Modeling of Tumor Induced Angiogenesis II Heather Harrington, Marc Maier & Lé Santha Naidoo Faculty Advisors: Panayotis Kevrekidis & Nathaniel Whitaker.

Slides:



Advertisements
Similar presentations
On-lattice agent-based simulation of populations of cells within the open-source Chaste framework Grazziela P. Figueredo Tanvi Joshi James Osborne Helen.
Advertisements

Protein Binding Phenomena Lecture 7, Medical Biochemstry.
Non-Specific Defenses April 3, Body fights disease in 2 ways 1. Non-specific defense system  Protects body from all foreign substances 2. Specific.
Using modified collagen scaffolds to promote angiogenesis for tissue engineering applications Julie M. Etheridge Tissue Engineering and Biomaterials Laboratory.
Image Segmentation some examples Zhiqiang wang
Project Macrophage Math Biology Summer School 2008 Jennifer Morrison & Caroline Séguin.
Graphs of Trig Functions
Modeling of Tumor Induced Angiogenesis III Heather Harrington, Marc Maier & Lé Santha Naidoo Faculty Advisors: Panayotis Kevrekidis, Nathaniel Whitaker,
Project Macrophage: Macrophages on the Move Heather More, Rachel Psutka, Vishaal Rajani.
Modeling Tumor Growth Katie Hogan 7 December 2006.
Lengths of Plane Curves
Modeling of Tumor Induced Angiogenesis Heather Harrington, Marc Maier & Lé Santha Naidoo Faculty Advisors: Panayotis Kevrekidis & Nathaniel Whitaker.
Figure 13.2 The Biology of Cancer (© Garland Science 2007) Hodgkins Lymphoma – A Cancer in Which 99% of the Tumor Cells Are Not Cancer Cells.
LECTURE UNIT 4.3 Normal Random Variables and Normal Probability Distributions.
What is Cancer? How it occurs and cell cycle regulation.
Dialogue Replaces Monologue:
CANCER. THE WORD CANCER  FEAR  DEATH  PAIN  SUFFERING  LOSS  ECONOMIC LOSS  DISFIGUREMENT  WHAT’S THE FUTURE.
Dynamic Equilibrium Constant small changes to help maintain homeostasis.
This is the graph of y = sin xo
Trigonometric Functions
1. Balance of proliferation & apoptosis. 2. Regeneration, scar formation & ECM.
Deformable Models Segmentation methods until now (no knowledge of shape: Thresholding Edge based Region based Deformable models Knowledge of the shape.
Condensed States of Matter
Cancer Uncontrolled cell growth. Cellular differentiation is the process by which a less specialized cell becomes a more specialized cell type. Occurs.
Nathaniel Whitaker Modeling Tumor Induced Angiogenesis University of Massachusetts Amherst.
CHAPTER Continuity Derivatives and the Shapes of Curves.
Heat Equation and its applications in imaging processing and mathematical biology Yongzhi Xu Department of Mathematics University of Louisville Louisville,
Mathematical Modelling of Cancer Invasion of Tissue: The Role of the Urokinase Plasminogen Activation System Mark Chaplain and Georgios Lolas Division.
Antiderivatives. Antiderivatives Definition A function F is called an antiderivative of f if F ′(x) = f (x) for all x on an interval I. Theorem.
C2: Exponential Functions Learning Objective: to be able to recognise a function in the form of f(x) = a x.
Immunotherapy for Treating Cancer Project A, proposed by Helen Byrne Rui Zhao, Peter Kim, and Natasha Li Advisor: Philip Maini.
The Mathematical Model Of Repressilator where i = lacl, tetR, cl and j = cl, lacl, tetR. α 0 : the number of protein copies per cell produced from a given.
POPULATION ECOLOGY. Density and Dispersion What is the density of a population? The number of individuals per unit area Dispersion is how they spread.
Multiscale Modeling of Avascular Tumor Growth Jelena Pjesivac-Grbovic Theoretical Division 7, LANL Ramapo college of New Jersey University of Tennessee,
Trigonometric Graphs.
Weak electrolyte Weak electrolytes are not fully ionized in solution, such as weak acids and bases. Degree of ionization (α): defined as the ratio of the.
Neoplasia VI DR OSAMA I NASSIF FRCPC CONSULTANT & ASSOCIATE PROFESSOR DEPARTMENT OF PATHOLOGY, KAUH Neoplasia VI DR OSAMA I NASSIF MD,FRCPC CONSULTANT.
Quantifying Growth Kinetics Unstructured model: assuming fixed cell composition. Applicable to balanced-growth condition: - exponential growth phase in.
ANGIOGENESIS Vasculogenesis: Embryonic development from endothelial precursors called ‘angioblasts’ Angiogenesis/ neovascularization: Process of blood.
The study of life “bio” meaning life “ology” meaning study of
AP BIOLOGY Chapter 8 Metabolism. The _____ Law of Thermodynamics states that energy can be transformed and transferred by NEVER created or destroyed Anabolic.
Fibroblast Growth Factors (FGFs)
 What is the density of a population?  The number of individuals per unit area  Dispersion is how they spread out in that area  What are the three.
Supermodel of melanoma dynamics Witold Dzwinel 1, Adrian Kłusek 1 and Oleg V. Vasilyev 2 AGH University of Science and Technology, Department of Computer.
Sprouting angiogenesis
Inflammation (1 of 5) Ali Al Khader, M.D. Faculty of Medicine
The ECM as a Spatial Organizer of
Two talks this week and next on morphogenesis
Chemical Kinetics And The Time-Dependent Diffusion Equation
Inducing Angiogenesis
Immunotherapy for Treating Cancer
Inducing Angiogenesis
생체계측 II Report # 송성진 Medical Instrumentation II.
Slit-Robo Cancer Cell Volume 4, Issue 1, Pages 1-2 (July 2003)
Extracellular Vesicles in Cancer: Cell-to-Cell Mediators of Metastasis
A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2). A function f is decreasing.
Lesson 11: Exponential Functions
Regulation of the Cell Cycle
Cancer.
Angiogenesis.
Macrophage Metabolism Shapes Angiogenesis in Tumors
Taking the Study of Cancer Cell Survival to a New Dimension
Environmental Carcinogenesis
Cellular Characteristics of Cancer Cells that Contribute to Metastasis
Florian Milde, Michael Bergdorf, Petros Koumoutsakos 
Process and mechanisms of blood vessel formation.
Jair Bar, MD, PhD, Glenwood D. Goss, MD, FCPSA, FRCPC 
Angiogenesis and Angiostatin
Gaddiel Yonathan Ouaknin, Pinhas Zvi Bar-Yoseph  Biophysical Journal 
Normalizing the tumor microenvironment
Presentation transcript:

