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Introduction to Mathematical Modeling in Biology with ODEs

Introduction to Mathematical Modeling in Biology with ODEs. Lisette de Pillis Department of Mathematics Harvey Mudd College. Mathematical Modeling and Mathematical Biology . What is Mathematical Modeling …and how do you spell it?

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Introduction to Mathematical Modeling in Biology with ODEs

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  1. Introduction to Mathematical Modeling in Biology with ODEs Lisette de Pillis Department of Mathematics Harvey Mudd College Lisette de Pillis HMC Mathematics

  2. Mathematical Modeling and Mathematical Biology • What is Mathematical Modeling …and how do you spell it? • “Mathematics consists of the study and development of methods for prediction” • The aim of Biology is “to find useful and verifiable descriptions and explanations of phenomena in the natural world” • Modeling = The use of mathematics as a tool to explain and make predictions of natural phenomena • MathematicalBiologyinvolves mathematically modeling biological phenomena Thanks: Cliff Taubes, 2001 Lisette de Pillis HMC Mathematics

  3. Mathematical Modeling Philosophy • Why are models useful: • Formulating precise ideasimplicit assumpltions less likely to “slip by” • Mathematics = concise language that encourages clarity of communication • Mathematical theorems and computational resources can be accessed Lisette de Pillis HMC Mathematics

  4. Mathematical Modeling Philosophy • Why are models useful (cont): • Can safely test hypotheses (eg, drug treatment), and confirm or reject • Can predict system performance under untested or untestable conditions • How models can be limited (trade-offs): • Easy math Unrealistic model • Realistic model Too many parameters • Caution: unrealistic conclusions possible Lisette de Pillis HMC Mathematics

  5. The Modeling Process Occam’s Razor* Model World Real World Interpret and Test (Validate) Formulate Model World Problem Model Results Solutions, Numerics Mathematical Model (Equations) *Occams’s Razor: “Entia non sunt multiplicanda praeter necessitatem” “Things should not be multiplied without good reason” Mathematical Analysis Lisette de Pillis HMC Mathematics

  6. Components of the Model World Model World Things whose effects are neglected Things that affect the model but whose behavior the model is not designed to study (exogenous or independent variables) Things the model is designed to study (endogenous or dependent variables) Lisette de Pillis HMC Mathematics

  7. The Five Stages of Modeling • Ask the question. • Select the modeling approach. • Formulate the model. • Solve the model. Validate if possible. • Answer the question. Lisette de Pillis HMC Mathematics

  8. Introduction to Continuous Models • One of simplest experiments in biology: Tracking cell divisions (eg, bacteria) over time. • Analogous dynamics for tumor cell divisions (what they learn in med school): A tumor starts as one cell The cell divides and becomes two cells Thanks: Leah Keshet, Ami Radunskaya Lisette de Pillis HMC Mathematics

  9. Introduction to Continuous Modeling Cell divisions continue… 22 cells 23 cells 24 cells Lisette de Pillis HMC Mathematics

  10. Ordinary Differential Equations (ODEs) • Mathematical equations used to study time dependent phenomena • A “differential equation” of a function = an algebraic equation involving the function and its derivatives • A “derivative” is a function representing the change of a dependent variable with respect to an independent variable. (Often thought of as representing a slope.) Lisette de Pillis HMC Mathematics

  11. Ordinary Differential Equations (ODEs) • Ex: If N (representing, eg, bacterial density, or number of tumor cells) is a continuous function of t(time), then the derivative of N with respect to t is another function, called dN/dt, whose value is defined by the limit process • This represents the change is N with respect to time. Lisette de Pillis HMC Mathematics

  12. Our Cell Division Model: Getting the ODE • Let N(t) = bacterial density over time • Let K = the reproduction rate of the bacteria per unit time (K > 0) • Observe bacterial cell density at times t and (t+Dt). Then N(t+Dt) ≈N(t) + KN(t) Dt • Rewrite: (N(t+Dt) – N(t))/Dt≈ KN(t) Total density at time t + increase in density due to reproduction during time interval Dt Total density at time t+Dt ≈ Lisette de Pillis HMC Mathematics

  13. Our Cell Division Model: Getting the ODE • Take the limit as Dt→ 0 “Exponential growth” (Malthus:1798) • Analytic solution possible here. • Implication: Can calculate doubling time Lisette de Pillis HMC Mathematics

  14. Analysis of Cell Division Model: Exponential • Find “population doubling time” t: • Point: doubling time inversely proportional to reproductive constant K and imply Taking logs and solving for t gives Lisette de Pillis HMC Mathematics

