introduction to mathematical modeling in biology with odes

June 2005 Lisette de Pillis HMC Mathematics

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?

– “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

– Mathematical Biology involves mathematically modeling biological phenomena

Thanks: Cliff Taubes, 2001


Mathematical Modeling Philosophy

• Why are models useful:– Formulating precise ideas implicit

assumpltions less likely to “slip by”– Mathematics = concise language that

encourages clarity of communication– Mathematical theorems and computational

resources can be accessed


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


The Modeling Process

Model World

Mathematical Model(Equations)

Real World

Occam’s Razor*

Interpret and Test(Validate)

Formulate ModelWorld Problem

Model Results

Mathematical Analysis

Solutions,Numerics

*Occams’s Razor: “Entia non sunt multiplicanda praeter necessitatem”“Things should not be multiplied without good reason”


Model World

Components of the 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)


The Five Stages of Modeling

1. Ask the question.

2. Select the modeling approach.

3. Formulate the model.

4. Solve the model. Validate if possible.

5. Answer the question.


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):

Thanks: Leah Keshet, Ami Radunskaya

A tumor starts as one cell The cell divides and becomes two cells


Introduction to Continuous Modeling

24 cells

Cell divisions continue…

22 cells

23 cells


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.)


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.

t

tNttN

dt

dNt

)()(lim

0


• 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) + K N(t) Dt

• Rewrite: (N(t+Dt) – N(t))/Dt ≈ KN(t)

Our Cell Division Model: Getting the ODE

Total density at time t+Dt

Total density at time t + increase in density due to reproduction during time interval Dt

≈


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

)N(N

eNtN Kt

0

)(

0

0

KNdTdN


• Find “population doubling time” t:

• Point: doubling time inversely proportional to reproductive constant K

Analysis of Cell Division Model: Exponential

KteNtN 0)( Ke2

2)( 0 NN

K)2ln( K/)2ln(

and

imply

Taking logs and solving for t gives


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

910

930 102

1210

1110

31cm

1110


Exponential Growth: Realistic?


Extending the Growth Model: Additional Assumptions + New System

• Reproductive rate K is 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.

CNdtdNdtdC

CNdtdN

NNCdtdN 0


• 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

tCeN

CN

CNtN 0

00

00

0)(

Analysis of Logistic Model for Cell Growth


Exponential versus Logistic Growth


Logistic Growth: Initial Conditions, Stability


Other Growth Models

• Power Law:

• Gompertz:

• Von Bertlanffy:

baNdtdN

)0(,1

)))(1(()(

:Solution

010

)1(1

NNbNC

CatbtNb

b

bNaNdtdN

agdtdg

gNdtdN

1ln:Alt

bbNtN at exp0)(:Solution

1 bNaNdtdN

)0(

)exp()exp(1

1)(:Solution

0

1

00

NNb

avttaNN

btN


Intrinsic Cell Growth Models: Comparisons

Von

Ber

tala

nffy

Logi

stic

Gom

pert

zP

ower

Law


Used to represent Inter- and Intra-Species Competition

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,

PopulationChange in Time

= Stuff Going In – Stuff Going Out

dNMkMcMdtdM

bNMkNaNdtdN

M

N

1:Predator

1 :Prey


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


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)


• 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))

A Two Population System

dEETcsdt

dET

rET 1)(

ETcbTaTdt

dT2)1(


Model Elements

dEETcsdt

dET

rET 1)(

ETcbTaTdt

dT2)1(

Stuff going in Stuff going outPopulation change in time


Model Elements

dEETcsdt

dET

rET 1)(

ETcbTaTdt

dT2)1(

Logistic GrowthMassAction

Michaelis-Menten


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


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


Dynamical Systems Analysis: When we cannot solve analytically

• Set ODEs to 0:

• Therefore:

• Solve for E and T curves (nullclines). Find points of overlap (intersections).

0

0

dtdT

dtdE

01

0

2

1

ETcbTaT

dEETcTrETs

Find equilibrium points


Analysis: the equilibria are determined by setting both differential equations to zero.

E-equation = 0T-equation = 0


Each stable equilibrium point has a basin of attraction


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…


Alternate Parameters: Tumor Dormancy with Immune System Evident

• Four equilibria - two stable• Dormancy: stable spiral

I m m u n e

Tumor


Alternate Parameters – Dangerous Regrowth with Immune System

• Creeping through to dangerous equilibrium:

Tumor

I m m u n e


Step 5: Answer the Question

• Question: Now do we see dormancy?

• Question: Now do we see aggressive regrowth in this model?

Yes!

Yes!


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.