Modeling of Tumor Induced Angiogenesis II Heather Harrington, Marc Maier & Lé Santha Naidoo Faculty Advisors: Panayotis Kevrekidis & Nathaniel Whitaker

Bio Recap Angiogenesis: The process of formation of capillary sprouts in response to external chemical stimuli which leads to the formation of blood vessels. Angiogenesis: The process of formation of capillary sprouts in response to external chemical stimuli which leads to the formation of blood vessels. Tumor Angiogenic Factors (TAFs): Stimuli secreted by Tumors Tumor Angiogenic Factors (TAFs): Stimuli secreted by Tumors Extra Cellular Matrix (ECM): The area in which cells interact with the Fibronectin(F). Extra Cellular Matrix (ECM): The area in which cells interact with the Fibronectin(F). Proteases (P): Secreted by tumor to attract cells and destroy Inhibitors. Promotes Angiogenesis. Proteases (P): Secreted by tumor to attract cells and destroy Inhibitors. Promotes Angiogenesis. Inhibitors: Prevent Cells from getting to tumor. Generated by fibronectin cells in the ECM to inactivate proteases. Inhibitors: Prevent Cells from getting to tumor. Generated by fibronectin cells in the ECM to inactivate proteases.

5 “Species” Dynamical Evolution Model (1 Dimension) (1) C t = D c ΔC – ∂/∂x(f F * ∂F/∂x) (1) C t = D c ΔC – ∂/∂x(f F * ∂F/∂x) - ∂/∂x(f T * ∂T/∂x) + ∂/∂x(f I * ∂I/∂x) + k 1 C(1-C) (2) T = e (-(x-L) ² /ε) (2) T = e (-(x-L) ² /ε) (3) F t = -k 2 PF (3) F t = -k 2 PF (4) P t = -k 3 PI + k 4 TC + k 5 T – k 6 P (4) P t = -k 3 PI + k 4 TC + k 5 T – k 6 P (5) I t = -k 3 PI (5) I t = -k 3 PI f T term represents chemotactic attraction of cells to tumor f F term represents haptotactic response to the Fibronectin f I term represents the “repulsive” effect of inhibitor gradients D c = Diffusion Coefficient f F = a 1 C f T = a 2 C/(1 + a 3 T) f I = a 4 C

