, U.S.A.
Coauthors: Sarah Hook
In this talk we will present techniques for the analysis and optimization of a mathematical model of the immune response to tumor antigen. The model consists of a system of delay differential equations, and is calibrated to experimental data from murine experiments performed specifically for the purpose of the development of the mathematical model. The goal of the model is to suggest dose and scheduling protocols that would maximize the cellular immune response. There is not a definitive answer to what constitutes the "best" response: is it the maximum peak response, the long-term levels, or the functionality of the immune cells? We therefore compare the results from several optimization techniques, with a few different objective functions. This is collaborative work with Dr. Sarah Hook, School of Pharmacy, University of Otago, Dunedin, New Zealand.
Date received: October 27, 2007
Minimizing the Tumor Size in Mathematical Models for Novel Cancer Treatments
by
Heinz Schättler
Department of Electrical and Systems Engineering, Washington University, St. Louis, USA
Coauthors: Urszula Ledzewicz, Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, USA and Alberto d’Onofrio. European Institute for Oncology, Milano, Italy
A simple mathematical model for tumor anti-angiogenesis combined with chemotherapy is considered as an optimal control problem. The model is based on the one by Hahnfeldt et al. (Cancer Research, 59, 1999) and the state of the system represents the primary tumor volume and the carrying capacity of the vasculature related to the endothelial cells. The nonlinear dynamics models how control functions representing angiogenic inhibitors effect the growths of these variables and now also includes a killing term on the primary tumor volume. The problem of how to schedule a priori given amounts of angiogenic inhibitors and cytotoxic agents so as to minimize the primary tumor volume is considered. Due to the multi-control aspect, even with simplified dynamical equations, this becomes a challenging problem mathematically and some initial results about the structure of optimal controls will be presented.
Date received: October 29, 2007
Calcium and Ducks
by
James Sneyd
University of Auckland
Oscillations in the concentration of calcium inside cells (practically every single cell in your body) control a large number of processes, ranging from muscular contraction, to saliva secretion, to gene expression, to cell differentiation. Because the underlying dynamics are so complicated and highly nonlinear, mathematical models are useful for helping us understand these oscillations. I'll present one example of how a mathematical model can help us understand some fundamental things about calcium oscillations, and help us design experiments to test our hypotheses. Conversely, I'll then show how these models can pose new and nontrivial mathematical questions. About ducks.
Date received: October 29, 2007
Evolution of repeats in microsatellite DNA and emergency of genetic disorders
by
Andrzej Swierniak
Silesian University of Technology
Coauthors: M. Kimmel, A.Polanski
Microsatellites are the shortest non-coding repeats of DNA which are composed of the repetitive sequences of 2 to 5 motifs (see e.g. Ramel, 1997). Formation of tandem repeats composed from such short units occurs most probably as a result of DNA replication errors in which slippage through strand occurs. The slippage of polymerase during replication leads to base pairs mismatching and, if not repaired , gives rise to elongation or shortening of the microsatellite with one or more repeated unit. The stability of the number of repeats in microsatellite sequence depends on the intact mismatch DNA repair. The changes in the number of repeats in microsatellites accompany some human genetic diseases. Disorders such as Hutington’s disease, spinocerebellar ataxia type 1, syndrome of fragile X chromosome, myotonic dystrophy and genetic diabetes are related to expansion of repeated units in microsatellites lying in the vicinity of some genes (Green, 1993).
We describe the time evolution of the distribution of the repeat loci in microsatellite DNA by a branching random walk with an absorbing boundary (Kimmel and Axelrod, 2002) and focus our interest on the stability analysis of the resulting model in the form of infinite dimensional system of linear differential equations. We follow the line of reasoning used previously in asymptotic analysis of drug resistance in cancer populations caused by gene amplification (Kimmel, Swierniak and Polanski, 1998). The techniques applied include Laplace transforms for the case of initial conditions with finite support and spectral analysis for respectively defined Banach operators in the case of infinite support.
