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Modeling Cancer-Immunology Dynamics
L.G. de PillisHarvey Mudd College
Claremont, California, USA
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Modeling Cancer-Immunology Dynamics
L.G. de PillisHarvey Mudd College
Claremont, California, USADenver, CO
Claremont, CA
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A Cooperative Endeavor
• Ami Radunskaya• Charles Wiseman• Dann Mallet• Seema Nanda• Angela Gallegos• Renee Fister• Sarah Hook• Yi Jiang• Weiqing Gu• Helen Moore• Yixin Guo• Erika Camacho• Elissa Schwartz
• Chris DeBoever• Helen Wu• Elizabeth Howe• Ajay Shenoy• Lindsay Crowl• Lorraine Thomas• Michael Vrable• Kathe Todd-Brown• Allison Wise• Tiffany Head• Kenji Kozai• Kenneth Maples• Anand Murugan• Todd Neal• James Moore• David Gross• Benjamin Preskill• Michael Daub
Lisette de PillisHarvey Mudd College
Claremont, California,USA
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Outline• Background: Cancer and genesis of this work• The original motivating problem• Controlling treatment• Immunology of cancer
– The Immune System Targets Cancer– Two Kinds of Immune Responses: Innate and Specific
• Extended model, include immune components
• Adding treatment• Specific cancer: B-CLL• Work in progress: Spatial tumor models
– Hybrid CA models– 3D PDE models
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What is Cancer?• Cancer is a cellular
disorder.• There are several
hundred types of cancer, but all have some general characteristics in common.
• It can begin with just one cell gone awry...
Thanks: cancer-info.com
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Cancer: Uncontrolled Growth• Cancer cells experience uncontrolled and
disorganized growth.• Cancer cells can divide “forever” but never
differentiate (vs normal cell 50x limit)
Thanks:www.sciencemuseum.org.uk
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• What: Simulation of tumor-immune dynamics:• Provide low-cost prediction, explanation.
• Why: Dr. Wiseman’s MoM group• Goals:
• Math model with range of dynamics, ability to simulate real laboratory and clinical data.
• Focus on immune-tumor interactions and treatment modeling.• Process, Method and Analysis:
• Model with differential equations and cellular automata. • Choose/create functions with empirical/biological fit to existing
experimental data.
Our Mathematical Model:What and Why
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Outline• Background: Cancer and genesis of this work• The original motivating problem• Controlling treatment• Immunology of cancer
– The Immune System Targets Cancer– Two Kinds of Immune Responses: Innate and Specific
• Extended model, include immune components
• Adding treatment• Specific cancer: B-CLL• Work in progress: Spatial tumor models
– Hybrid CA models– 3D PDE models
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• Why might a tumor grow when it is treated, and shrink when it is not? That is, In the clinic, what causes asynchronous response to chemotherapy?
• Note: The 2 population models of the time did not answer this question…We needed to extend the models.
The first question we investigated…
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Could competition for resources cause the asynchronous response?
• Develop a three then four population model (dePillis and Radunskaya, 2001, 2003): include normal cell competition andchemo
• Why: Gives more realistic response to chemotherapy treatments: allows for delayed response to chemotherapy
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dE dt = s + ρET (a + T) − c1ET − d1EdT dt = r1T(1− b1T) − c2ET − c3TNdN dt = r2N(1− b2N) − c4TN
Three Population Mathematical Model
• Combine Effector (Immune), Tumor,
Normal Cells
Note: There is always a tumor-free equilibrium at (s/d,0,1)
Stuff going in Stuff going outPopulation change in time
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Analysis: Finding Null Surfaces
• Curved Surface:
• PlanesdE dt = 0 ⇒ E =
s(A + T)c1T(A + T) + d1(A + T) − rT
dT dt = 0 ⇒ T = 0 or T =1b1
−c2
b1r1
⎛ ⎝ ⎜
⎞ ⎠ ⎟ E −
c3
b1r1
⎛ ⎝ ⎜
⎞ ⎠ ⎟ N
dN dt = 0 ⇒ N = 0 or N =1b2
−c4
b2r2
⎛ ⎝ ⎜
⎞ ⎠ ⎟ T
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Null Surfaces: Immune, Tumor, Normal cells
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Analysis: Determining Stability of Equilibrium Points
• Linearize ODE’s about (eg, tumor-free) equilibrium point
• Solve for system eigenvalues:
λ1 = −d1 < 0 Always Negativeλ2 = −r2 − c2 b2 < 0 Always Negativeλ3 = r1 − c3s d1 − c2 b2 Positive or Negative
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CoExisting Equilibria Map: Parameter Spaceρ − s
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Cell Response to Chemotherapy• To add drug response term to each DE,
create new DE describing drugAmount of cell kill for given amount of drug u:
F(u) = ai(1− e−ku)