  15. Exponential Growth Implications: 1 Day Doubling • Doubling time t=ln(2)/K • Suppose K=ln(2), so t=1, ie, cell popn doubles in 1 day. • : In 30 days, 1 cell →→ detectable population • is about a sphere (bag) • is about a 100 grams (1/10 kilo) of tumor (bag) • Tumor will reach 100 grams between days 36 and 37. • One week later, tumor weighs a kilo (at around cells) and is lethal. • 90% tumor removal of cells leaves 10 billion cells. • 99% removals leaves 1 billion cells. • Every cancer cell must be killed to eliminate the tumor Lisette de Pillis HMC Mathematics

  16. Exponential Growth: Realistic? Lisette de Pillis HMC Mathematics

  17. Extending the Growth Model: Additional Assumptions + New System • Reproductive rate Kis proportional to the nutrient concentration, C(t): so K(C)=kC • a units of nutrient are consumed in producing 1 unit of pop’n increment → system of equations: • Simplify the system of ODEs (collapse): • Logistic Growth Law! • Note: equiv to assuming K=K(N)=C0- aN, ie K is density dependent. Lisette de Pillis HMC Mathematics

  18. Analysis of Logistic Model for Cell Growth • Solution: • N0 = initial population • kC0 = intrinsic growth rate • C0/a = carrying capacity • For small popn levels N, N grows about “exponentially”, with growth rate r ≈ kC0 • As time t→ ∞, N → N(∞)=C0/a • This “self limiting” behavior may be more realistic for longer times Lisette de Pillis HMC Mathematics

  19. Exponential versus Logistic Growth Lisette de Pillis HMC Mathematics

  20. Logistic Growth: Initial Conditions, Stability Lisette de Pillis HMC Mathematics

  21. Other Growth Models • Power Law: • Gompertz: • Von Bertlanffy: Lisette de Pillis HMC Mathematics

  22. Intrinsic Cell Growth Models: Comparisons Logistic Power Law Von Bertalanffy Gompertz Lisette de Pillis HMC Mathematics

  23. Dynamic Population Model Formulation: General Approach • Balance (Conservation): • Law of Mass Action: Encounters between populations occur randomly, and the number of encounters is proportional to the product of the populations, eg, Population Change in Time Stuff Going In – Stuff Going Out = Used to represent Inter- and Intra-Species Competition Lisette de Pillis HMC Mathematics

  24. Formulating a 2-Population Model: Tumor-Immune Interactions • Step 1 - Ask the Question: How does the immune system affect tumor cell growth? Could it be responsible for “dormancy” followed by aggressive recurrence? • Step 2 - Select the Modeling Approach: Track tumor and immune populations over time → Employ ODEs Lisette de Pillis HMC Mathematics

  25. Formulating a 2-Population Model: Tumor-Immune Interactions • Step 3 - Forumlate the Model: • Identify important quantities to track: • Dependent Variables: • E(t)=Immune Cells that kill tumor cells (Effectors) (#cells or density) • T(t)=Tumor cells (#cells or density) • Independent Variable: t (time) • Specify Basic Assumptions: • Effectors have a constant source • Effectors are recuited by tumor cells • Tumor cells can deactivate effectors (assume mass action law) • Effectors have a natural death rate • Tumor cell population grows logistically (includes death already) • Effector cells kill tumor cells (assume mass action law) Lisette de Pillis HMC Mathematics

  26. A Two Population System • Rate parameters (units) • s=constant immune cells source rate (#cells/day) • s=steepness coefficient (#cells) • r=Tumor recruitment rate of effectors (1/day) • c1=Tumor deactivation rate of effectors (1/(cell*day)) • d=Effector death rate (1/day) • a=intrinsic tumor growth rate (1/day) • 1/b=tumor population carrying capacity (#cells) • c2=Effector kill rate of tumor cells (1/(cell*day)) Lisette de Pillis HMC Mathematics

  27. Model Elements Population change in time Stuff going in Stuff going out Lisette de Pillis HMC Mathematics

  28. Model Elements Michaelis- Menten Mass Action Logistic Growth Lisette de Pillis HMC Mathematics

  29. Step 4: Solve the System • Must treat system as a whole • In general, a closed-form solution does not exist • Solution approaches: • Dynamical systems analysis (find general system features) • Numerical (find example system solutions) • Next up: Finding general system features Lisette de Pillis HMC Mathematics