Continue the modeling cycle…


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


Three Population Mathematical Model

• Combine Effector (Immune), Tumor,

Normal Cells

TNcNbNrdtdN

TNcETcTbTrdtdT

EdETcTAETsdtdE

422

3211

11

)1(

)1(

)(

Note: There is always a tumor-free equilibrium at (s/d,0,1)

Stuff going in Stuff going outPopulation change in time


Analysis: Finding Null Surfaces

Trb

c

bNNdtdN

Nrb

cE

rb

c

bTTdtdT

rTTAdTATc

TAsEdtdE

22

4

2

11

3

11

2

1

11

1or 00

1or 00

)()(

)(0

• Curved Surface:

• Planes


Null surfaces: Immune, Tumor, Normal cells


Analysis: Determining Stability of Equilibrium Points

• Linearize ODE’s about (eg, tumor-free) equilibrium point

• Solve for system eigenvalues:

Negativeor Positive

Negative Always 0

Negative Always 0

221313

2222

11

bcdscr

bcr

d


CoExisting Equilibria Map: Paremeter Space s

Region 4: Stable @ (E=0.4, T=0.6, N=0.4) Unstable @ (E=0.8, T=0.2, N=0.8)


Time Series Plots

• Creeping Through to Dangerous Equilibrium:


Evolution in Time: Increasing Initial Immune Strength

Initial Immune Strength Range: 0.0 < E(0) < 0.3Basin Boundary Range: 0.12 < E(0) < 0.15

Stable Equilibrium - Co-Existing: E=0.4, T=0.6, N=0.4Stable Equilibrium - Tumor Free: E=1.65, T=0, N=1.0

Time vs Tumor Time vs Normal Time vs Immune


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: )1()( ku

i eauF


• 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

Normal,Tumor & Effector cells with Chemotherapy

udtvdtdu

NeaTNcNbNrdtdN

TeaTNcETcTbTrdtdT

EeaEdETcTArETsdtdE

u

u

u

2

3422

23211

111

)(

)1()1(

)1()1(

)1()(

)(tv


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.


Tumor Growth - No Medication

E(0) = 0.1E(0) = 0.15


Tumor Growth - Traditional Pulsed Chemotherapy

I(0) = 0.15 I(0) = 0.1


Compare to chemotherapy based on Optimal Control Theory:

I(0) = 0.15 I(0) = 0.1



• More questions:

• Can we validate the model?

• Are there experimental data against which we can compare model components?

• If we find data, can we modify our dynamics?


Conventional effector-target interaction term:cNTcellsNK-by Lysis CellTarget of Rate

Tumor Cell Lysis by NK-Cells: Fit to Mouse Data

Mass Action Law: Does it Fit Data?


Tumor Cell Lysis by CD8+T-Cells: Fit to Mouse DataConventional product (power) form not necessarily a good fit for CD8+T-Target interactions

TTLs

TLd

eL

eL

)/(

)/(cells-Tby Lysis CellTarget of Rate

NEW EFFECTOR to TARGET LYSIS LAW:

Rational Law a Better Fit for CD8+T Cells


Ratio Dependence: A Predator-Prey Model

Refs:Akcakaya et al. Ecology Apr 1995Abrams and Ginzberg TREE Aug 2000

LLLTgfdT

dL

LLTgTTfdtdT

)),,((

),()(

2

1

g(T,L) = “Functional Response”

f2(T,L) = “Numerical Response”

Ratio Dependence


Ratio Dependence: A Predator-Prey Model

Our “Functional Response:”

LTL

sT

LdLTg

),(

Our “Numerical Response:”

A Michaelis-Menten type function of g(T,L).

Ratio Dependence


Tumor Cell Lysis by CD8+T-Cells: Fit to Mouse DataConventional power form not necessarily a good fit for CD8+T-Target interactions

Close Up: POWER vs RATIONAL LAWS:

Power vs Rational:Non-Ligand-Transduced Ligand-Transduced

Power vs Rational:

Ratio Dependence: Good Fit to Data


CD8-Tumor Lysis Equations: Error Comparison

Goodness of Fit for CD8+T-cell Lytic Activity: Comparing the residuals (error) of the conventionalconventional product formproduct form with the new rational form.


