Transcript Introduction to Mathematical Modeling in

Introduction to Mathematical
Modeling in Biology with ODEs
Lisette de Pillis
Department of Mathematics
Harvey Mudd College
June 2005
Lisette de Pillis HMC Mathematics
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
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Lisette de Pillis HMC Mathematics
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
June 2005
Lisette de Pillis HMC Mathematics
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
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Lisette de Pillis HMC Mathematics
The Modeling Process
Occam’s Razor*
Model World
Real World
Interpret and Test
(Validate)
Model
Results
Solutions,
Numerics
*Occams’s Razor:
“Entia non sunt multiplicanda
praeter necessitatem”
“Things should not be
multiplied without good reason”
June 2005
Formulate Model
World Problem
Mathematical Model
(Equations)
Lisette de Pillis HMC Mathematics
Mathematical
Analysis
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)
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The Five Stages of Modeling
1.
2.
3.
4.
5.
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Ask the question.
Select the modeling approach.
Formulate the model.
Solve the model. Validate if possible.
Answer the question.
Lisette de Pillis HMC Mathematics
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
June 2005
The cell divides and becomes two cells
Lisette de Pillis HMC Mathematics
Thanks: Leah Keshet,
Ami Radunskaya
Introduction to Continuous Modeling
Cell divisions continue…
22 cells
23 cells
24 cells
June 2005
Lisette de Pillis HMC Mathematics
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.)
June 2005
Lisette de Pillis HMC Mathematics
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
dN
N (t  t )  N (t )
 lim
t 0
dt
t
• This represents the change is N with respect
to time.
June 2005
Lisette de Pillis HMC Mathematics
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) + K N(t) Dt
Total density
at time t+Dt
Total density at time t + increase in
≈ density due to reproduction during time
interval Dt
• Rewrite: (N(t+Dt) – N(t))/Dt ≈ KN(t)
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Lisette de Pillis HMC Mathematics
Our Cell Division Model: Getting the ODE
• Take the limit as Dt → 0
dN dT  KN
“Exponential growth” (Malthus:1798)
• Analytic solution possible here.
N (t )  N 0 e
Kt
N 0  N( 0 )
• Implication: Can calculate doubling time
June 2005
Lisette de Pillis HMC Mathematics
Analysis of Cell Division Model: Exponential
• Find “population doubling time” t:
N ( ) N0  2
imply
N (t )  N0e
K
2e
and
Kt
Taking logs and solving for t gives
ln(2)  K
  ln(2) / K
• Point: doubling time inversely proportional to
reproductive constant K
June 2005
Lisette de Pillis HMC Mathematics
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.
•
•
•
•
•
2 10 : In 30 days, 1 cell →→ detectable population
3
109 is about a 1cm sphere (bag)
1011 is about a 100 grams (1/10 kilo) of tumor (bag)
30
9
Tumor will reach 100 grams between days 36 and 37.
One week later, tumor weighs a kilo (at around 1012
cells) and is lethal.
11
• 90% tumor removal of 10 cells leaves 10 billion cells.
• 99% removals leaves 1 billion cells.
• Every cancer cell must be killed to eliminate the tumor
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Exponential Growth: Realistic?
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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:
dN dt   CN
dC dt   dN dt   CN
• Simplify the system of ODEs (collapse):
dN dt   C0   N  N
•
Logistic Growth Law!
• Note: equiv to assuming K=K(N)=C0- aN, ie K is density
dependent.
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Analysis of Logistic Model for Cell Growth
• Solution:
C 
N (t )  N 0  0 
 

 C0 
  C0 t 
 N 0     N 0  e

  



• 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
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Exponential versus Logistic Growth
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Logistic Growth: Initial Conditions, Stability
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Other Growth Models
• Power Law:
dN dt  aNb
Solution :
N (t )  ((1  b)(at  C ))1 (1b )
C  N 01b 1  b ,
• Gompertz:
N 0  N (0)
dN dt  gN
dg dt  ag
Alt : dN dt  aN ln1 bN 


Solution: N (t )  bN0 expat  b
• Von Bertlanffy: dN dt  aN  bN 1
1
1
exp(avt) 