  30. Dynamical Systems Analysis: When we cannot solve analytically • Find equilibrium points (set ODEs to 0): plot nullclines and find intersections • Find stability properties of equilbrium points (if nonlinear: must linearize) • Trace possible trajectories in phase diagram Lisette de Pillis HMC Mathematics

  31. Dynamical Systems Analysis: When we cannot solve analytically Find equilibrium points • Set ODEs to 0: • Therefore: • Solve for E and T curves (nullclines). Find points of overlap (intersections). Lisette de Pillis HMC Mathematics

  32. Analysis: the equilibria are determined by setting both differential equations to zero. E-equation = 0 T-equation = 0 Lisette de Pillis HMC Mathematics

  33. Each stable equilibrium point has a basin of attraction Lisette de Pillis HMC Mathematics

  34. Step 5: Answer the Question • Question: Do we see dormancy? • Question: Do we see aggressive regrowth in this model? • Not yet: How about with different parameters? Let’s see… Lisette de Pillis HMC Mathematics

  35. Alternate Parameters: Tumor Dormancy with Immune System Evident • Four equilibria - two stable • Dormancy: stable spiral T u m o r I m m u n e Lisette de Pillis HMC Mathematics

  36. Alternate Parameters – Dangerous Regrowth with Immune System • Creeping through to dangerous equilibrium: T u m o r I m m u n e Lisette de Pillis HMC Mathematics

  37. Step 5: Answer the Question • Question:Now do we see dormancy? • Question:Now do we see aggressive regrowth in this model? Yes! Yes! Lisette de Pillis HMC Mathematics

  38. Continue the modeling cycle… Step 1: Ask a New Question • New Question: In the clinic, what causes asynchronous response to chemotherapy? • Note: The current 2 population model does not answer this question…We need to extend the model. Lisette de Pillis HMC Mathematics

  39. Extend the Model Further - More Realism: Adding Normal Cells (Competition) • Turn the two population model into a three population model (dePillis and Radunskaya, 2001, 2003) • Why: Gives more realistic response to chemotherapy treatments, eg, allows for delayed response to chemotherapy Lisette de Pillis HMC Mathematics

  40. Three Population Mathematical Model Population change in time Stuff going in Stuff going out • Combine Effector (Immune), Tumor, Normal Cells Note: There is always a tumor-free equilibrium at (s/d,0,1) Lisette de Pillis HMC Mathematics

  41. Analysis: Finding Null Surfaces • Curved Surface: • Planes Lisette de Pillis HMC Mathematics

  42. Null surfaces: Immune, Tumor, Normal cells Lisette de Pillis HMC Mathematics

  43. Analysis: Determining Stability of Equilibrium Points • Linearize ODE’s about (eg, tumor-free) equilibrium point • Solve for system eigenvalues: Lisette de Pillis HMC Mathematics

  44. CoExisting Equilibria Map: Paremeter Space Region 4: Stable @ (E=0.4, T=0.6, N=0.4) Unstable @ (E=0.8, T=0.2, N=0.8) Lisette de Pillis HMC Mathematics

  45. Time Series Plots • Creeping Through to Dangerous Equilibrium: Lisette de Pillis HMC Mathematics

  46. Evolution in Time: Increasing Initial Immune Strength Initial Immune Strength Range: 0.0 < E(0) < 0.3 Basin Boundary Range: 0.12 < E(0) < 0.15 Time vs Tumor Time vs Normal Time vs Immune Stable Equilibrium - Co-Existing: E=0.4, T=0.6, N=0.4 Stable Equilibrium - Tumor Free: E=1.65, T=0, N=1.0 Lisette de Pillis HMC Mathematics

  47. Cell Response to Chemotherapy • Idea: Add drug response term to each DE, create DE describing drug Amount of cell kill for given amount of drug u: Lisette de Pillis HMC Mathematics

  48. Normal,Tumor & Effector cells with Chemotherapy • Four populations: • Goal: control dose to minimize tumor • See: “A Mathematical Tumor Model with Immune Resistance and Drug Therapy: an Optimal Control Approach”, Journal of Theoretical Medicine, 2001 Lisette de Pillis HMC Mathematics

  49. Continuing the Modeling Process • Ask new questions: Example – Are there better treatments that can cure when traditional treatments do not? • How to use our model: Experiment with timings, Apply optimal control. Lisette de Pillis HMC Mathematics

  50. Tumor Growth - No Medication E(0) = 0.15 E(0) = 0.1 Lisette de Pillis HMC Mathematics

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