Green H. (1993), Human genetics diseases due to codon reiteration: relationship to evolutionary mechanism. Cell, 74, 955-956
Kimmel M. and Axelrod D.E. (2002), Branching Processes in Biology, Springer Verlag, New York
Kimmel M., Swierniak A. and Polanski A. (1998), Infinite dimensional model of evolution of drug resistance of cancer cells, J. Math. Syst. Estim. Contr., 8, 1-16
Ramel C. (1997), Mini- and microsatellites., Env. Health Persp., 105, 781-789
Date received: October 11, 2007
SVD based analysis of DNA microarray data
by
Michal Swierniak
Department of Nuclear Medicine and Endocrine Oncology, Institute of Oncology, M. Sklodowska-Curie Memorial Cancer Centre, Gliwice Branch
Coauthors: dr. Krzysztof Simek - The Silesian University of Technology,
dr. Michal Jarzab - Institute of Oncology,Cancer Centre, Gliwice Branch
The aim of this talk is to show how some techniques based on the Singular Value Decomposition may be used in DNA microarray analysis. Since usually a number of rows in the microarray matrix (number of genes) is much greater than a number of columns (number of samples) SVD seems to be the most proper method for investigation of basic trends in the data. We describe algorithms based on SVD which may be used to select a set of genes with the most important significance of the data and demonstrate how they may be used in unsupervised classification of the patterns and discovery of new classes. Moreover we present results of the oligonucleotide microarray experiments for thyroid carcinomas. We discuss different rules of gene selection and compare the results with the ones previously published. Moreover we discuss some biological issues resulting from the presented analysis.
Date received: October 25, 2007
Modelling of Cancer Treatment
by
Graeme Wake
Centre for Mathematics in Industry, Massey University Auckland, New Zealand
Coauthors: Bruce Baguley, University of Auckland
Britta Basse, University of Canterbury,
Ronald Begg, Massey and Canterbury Universities
Bruce van-Brunt, Massey University
David Wall, Massey University
Improved treatment of cancer is one of the most important challenges for medical science. Tailoring treatment for individual patients has long been an objective for oncologists. While many biological techniques and mathematical models have been devised to predict the course of treatment, none have applied routinely to clinical oncology. Our model, which describes the complexities of the responses of tumour cells over time to both anticancer drugs and radiation, has considerable impact on our ability to advance individualisation of cancer therapy. This process is in advanced stages of implementation. Over the last few years, we have developed sophisticated mathematical equations describing the behaviour of cancer cells as they progress through the cell division cycle. Which stage in the cycle the cells are actually in, can be differentiated by their DNA content and this enables model outcomes to be compared directly to experimental results. These equations describe the response of human tumours to chemotherapy and radiotherapy. Firstly we incorporate programmed cell death (apotosis) into the model. We then consider perturbations of model parameters by treatment and compare model results with data. This research will provide significant new analytical and computational insights into the area of non-local equations, where cause and effect are separated in space and time, as well as underpinning support to oncologists concerned with treatment, drug companies producing drugs and management of clinics. The support of the NZIMA by the award of a Maclaurin Fellowship to assist in the development of this work is gratefully acknowledged.
Date received: September 20, 2007
A mathematical model of airway and pulmonary arteriole smooth muscle.
by
Inga Wang
University of Auckland
Coauthors: Antonio Z. Politi, Nessy Tania, Yan Bai, Michael J. Sanderson and James Sneyd
Airway hyper-responsiveness (AHR) is a major characteristic
of asthma and is believed to result from the excessive contraction
of airway smooth muscle cells (SMCs). However, the identification of
the mechanisms responsible for AHR is hindered by our limited
understanding of how calcium, myosin light chain kinase
(MLCK) and myosin light chain phosphatase (MLCP) interact to
regulate airway SMC contraction. In this talk, I will present a modified
Hai-Murphy cross-bridge model of SMC contraction that incorporates the calcium regulation of the MLCK and MLCP.
Date received: November 1, 2007
The lipid bilayer at the mesoscale: a physical continuum model
by
Phil Wilson
University of Canterbury
Coauthors: Huaxiong Huang (York University, Canada)
Shu Takagi (The University of Tokyo, Japan)
Cell membranes are the most abundant cellular structure in all living matter. Their core component is a soft, strong, self-assembling sheet called the lipid bilayer. Multiscale simulations of blood flow depend on lipid bilayer properties because such bilayers surround red blood cells and contribute significantly to their modes of deformation. Small patches of the bilayer can be simulated for short times with discrete numerical methods such as Molecular Dynamics. The interaction of neighbouring red blood cells can be simulated with continuum dynamical methods. However, there is as yet no robust way to transfer microscale information to the macroscale. In this talk we discuss one such potential mesoscale filter. This continuum model is based on minimising the free energy of a mixture of lipid and water molecules. The model extends previous work by (a) formulating a more physical model of the hydrophobic effect, (b) clarifying the meaning of the model parameters through numerical solutions, (c) outlining a method for determining parameter values based on a quantitative comparison of numerical results with physical experimental data.
Date received: October 14, 2007