Solution : N (t )  N 0  N 0 1  exp(at ) 


b
b



N 0  N (0)
June 2005
Lisette de Pillis HMC Mathematics
June 2005
Logistic
Von Bertalanffy
Gompertz
Power Law
Intrinsic Cell Growth Models: Comparisons
Lisette de Pillis HMC Mathematics
Dynamic Population Model Formulation:
General Approach
• Balance (Conservation):
Population
Change in Time
= Stuff Going In – Stuff Going Out
• Law of Mass Action: Encounters between populations
occur randomly, and the number of encounters is
proportional to the product of the populations, eg,
Prey:
dN dt  aN1  N k N   bNM
Predator: dM dt  cM 1  M kM   dNM
Used to represent Inter- and Intra-Species Competition
June 2005
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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
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Lisette de Pillis HMC Mathematics
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)
June 2005
Lisette de Pillis HMC Mathematics
A Two Population System
dE
 s  rET ( T )  c1 ET  dE
dt
dT
 aT (1  bT )  c2 ET
dt
• Rate parameters (units)
• s=constant immune cells source rate (#cells/day)
• s=steepness coefficient (#cells)
•
•
•
•
•
•
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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
Model Elements
Population change in time
Stuff going in
Stuff going out
dE
 s  rET ( T )  c1 ET  dE
dt
dT
 aT (1  bT )  c2 ET
dt
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Lisette de Pillis HMC Mathematics
Model Elements
Mass
Action
Logistic Growth
MichaelisMenten
dE
 s  rET ( T )  c1 ET  dE
dt
dT
 aT (1  bT )  c2 ET
dt
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Lisette de Pillis HMC Mathematics
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
June 2005
Lisette de Pillis HMC Mathematics
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
June 2005
Lisette de Pillis HMC Mathematics
Dynamical Systems Analysis: When we cannot
solve analytically
Find equilibrium points
• Set ODEs to 0:
• Therefore:
dE dt  0
dT dt  0
s  rET   T   c1 ET  dE  0
aT 1  bT   c2 ET  0
• Solve for E and T curves (nullclines). Find
points of overlap (intersections).
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Lisette de Pillis HMC Mathematics
Analysis: the equilibria are determined by setting
both differential equations to zero.
E-equation = 0
T-equation = 0
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Lisette de Pillis HMC Mathematics
Each stable equilibrium point has a
basin of attraction
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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…
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Lisette de Pillis HMC Mathematics
Alternate Parameters: Tumor Dormancy with
Immune System Evident
• Four equilibria - two stable
• Dormancy: stable spiral
T
u
m
o
r
Immune
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Lisette de Pillis HMC Mathematics
Alternate Parameters – Dangerous Regrowth
with Immune System
• Creeping through to dangerous equilibrium:
T
u
m
o
r
Immune
June 2005
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Step 5: Answer the Question
• Question: Now do we see dormancy?
Yes!
• Question: Now do we see aggressive
regrowth in this model?
Yes!
June 2005
Lisette de Pillis HMC Mathematics
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.
June 2005
Lisette de Pillis HMC Mathematics
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
June 2005
Lisette de Pillis HMC Mathematics
Three Population Mathematical Model
Population change in time
Stuff going in
Stuff going out
• Combine Effector (Immune), Tumor,
Normal Cells
dE dt  s   ET ( A  T )  c1 ET  d1 E
dT dt  r1T (1  b1T )  c2 ET  c3TN
dN dt  r2 N (1  b2 N )  c4TN
Note: There is always a tumor-free equilibrium at (s/d,0,1)
June 2005
Lisette de Pillis HMC Mathematics
Analysis: Finding Null Surfaces
• Curved Surface:
s( A  T )
dE dt  0  E 
c
T
(
A

T
)

d
(
A

T
)