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• Four populations:
• Chemotherapy dose to treat tumor• See: “A Mathematical Tumor Model with Immune Resistance and Drug
Therapy: an Optimal Control Approach”, Journal of Theoretical Medicine, 2001
v(t)
Normal,Tumor & Immune Cells with Chemotherapy
dE dt = s + rET (A + T) − c1ET − d1E − a1(1− e−u)EdT dt = r1T(1− b1T) − c2ET − c3TN − a2(1− e−u)TdN dt = r2N(1− b2N) − c4TN − a3(1− e−u)Ndu dt = v(t) − d2u
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Question Answered – Asynchronous (Delayed) Response happens with Immune
System and Normal Cells
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Outline• Background: Cancer and genesis of this work• The original motivating problem• Controlling treatment• Immunology of cancer
– The Immune System Targets Cancer– Two Kinds of Immune Responses: Innate and Specific
• Extended model, include immune components
• Adding treatment• Specific cancer: B-CLL• Work in progress: Spatial tumor models
– Hybrid CA models– 3D PDE models
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• 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
Second Question: Can we find a better chemotherapy schedule?
dE dt = s + rET (A + T) − c1ET − d1E − a1(1− e−u)EdT dt = r1T(1− b1T) − c2ET − c3TN − a2(1− e−u)TdN dt = r2N(1− b2N) − c4TN − a3(1− e−u)Ndu dt = v(t) − d2u
v(t)
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•Objective function options:•Minimize a combination of total tumor and final tumor burden.
•Minimize amount of drug given, maximize the number of effector cells.
•Constraint options:•Always keep circulating lymphocytes above a given threshold.•Treat only when circulating lymphocytes are above a threshold.•Fix total amount of drug given.
(Experiment with different options …)
Optimal Control: Therapy Designprovides a theoretical framework to solve the problem: maximize or minimize
X (objective) while making sure that Y is … (constraint)
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Basic Optimal Control Problem:• Let (Effector,Tumor,Normal)=• Find control variable v(t) that minimizes
objective functional
• subject to state equations with IC’s
• and inequality constraint
This problem admits Bang-Bang solutions (on or off)
J[x,v] = K1 ⋅ x2(t f ) + K2 ⋅ x2(t)dtt0
t f
∫
dx dt = f (x(t),v(t), t), x(t0) = x0
g(x(t),v(t)) = x3(t) − .75 ≥ 0 t ∈[t0,t f ]
x = (x1, x2, x3)
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Basic Optimal Control Solution• Pontryagin’s Max/Min Theorem: If
Hamiltonian H is
• where only when• is the integrand of the objective J• then v(t) is a candidate for a max/min of J
if we can find co-state variables psatisfying
• and v(t) is such that
H = θ + (pT ⋅ f ) + ηgη(t) > 0 g(x(t),v(t)) = 0
θ
dpi
dt= −
∂H∂xi
, pi(t f ) = ∂J ∂xi |t f
∂H ∂v = 0
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Bang-Bang Solutions
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Optimal Control Solutions
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Tumor Growth - No Medication
E(0) = 0.1E(0) = 0.15
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I(0) = 0.15 I(0) = 0.1
Tumor Growth - Pulsed Chemo
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I(0) = 0.15 I(0) = 0.1
Tumor Growth - Optimal Control Chemo
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Tumor Growth - Optimal Control Chemotherapy
Current work: New models with quadratic and linearOptimal control: Analysis
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Tumor Growth - Optimal Control ChemotherapySingle Quadratic Control: No Singularities
• T(t), tumor cells• N(t), natural killer
effector cells• C(t), circulating
lymphocytes• M(t), chemotherapy in
patient• v_M(t), chemotherapy
drug dose
J(VM ) = T(t) + εM VM2 (t)( )
0
t f
∫ dt
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Tumor Growth - Optimal Control ChemotherapySingle Quadratic Control: No Singularities
J(VM ) = T(t) + εM VM2 (t)( )
0
t f
∫ dt
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Tumor Growth – Single Linear Optimal Control of Chemotherapy
Determining Singular Regions
J = (T(t) + εM VM (t)( )0
t f
∫ dt
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Tumor Growth – Single Linear Optimal Control of Chemotherapy
J = (T(t) + εM VM (t)( )0
t f
∫ dt J = T(t)0
t f
∫ dt, Constraint: N>0.1N(0)
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Outline• Background: Cancer and genesis of this work• The original motivating problem• Controlling treatment• Immunology of cancer
– The Immune System Targets Cancer– Two Kinds of Immune Responses: Innate and Specific
• Extended model, include immune components
• Adding treatment• Specific cancer: B-CLL• Work in progress: Spatial tumor models
– Hybrid CA models– 3D PDE models
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The next question:
• What role can immunotherapy and vaccine therapy play in cancer treatment?
