Mathematical Models of Tumor Invasion
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Transcript of Mathematical Models of Tumor Invasion
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Mathematical Models of Tumor InvasionSean Davis
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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
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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
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General Equations
Kinetics
Acceleration in x-direction
Acceleration in y-direction
Diffusion Coefficients
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Linear StabilityDiffusion is a form of stabilisation
Diffusion - Driven Instability
Grasshoppers and Fire
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Non-Dimensionalized Equations
Diffusion Ratio
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Method of Lines
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Simulation Results
Initial
Half Way
Quarter way
At t = 10
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Cancer ModelKey Insight: Transformation of Tumors - Makes more Acid
Aligned with Empirical Studies (Accuracy)
Reaction Diffusion
Population Density
Predator - Prey
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Predator Prey Models
Growth due to birthDeath due to predations
Growth due to predation Death
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Population Density
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Cancer Model
Carrying Capacity CompetitionDeath due to pH levels
Diffusion
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Cancer Model
Source Sink Diffusion
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Non Dimensionalized model
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Fixed Points
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Extensions
Degradation
Production DecayDiffusion
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Low pH
MMP Initial
MMP 1/3
MMP Final
Tumor Initial
Tumor 1/3
Tumor Final
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Medium pHMMPInitial
MMP Final
Tumor Initial
Tumor Final
MMP1/3 Tumor 1/3
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High pH
MMP Initial
MMP Final Tumor Final
Tumor Initial
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Cellular Automata What happens when there aren’t very many cells
Can’t use continuous global model
Instead compute cells individually
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StatesNormal (and Quiescent Normal)
Tumor (and Quiescent Tumor)
Micro-Vessel
Vacant
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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
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Glucose and Hydrogen Equations
Hydrogen concentration Constant depending on state of automaton Element
Diffusion CoefficientGlucose concentration
constant depending on state of automaton element
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Boundary Conditions
Glucose Concentration next to Vessel Wall
Permeability of wall
Serum Glucose and Hydrogen
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Low Vascular Densities
Initial At t = 4
At t = 6At t = 10
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High Vascular Densities
T = 2 T = 4
T = 8 T = 10
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Questions?
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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.