rT
1
1
• Planes
1  c2   c3 
dT dt  0  T  0 or T     E    N
b1  b1r1   b1r1 
1  c4 
T
dN dt  0  N  0 or N   
b2  b2 r2 
June 2005
Lisette de Pillis HMC Mathematics
Null surfaces: Immune, Tumor, Normal cells
June 2005
Lisette de Pillis HMC Mathematics
Analysis: Determining Stability of Equilibrium
Points
• Linearize ODE’s about (eg, tumor-free)
equilibrium point
• Solve for system eigenvalues:
1  d1  0
Always Negative
2  r2  c2 b2  0
Always Negative
3  r1  c3 s d1  c2 b2 Positive or Negative
June 2005
Lisette de Pillis HMC Mathematics
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)
June 2005
Lisette de Pillis HMC Mathematics
Time Series Plots
• Creeping Through to Dangerous
Equilibrium:
June 2005
Lisette de Pillis HMC Mathematics
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
June 2005
Lisette de Pillis HMC Mathematics
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:
 ku
i
F (u )  a (1  e
June 2005
Lisette de Pillis HMC Mathematics
)
Normal,Tumor & Effector cells with Chemotherapy
• Four populations:
dE dt  s  rET ( A  T )  c1 ET  d1 E  a1 (1  e  u ) E
u
dT dt  r1T (1  b1T )  c2 ET  c3TN  a2 (1  e )T
u
dN dt  r2 N (1  b2 N )  c4TN  a3 (1  e ) N
du dt  v(t )  d 2u
• Goal: control dose v (t ) to minimize tumor
• See: “A Mathematical Tumor Model with Immune Resistance and Drug
Therapy: an Optimal Control Approach”, Journal of Theoretical Medicine,
2001
June 2005
Lisette de Pillis HMC Mathematics
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.
June 2005
Lisette de Pillis HMC Mathematics
Tumor Growth - No Medication
E(0) = 0.15
June 2005
E(0) = 0.1
Lisette de Pillis HMC Mathematics
Tumor Growth - Traditional Pulsed Chemotherapy
I(0) = 0.15
June 2005
Lisette de Pillis HMC Mathematics
I(0) = 0.1
Compare to chemotherapy based on
Optimal Control Theory:
I(0) = 0.15
June 2005
I(0) = 0.1
Lisette de Pillis HMC Mathematics
Continuing the Modeling Process
• 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?
June 2005
Lisette de Pillis HMC Mathematics
Mass Action Law: Does it Fit Data?
Tumor Cell Lysis by NK-Cells: Fit to Mouse Data
Conventional effector-target interaction term:
Rate of T argetCell Lysis by NK-cells  cNT
June 2005
Lisette de Pillis HMC Mathematics
Rational Law a Better Fit for CD8+T Cells
Tumor Cell Lysis by CD8+T-Cells: Fit to Mouse Data
Conventional product (power) form not necessarily a good fit for CD8+T-Target interactions
NEW EFFECTOR to TARGET LYSIS LAW:
( L / T )eL
Rate of T argetCell Lysis by T- cells  d
T
s  ( L / T )eL
June 2005
Lisette de Pillis HMC Mathematics
Ratio Dependence
Ratio Dependence: A Predator-Prey Model
dT
 f1 (T )T  g (T , L) L
dt
dL
 f2 ( g (T , L), L)  m L
dT
g(T,L) = “Functional Response”
f2(T,L) = “Numerical Response”
Refs: June 2005
Akcakaya et al. Ecology Apr 1995
Abrams and Ginzberg TREE Aug 2000
Lisette de Pillis HMC Mathematics
Ratio Dependence
Ratio Dependence: A Predator-Prey Model
Our “Functional Response:”