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Cancer Immunology in the News
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Treatment: Day 0 - Anti CD-3 10-75 mcg iv/60 minDay 1 - Cyclophosphamide 300 mg/m^2Day 28 - Re-evaluate, MRI, re-treat
Immunotherpay: Clinical Response to Anti-CD3Cancer Immunotherapy
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VACCINES and IMMUNOTHERAPY•Immunotherapy boosts immune resistance with biological response modifiers•Vaccine Therapy (special case) boosts immune resistance with modified tumor challenge•Use: Vaccines used mainly therapeutically, not yet preventatively.•Sometimes Only Option: When chemo won’t work. Certain cancers good candidates, eg, melanoma,glioma•Benefits: Low toxicity, potentially high efficacy
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Thanks: National Cancer Institute
Immune System Targets Cancer
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Experimental Data: Basis for ODE ModelsMouse Data: Basis for Preliminary VaccineTherapy in Model.The Diefenbach et al.[2] study
Human Data: Basis for Immunotherapy in Model:The Rosenberg et al. [4] study
Data Evidence
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Diefenbach mouse trials with various vaccination strategies.
Mouse Lab Data: Preventative Vaccination
Mice “vaccinated” with
ligand transduced cells,
then rechallenged with
control-transduced cells,
were proteced.reprinted from Nature, 2001;41:165-171
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Diefenbach mouse trials with varying tumor challenge levels.
Mouse Lab Data:CD8 vs NK Protection
Black circles:RMA-Rae1b
Ligand Transduced Cells
reprinted from Nature, 2001;41:165-171
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How the Immune System Works
• Huge army of “defender cells”: White Blood Cells
• Body creates about 1000 million per day
• Natural Immunity: Regular Patrols (“Secret handshake”)
• Specific Immunity:Activated After Invasion (“Glove sniffing dog”)
Coloured electron micrograph of a white blood cell.National Medical Slide Bank/Wellcome Photo Library
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The Immune System
Thanks: The Biology Project
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NK Cell Killing Cancer Cell Aspects of Tumor Immune Response: NK cells
Thanks: http://mediafreaks.com
Thanks: http://www.media-freaks.com/casestudies/eexcel_cdrom/
NKsrecognizeself (MHC-I expressed)
Down-regof MHC-I (as with certain cancer cells) allows NK-tumor lysis
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Innate Immune Response to Cancer (Natural Killer Cell = NK)
NK recognizes “self” and attacks “non-self”(the cancer): Secret Handshake
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Specific Immune Response to Cancer
• T-cell (CTL, CD8+T-cell) recognizes and attacks cancer: “Glove Sniffing Dog”
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T-cell Attacking Cancer-cell Movie
Thanks: CellsAlive.com
QuickTime™ and aVideo decompressor
are needed to see this picture.
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T-Cells Killing a Cancer Cell• Before
A fully intact cancer cell surrounded by the immune system’s killer T-cells. Notice the tentacles of the cancer cell.
• After The cancer cell is completely flattened and totally destroyed.
Thanks: cancer-info.com
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Outline• Background: Cancer and genesis of this work• The original motivating problem• Controlling treatment• Immunology of cancer
– The Immune System Targets Cancer– Two Kinds of Immune Responses: Innate and Specific
• Extended model, include immune components
• Adding treatment• Specific cancer: B-CLL• Work in progress: Spatial tumor models
– Hybrid CA models– 3D PDE models
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New Mathematical Model ComponentsCell Populations:•Tumor Cells:T(t).
•Natural Killer (NK) Cells: N(t).Nonspecific. Always present, stimulated by the presence of tumor cells.