CD8-Tumor Cell Lysis Equations: Fit to Human Data

TTLs

TLd

eL

eL

)/(

)/(cells-Tby Lysis CellTarget of Rate

NEW EFFECTOR to TARGET LYSIS LAW applies to HUMAN DATA:



• The new dyamics require a new model:

• Develop model with different populations to track:

• Specific Immune Cells (CD8+T, Rational Kill)• Nonspecific Immune Cells (NK, Mass-Action Kill)• Tumor Cells

• Test new treatments


T

sD

qLTLDk

DjmL

dt

dL

pNTNTh

TgfNe

dt

dN

dDcNTbT)aT(1dt

dT

eLT

L

2

2

2

2

Where eLT

L

Logistic Growth

NK-Tumor Kill:Power Law

CD8-Tumor Kill:Rational Law

Immune Recruitment:Michaelis-Menten Kinetics

New Model Equations: Two Immune Populations, Ratio Dependent Kill Term


Parameters a, b, c, d, s, and eL were fit from published experimental data. All other parameters were estimated or taken from the literature.

Circulating lymphocytes

Rate of drug administration and decay

No IL2

IL-2 boost

System of Model Equations: Additional Treatment


Treatment: Chemotherapy Alone, Cancer Escapes

Healthy Immune System.

Twice Tumor Burden T0=2x107

Multiple Chemotherapy Doses.

Simulation parameters: human, with chemo, no vaccine, u small

Bolus chemotherapy every 10 days


Treatment: Vaccine Therapy Alone, Cancer Escapes



Single Vaccine Dose.

Simulation parameters: human, vaccine alone, u small


Treatment: Vaccine and Chemo Combined

Cancer Is Controlled



Single Vaccine Dose.

Simulation parameters: human, with chemo, with vaccine, u small

Three Chemotherapy Doses


Equilibrium points of 4-population system (no treatment) are found at points where the values of LE1 from equation (20) intersect with the solutions L of equation (21). These points of intersection can be found numerically, yielding equilibrium

point(s) (TE,NE,LE,CE).


Stability: Zero Tumor Equilibrium


Stability: Specific Parameter Set

• With the specific parameter set:– The zero tumor equilibrium is unstable – There is only one non-zero tumor equilibrium,

and it is stable.

• Point: – The tumor-free equilibrium is unstable, while

the high-tumor equilibrium is stable: Only a change in system parameter values may permit permanent removal of the tumor → Immunotherapy/Vaccine is one way to do this


Bifurcation diagram: the effect of varying the NK-kill rate, c.


Sensitivity to Initial Conditions after Bifurcation Point. C*=0.9763


Bifurcation Diagram: CD8+T Parameter j


Basin of Attraftion of zero−tumor and high−tumor equilibria


Bifurcation Analysis: Basins of Attraction

The barrier separates system-states which evolve towards the low-tumor-burden equilibrium from those which evolve towards the high tumor-burden state.

With Immunotherapy

With Chemotherapy

No therapy

This barrier moves with therapy


•Add spatial heterogeneity: non-uniform tissue, morphology-dependent.

•Cellular automata: discrete, probabilistic, and/or hybrids.

Additional Model Extensions – Extending from ODEs to PDEs and Cellular Automata


Deterministic & Probabilistic:2D and 3D

Image Courtesy http://www.ssainc.net/images/melanoma_pics.GIF

http://www.lbah.com/Rats/rat_mammary_tumor.htm

http://www.lbah.com/Rats/ovarian_tumor.htm

http://www.loni.ucla.edu/~thompson/HBM2000/sean_SNO2000abs.html

Spatial Tumor Growth Modeling


Microenvironment Simulations: Entire System.

Modeling Tumor Growth and TreatmentL.G. de Pillis & A.E. RadunskayaL.G. de Pillis & A.E. Radunskaya


Final Thoughts on Modeling

• “All models are wrong…some are useful”, Box and Draper, 1987

• “All decisions are based on models…and all models are wrong”, Sterman, 2002

• “Although knowledge is incomplete, nonetheless decisions have to be made. Modeling…takes place in the effort to plan clinical trials or understand their results. Formal modeling should improve that effort, but cautious consideration of the assumptions is demanded”, Day, Shackness and Peters, 2005


Thanks for listening!

Lisette de Pillis

[email protected]


BLANK


CA Simulation: Movie - a snapshot every 20 days for 200 days showing tumor growth and necrosis.

QuickTime™ and aH.263 decompressor

are needed to see this picture.


QuickTime™ and aDV/DVCPRO - NTSC decompressor

are needed to see this picture.

The tumor affects the acidity of the micro-environment:


•Add nano-vaccines (based on mouse models).

•What are some of the questions under discussion? (Dose, treatment scheduling, where to administer the vaccine)

•What parameters might be good indicators of successful response to treatment?

Modeling Tumor Growth and TreatmentA.E. RadunskayaA.E. Radunskaya

What next ?

What do we have?

•A mathematical model which simulates some of the main features of tumor growth: hypoxia, high acidity, necrosis.

•A model which allows the addition of other cells (immune cells) and/or small molecules (drugs, vaccines).

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