Mathematical Models of Tumor InvasionSean Davis

OverviewGoal: Examine how a mathematical idea, matures and understands natural phenomena, aside from physics

Cellular Automata / Discrete and Local Cancer Models

Reaction Diffusion Equations / Continuous Global Cancer Models

Reaction Diffusion EquationsMathematical framework for describing how entities interact as they change in time and space

Applications:

Embryology

Tumors

Pattern Development of Mollusc Shells

Epidemic Models

General Equations

Kinetics

Acceleration in x-direction

Acceleration in y-direction

Diffusion Coefficients

Linear StabilityDiffusion is a form of stabilisation

Diffusion - Driven Instability

Grasshoppers and Fire

Non-Dimensionalized Equations

Diffusion Ratio

Method of Lines

Simulation Results

Initial

Half Way

Quarter way

At t = 10

Cancer ModelKey Insight: Transformation of Tumors - Makes more Acid

Aligned with Empirical Studies (Accuracy)

Reaction Diffusion

Population Density

Predator - Prey

Predator Prey Models

Growth due to birthDeath due to predations

Growth due to predation Death

Population Density

Cancer Model

Carrying Capacity CompetitionDeath due to pH levels

Diffusion

Cancer Model

Source Sink Diffusion

Non Dimensionalized model

Fixed Points

Extensions

Degradation

Production DecayDiffusion

Low pH

MMP Initial

MMP 1/3

MMP Final

Tumor Initial

Tumor 1/3

Tumor Final

Medium pHMMPInitial

MMP Final

Tumor Initial

Tumor Final

MMP1/3 Tumor 1/3

High pH

MMP Initial

MMP Final Tumor Final

Tumor Initial

Cellular Automata What happens when there aren’t very many cells

Can’t use continuous global model

Instead compute cells individually

StatesNormal (and Quiescent Normal)

Tumor (and Quiescent Tumor)

Micro-Vessel

Vacant

RulesEach Automaton Element is placed on an (N x N) Grid.

If the element is a Tumor or a Normal Cell and the pH are above a min they live

If it is also above a “reproduction threshold” then they have the oppurunity to reproduce

Provided one of their neighbours have enough glucose

Glucose and Hydrogen Equations

Hydrogen concentration Constant depending on state of automaton Element

Diffusion CoefficientGlucose concentration

constant depending on state of automaton element

Boundary Conditions

Glucose Concentration next to Vessel Wall

Permeability of wall

Serum Glucose and Hydrogen

Low Vascular Densities

Initial At t = 4

At t = 6At t = 10

High Vascular Densities

T = 2 T = 4

T = 8 T = 10

Questions?

BibliographyAALPEN A. PATEL, EDWARD T. GAWLINSKI, SUSAN K. LEMIEUX, ROBERT A. GATENBY, A Cellular Automaton Model of Early Tumor Growth and Invasion: The Effects of Native Tissue Vascularity and Increased Anaerobic Tumor Metabolism, Journal of Theoretical Biology, Volume 213, Issue 3, 7 December 2001, Pages 315-331, ISSN 0022-5193, http://dx.doi.org/10.1006/jtbi.2001.2385.

Boyce, William E. and DiPrima Richard C. Elementary Differential Equations. 9th Edition. John Wiley & Sons Inc. 2009. Print

Burden Richard L Faire J. Douglase. Numerical Analysis 34d Edition. Boston: PWS Publishers, 1981. Print.

de Vries Gerda, Hillen Thomas, Lewis Mark, Johannes Muller, Schonfisch Birgitt. A Course in Mathematical Biology: Quantitative modeling with 1Mathematical and Computational Methods. Society for Industrial and Applied Mathematics, 2006, Print.

Martin, Natasha K. et al. “Tumour-Stromal Interactions in Acid-Mediated Invasion: A Mathematical Model.” Journal of theoretical biology 267.3 (2010): 461–470. PMC. Web. 6 Dec. 2015.

Murray J.D. Mathematical Biology Second, Corrected Edition. Springer, 1991

Gatenby, Robert A. and Gawlinski Edward T. “A Reaction-Diffusion Model of Cancer Invasion. Cancer Research 56 (1996): 5745-5753 (Web) Accessed December 5th 2015.