L
g (T , L)  d
L 

sT    L
T 
Our “Numerical Response:”
A Michaelis-Menten type function of g(T,L).
June 2005
Lisette de Pillis HMC Mathematics
Ratio Dependence: Good Fit to Data
Tumor Cell Lysis by CD8+T-Cells: Fit to Mouse Data
Conventional power form not necessarily a good fit for CD8+T-Target interactions
Close Up: POWER vs RATIONAL LAWS:
Power vs Rational:
Power vs Rational:
Non-Ligand-Transduced
Ligand-Transduced
June 2005
Lisette de Pillis HMC Mathematics
CD8-Tumor Lysis Equations: Error Comparison
Goodness of Fit for CD8+T-cell Lytic Activity: Comparing the residuals (error)
of the conventional product form with the new rational form.
June 2005
Lisette de Pillis HMC Mathematics
CD8-Tumor Cell Lysis Equations: Fit to Human Data
NEW EFFECTOR to TARGET LYSIS LAW
applies to HUMAN DATA:
( L / T ) eL
Rate of T argetCell Lysis by T - cells  d
T
s  ( L / T ) eL
June 2005
Lisette de Pillis HMC Mathematics
Continuing the Modeling Process
• 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
June 2005
Lisette de Pillis HMC Mathematics
New Model Equations: Two Immune Populations,
Ratio Dependent Kill Term
dT
 aT(1  bT)  cNT  dD
dt
2
dN
T
 e  fN  g
 pNT
2 N
dt
h T
dL
D2
  mL  j
 qLT
2 L
dt
k D
Where D 
June 2005
L T eL
eL T
L
s  T
Lisette de Pillis HMC Mathematics
Logistic Growth
NK-Tumor Kill:
Power Law
CD8-Tumor Kill:
Rational Law
Immune Recruitment:
Michaelis-Menten
Kinetics
System of Model Equations: Additional Treatment
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
IL-2 boost
June 2005
No IL2
Lisette de Pillis HMC Mathematics
Treatment: Chemotherapy Alone, Cancer Escapes
Healthy Immune System.
Twice Tumor Burden T0=2x107
Multiple Chemotherapy Doses.
Bolus chemotherapy every 10 days
June 2005
Lisette de Pillis HMC Mathematics
Simulation parameters: human, with chemo, no vaccine, u small
Treatment: Vaccine Therapy Alone, Cancer Escapes
Healthy Immune System.
Twice Tumor Burden T0=2x107
Single Vaccine Dose.
June 2005
Lisette de Pillis HMC Mathematics
Simulation parameters: human, vaccine alone, u small
Treatment: Vaccine and Chemo Combined
Cancer Is Controlled
Healthy Immune System.
Twice Tumor Burden T0=2x107
Single Vaccine Dose.
Three Chemotherapy Doses
June 2005
Lisette de Pillis HMC Mathematics
Simulation parameters: human, with chemo, with vaccine, u small
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).
June 2005
Lisette de Pillis HMC Mathematics
Stability: Zero Tumor Equilibrium
June 2005
Lisette de Pillis HMC Mathematics
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
June 2005
Lisette de Pillis HMC Mathematics
Bifurcation diagram: the effect of varying the NK-kill rate, c.
June 2005
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Sensitivity to Initial Conditions after Bifurcation Point.
C*=0.9763
June 2005
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Bifurcation Diagram: CD8+T Parameter j
June 2005
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Basin of Attraftion of zero−tumor and high−tumor equilibria
June 2005
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Bifurcation Analysis: Basins of Attraction
The barrier separates
system-states which evolve
towards the low-tumorburden equilibrium from
those which evolve towards
the high tumor-burden state.
No therapy
With
Chemotherapy
June 2005
This barrier moves
with therapy
With Immunotherapy
Lisette de Pillis HMC Mathematics
Additional Model Extensions – Extending from
ODEs to PDEs and Cellular Automata
•Add spatial heterogeneity: non-uniform
tissue, morphology-dependent.
•Cellular automata: discrete, probabilistic,
and/or hybrids.
June 2005
Lisette de Pillis HMC Mathematics
Spatial Tumor Growth Modeling
Deterministic & Probabilistic:2D and 3D
http://www.lbah.com/Rats/ovarian_tumor.htm
http://www.lbah.com/Rats/rat_mammary_tumor.htm
June 2005
Lisette de Pillis HMC Mathematics
Image Courtesy http://www.ssainc.net/images/melanoma_pics.GIF
http://www.loni.ucla.edu/~thompson/HBM2000/sean_SNO2000abs.html
Modeling Tumor Growth and Treatment
L.G. de Pillis & A.E. Radunskaya
Microenvironment Simulations: Entire System.
June 2005
Lisette de Pillis HMC Mathematics
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
June 2005
Lisette de Pillis HMC Mathematics
Thanks for listening!
Lisette de Pillis
[email protected]
June 2005
Lisette de Pillis HMC Mathematics
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June 2005
Lisette de Pillis HMC Mathematics
CA Simulation: Movie - a snapshot every 20 days for 200
days showing tumor growth and necrosis.
QuickTime™ and a
H.263 decompressor
are needed to see this picture.
June 2005
Lisette de Pillis HMC Mathematics
The tumor affects the acidity of the micro-environment:
QuickTime™ and a
DV/DVCPRO - NTSC decompressor
are needed to see this picture.
June 2005
Lisette de Pillis HMC Mathematics
Modeling Tumor Growth and Treatment
A.E. Radunskaya
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).
What next ?
•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
responseLisette
to treatment?
June 2005
de Pillis HMC Mathematics