•CD8+T Cells: L(t). Specific. Cytolytic activity and cell proliferation are increased by the presence of tumor cells. Image courtesy
http://www.wellesley.edu/Chemistry/Chem101/antibiotics/immune.html
Image courtesy http://www.immuneresources.com/cancer.htm
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Mathematical Model Flow Diagram
R = RecruitmentI = InactivationF = Fractional Cell Kill
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cNT=cellsNK-by Lysis CellTarget of Rate
Tumor Cell Lysis by NK-Cells: Fit to Mouse Data
Mathematically Modeling the Innate Immune (Secret Handshake) Killing of Cancer Cells
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Tumor Cell Lysis by CD8+T-Cells: Fit to Mouse DataConventional vs DePillis-Rad Laws
Ligand-Transduced Cancer Cells
Mathematically Modeling the Specific (Glove Sniffing) Killing of Cancer Cells:
New “de Pillis-Radunskaya Law”
TTLs
TLd ν
ν
)/()/(
:cells-Tby Lysis CellTarget of Rate
+
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TTLs
TLd ν
ν
)/()/(cells-Tby Lysis CellTarget of Rate
+=
NEW DE PILLIS-RAD LAW also applies to HUMAN DATA:
Specific (Glove Sniffing) Killing of Cancer CellsFollows New Mathematical Law
De Pillis-Radunskaya Law
Tumor Cell Lysis by CD8+T-Cells: Validated with Human Data
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More Good Fit Evidence: Fit to Other Mouse DataData from Antoni Ribas, UCLA, fit to raw chromium release assay data.
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Elements in Mathematical Model Equations
( )T
sD
qLTLDk
DjmLdtdL
pNTNTh
TgfNedtdN
dDcNTbT)aT(1dtdT
lT
L
2
2
2
2
+=
−+
+−=
−+
+−=
−−−=
Where ( ) lT
L
Logistic Growth
NK-Tumor Kill:Power LawCD8-Tumor Kill:Rational LawImmune Recruitment:Michaelis-MentenKinetics
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•Questions:•How do simulation outcomes vary as the parameters are varied?
•Which parameters are the best predictors of successful outcomes?
•One Answer: •Need Sensitivity Analysis
Are Some Parameters More Important than Others?
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Model Simulations: Traditional Sensitivity Analysisone parameter is changed at a time
Simulation parameters: human, no chemo
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Uncertainty Analysis: Latin Hypercube Samplingall parameters are varied simultaneously
[3] S.M. Blower and H. Dowlatabadi,“Sensitivity and Uncertainty Analysis of Complex Models of Disease Transmission: an HIV Model, as an Example. International Statistical Review (1994), 62,2,pp.229-243.
Sensitivity Analysis (LHS)Method: Latin Hypercube Sampling (LHS) [3].
Outcome: the uncertainty in the predicted tumor size grows over time.
Details: 10,000 sample parameter sets were randomly selected in a range centered around the estimated values, and each parameter was varied independently over its own range. Median tumor size over time is depicted by the solid blue line. Upper and lower quartiles are shown by green lines. Full range of outcomes given by red bars.
Comment: While the uncertainty in the prediction grows over time, it is clear that the distribution of tumor sizes is not uniform, but rather is concentrated at the lower tumor levels.
Simulation parameters: human, no chemo, 5% range or reported ranges,
truncated normal distribution
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Model Simulations: Latin Hypercube SamplingPRCC Results• PRCCs (partially ranked correlation coefficients): measure outcome's sensitivity to each parameter. Bar graph: Relative ranking of the six most sensitive parameters with respect to tumor size.
• Parameters d and eL: represent overall tumor-cell lysis rate and the strength of the immune-tumor interaction, respectively. Both can be estimated from patient data, as in this example. Parameter a represents tumor growth rate
• Predictions: Tumor aggressiveness as well as patient specific immune strength may predict patient response to immunotherapy treatment.
Significance w/ Student’s T:
P-values all less than 0.00001
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Validation: Simulating Vaccine in Mouse Model
Validation: These In Silico Experiments Mirror In Vivo Mouse Experiments
See: dePillis et al, Cancer Res, 65(17), 2005
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Outline• Background: Cancer and genesis of this work• The original motivating problem• Controlling treatment• Immunology of cancer
– The Immune System Targets Cancer– Two Kinds of Immune Responses: Innate and Specific
• Extended model, include immune components
• Adding treatment• Specific cancer: B-CLL• Work in progress: Spatial tumor models
– Hybrid CA models– 3D PDE models
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Experimenting with Treatments
• Must Extend the Model to Examine:– Chemo Alone– Immunotherapy Alone– Combined Therapy
• To Simulate Dudley’s Human Data• Add IL-2 immunotherapy• Add Circulating Lymphocytes to track
“health”
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Multi-Population Model Schematic
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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 Treatments Possible
See dePillis et al, J. Theor. Biol., 2005
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Bifurcation diagram: the effect of varying the NK-kill rate, c.