After Discretization We Get… C (n, k+1) = P r C (n-1, k) + P s C (n,k) + P l C (n+1, k) C (n, k+1) = P r C (n-1, k) + P s C (n,k) + P l C (n+1, k) F (n, k+1) = F (n,k) *(1 – Δt k 2 P (n,k) ) F (n, k+1) = F (n,k) *(1 – Δt k 2 P (n,k) ) P (n, k+1) = P (n, k) (1 – Δt k 6 – Δt k 3 I (n,k) P (n, k+1) = P (n, k) (1 – Δt k 6 – Δt k 3 I (n,k) + T (n,k) (Δt k 4 C (n,k) + Δt k 5 ) I (n, k+1) = I (n,k) (1 – Δt k 3 P (n,k) ) I (n, k+1) = I (n,k) (1 – Δt k 3 P (n,k) ) T = e -(x – L)²/ε (constant) T = e -(x – L)²/ε (constant)

1 - D results Near Tumor Far from Tumor No inhibitor

Adding an Inhibitor Near tumorFar from tumor weak inhibitor

Another Inhibitor Near tumorFar from tumor Strong Inhibitor

Replenished Inhibitor Examples Near tumor Far from tumor Weak Inhibitor

Replenished cont… Near Tumor Far from tumor Strong Inhibitor

5 Species Dynamic Evolution 2 Dimensional Model (1) C t = D c ΔC – (f F * F) - (f T * T) (1) C t = D c ΔC – (f F * F) - (f T * T) + (f I * I) + k 1 C(1-C) (2) T = e (-(x-L) ² /ε) (2) T = e (-(x-L) ² /ε) (3) F t = -k 2 PF (3) F t = -k 2 PF (4) P t = -k 3 PI + k 4 TC + k 5 T – k 6 P (4) P t = -k 3 PI + k 4 TC + k 5 T – k 6 P (5) I t = -k 3 PI (5) I t = -k 3 PI

After Discretization (2 Dimensions)… C (n, m, k+1) = P r C (n-1, m, k) + P l C (n+1, m, k) C (n, m, k+1) = P r C (n-1, m, k) + P l C (n+1, m, k) + P s C (n, m, k) + P u C (n, m-1, k) + P d C (n, m+1, k) F (n, m, k+1) = F (n, m, k) *(1 – Δt k 2 P (n, m, k) ) F (n, m, k+1) = F (n, m, k) *(1 – Δt k 2 P (n, m, k) ) P (n, m, k+1) = P (n, m, k) (1 – Δt k 6 – Δt k 3 I (n, m, k) P (n, m, k+1) = P (n, m, k) (1 – Δt k 6 – Δt k 3 I (n, m, k) + T (n, m, k) (Δt k 4 C (n, m, k) + Δt k 5 ) I (n, m, k+1) = I (n, m, k) (1 – Δt k 3 P (n, m, k) ) I (n, m, k+1) = I (n, m, k) (1 – Δt k 3 P (n, m, k) ) T = e -[(x – L)² + (y-L) ²]/ε (constant) T = e -[(x – L)² + (y-L) ²]/ε (constant)

2 – D Results Near Tumor – No Inhibitor

Far from Tumor – No Inhibitor

Near Tumor – Weak inhibitor

Far from Tumor – Weak Inhibitor

Angiogenesis in the Cornea ∂C/∂t = DΔC - k C – u L C ∂C/∂t = DΔC - k C – u L C D = Diffusion Coefficient C = Tumor Angiogenic Factors (TAF) D = Diffusion Coefficient C = Tumor Angiogenic Factors (TAF) k = rate constant of inactivation u = rate constant of uptake k = rate constant of inactivation u = rate constant of uptake L = total vessel length per unit area ΔC = ∂²C/∂x² + ∂²C/∂y² L = total vessel length per unit area ΔC = ∂²C/∂x² + ∂²C/∂y² f(C) = f(C) = C t = Threshold Concentration α = constant that controls shape of the curve C t = Threshold Concentration α = constant that controls shape of the curve n = S max f(C) Δl Δt n = S max f(C) Δl Δt (probability for the formation of 1 sprout from a vessel segment) (probability for the formation of 1 sprout from a vessel segment) S max = rate constant that determines max probability of sprout formation S max = rate constant that determines max probability of sprout formation 0, 0 ≤ C ≤ C t 1 – e -α(C – C t ), Ct ≤ C

Sprout Growth = P + (1-P) E = direction of growth in previous time step E = direction of growth in previous time step G = Direction of concentration gradient of TAF G = Direction of concentration gradient of TAF P = Persistance ratio P = Persistance ratio Δl = V max f(C) Δt(Length increase of sprouts) Δl = V max f(C) Δt(Length increase of sprouts) V max = maximum rate of length increase V max = maximum rate of length increase E x T E xo T G xo T cos θ sin θ E y E yo G yo -sin θ cos θ

Cornea Graphs

Progress & Goals 1-Dimensional Model with “random walker cells” 2-Dimensional Model of Angiogenesis Modeling Angiogenesis in the Cornea (ignoring inhibitors) – In Progress Angiogenesis in the Cornea with Inhibitors and perhaps other factors