See dePillis et al, J. Theor. Biol., 2005
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Sensitivity to Initial Conditions after Bifurcation Point. C*=0.9763
Sensitive to ICsafter bifurcation
Beforebif, 0 unstable
After bif,0 stable
See dePillis et al, J. Theor. Biol., 2005
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Mixed Therapy - Mouse Params
No treatment Pulsed Chemo
TIL treatment TIL with Pulsed Chemo
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Mixed Therapy - Human Params
Top left: Pulsed chemo fails on 10^6 tumor (healthy immune). Top right: TIL and IL2 fail. Middle left and right: Combo therapy kills tumor. (Right has more aggressive immunotherapy)
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•Stable Zero Tumor Equilibrium:•Immune system keeps tumor under control
•Stable High Tumor Equilibrium: •Immune system too weak to control tumor
•Unstable Intermediate Tumor Equilibrium:•System wants to move toward high or zero
Bifurcation Analysis: Basins of Attraction(ODE model with IL-2)
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Basin of Attraction of zero-tumor andhigh-tumor equilibria
See dePillis et al, J. Theor. Biol., 2005
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Bifurcation Analysis: Basins of AttractionThe barrier separates system-states which evolve toward the low-tumor equilibrium from those which evolve toward the high tumor state.
With Immunotherapy
With Chemotherapy
No therapy
This barrier moves with therapy
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Outline• Background: Cancer and genesis of this work• The original motivating problem• Controlling treatment• Immunology of cancer
– The Immune System Targets Cancer– Two Kinds of Immune Responses: Innate and Specific
• Extended model, include immune components
• Adding treatment• Specific cancer: B-CLL• Work in progress: Spatial tumor models
– Hybrid CA models– 3D PDE models
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B-CLL model:A work in progress…
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Image: http://www.vimm.it/research/images/semenzato_fig1.jpg
B-Cell Chronic Lymphocytic Leukemia (B-CLL)
• Fuzzy characterization: over 5K cells / μ-liter one measure
• Precise cause unknown• No cure• High proliferation rate more serious than high
numbers of cells• NK cells, helper T cells and cytotoxic T cells
may all play a role in stemming the growth of B-CLL
• B-CLL: Cancer of the immune system. Characterized by the accumulation of large numbers of white blood cells (B cells) in the blood, bone marrow, spleen and lymph nodes.
• Current understanding: B-CLL cells derive from mature antigen-stimulated B-cells that are immunologically competent.
Subcellular localization of HS1 analyzed by confocal microscopy. HS1 is uniformly distribuited in the cytosol of normal B cells, while it shows a nuclear spotting distribution in B-CLL cells.
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Example Model Equations
sB
kaTdtdT
dBNBNbB +(r-dB)BdtdB
L
++=
−−
α
= dBTBT
dNBNBbNdtdN
−= dNN−
Growth or SourceDeath
Immune Cell KillingImmune Recruitment
− dTT dTBTB−B L TH
p
sB
aTdt
dTHL
++= − dTHTH
B L THp
bTH
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Parameter Choice: bB• Represents: B-CLL creation
– A fraction of antigenically experieced immunologically competent B cells.
– A constant source of newly mutated cells.• Units: cells/μL per day• Numerical value(s) used: 70 (range: [10,80])• Source: Numerical Bifurcation Study
Unstable
Stable
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Numerical Simulations
bB=68 bB=100
Bifurcation at bB = 70Unstable
Stable
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Numerical Simulations
Solution at Steady State Sudden Immune Depletion
dBNBNbB +(r-dB)BdtdB
−−= dBTBT
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* Messmer et al., J. Clin. Invest. 2005r range:[0.0011,0.0176], mean:0.004636dB range:[-3.9e-3,2.14e-2]
Fitting for parameters (-r+dB)Patients 331 &360*
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Treatment Possibilities: • Keating (2003) suggests:Chemotherapy, Single and Combination
alkylating agent
purine analog
Combination therapies
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Combination Therapy:
Fludarabine & Rituximab
Making it tougher:bB=.5%, leukemic cell doubling time 205 days, 50%fludarabine resistant cells, low CD20 expression
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Fludarabine vs RituximabRituximab
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Combination Therapy
Concurrent treatment best:93% lysis, killed 87700 cells/microliter of blood. Still below pre-treatment levels after 5 years.
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Outline• Background: Cancer and genesis of this work• The original motivating problem• Controlling treatment• Immunology of cancer
– The Immune System Targets Cancer– Two Kinds of Immune Responses: Innate and Specific
• Extended model, include immune components
• Adding treatment• Specific cancer: B-CLL• Work in progress: Spatial tumor models
– Hybrid CA models– 3D PDE models
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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
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Goals for Spatial Modeling• To model:
– Nutrient dependent tumor growth in 2D– Immune system dynamics
• To explore:– Effects of the immune response– Effects of tumor “gluttony”– Effects of tumor adhesivity– Dynamic Energy Budget concept (DEB)– Effects of microenvironment
• Build from:– Immunology literature, ODE concepts
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Approach: hybrid cellular automata
• Laws of evolution are written as partial differential equations or discrete rules, either stochastic or deterministic.
• Typically, all rules are eventually discretized for numerical solution.
• Inherent in these models: two time scales, one for the (fast) diffusion of small molecules, one for the evolution of cell populations.
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o Include Tumor cells (living and necrotic), Immune cells (NK and CTL), and normal Host cells.
o Two types of nutrients: one for Maintenance and one necessary for cell division (N).
o Nutrients diffuse from a (constant) source: blood vessels at the upper and lower edge of the computational domain and are consumed by living cells.
o NK cells are constantly replenished in order to maintain relatively constant population.
o CTLs are recruited when tumor cells are lysed or recognized by the immune system.
o Tumor cells die, proliferate and migrate, affected by local nutrient concentrations.
In Particular…
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Cellular automata - the idea…• DePillis/Mallet/Radunskaya models
work w/ 2d, but can also be 3d• Grid of elements where cells can be
located• Discrete time steps, cells:
• Move• Divide• Interact• Die• Signal• Consume nutrients• etc
T
Tmr
NK
Natural/regularhost cells
NK
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Model I• Hybrid PDE/CA model• 2D spatial domain• Nutrient
sources• Initial cell scattering• Stochastic cell rules• Explore: Effect of varying
nutrient consumption rates
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Nondimensional Nutrient PDEs
• N: nutrients required for proliferation• M: nutrients required for survival• H: host cells• T: tumor cells• NK: NK immune cells• L: CD8+T lymphocytes
Dimensionless rate of consumption Tumor excess consumption factor
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Cell rules• Evolution: according to probabilistic rules• All cells: consume nutrients• Tumor cells: move, divide, die (from
insufficient nutrient, or from immune cell attack)
• NK cells: move randomly, kill tumor, induce CTL recruitment; one tumor cell kill allowed.
• CTL cells: move preferentially toward tumor, kill tumor, induce further CTL recruitment; multiple tumor cell kills allowed.
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Spherical Growth: Lower nutrient consumption rates
• No immune system…exp./lin. tumour growth
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Papillary growth: Higher nutrient consumption rates
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Spatial Tumor Growth: one nutrient, one blood vessel•Nutrients diffuse from blood vessel (at top) in a continuous model (PDE).•Cells proliferate according to a probabilistic model based on available nutrients.
λ
α
Cancer to normal cell consumption factor
Normal & cancer cell consumption
coefficient
A blood vessel runs along the top of each square
Papillary versus Spherical
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Spatial Tumor Growth•Chemotherapy Experiments: Every Three Weeks
Spatial Features: Add Chemo
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Spatial Tumor Growth•Chemotherapy Experiments: Every Two Weeks
Spatial Features: Add Chemo
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• High recruitment
• Low recruitment
Effects of CTLRecruitment to
Tumour locations
Add CTL recruitment to tumor
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Spatial Tumor Growth•NK and CD8 Immune Activity
Thanks: Dann Mallet
Simulation 2: TumorSimulation 1: Tumor
QuickTime™ and a decompressor
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Immune cell infiltration: Qualitative agreement with
biological experiment
Ovarian carcinoma. Tumor cells (blue) infiltrated by immune cells (gray). Thanks: Zhang et al 2003
Simulation. Tumor cells (white) infiltrated by immune cells (black).
Mallet & de Pillis JTB 239, 2006
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Radiation Treatment: in progresso Goal of radiation: create enough DNA
double-strand breaks to cause cell death.
o Standard model: Linear Quadratic (LQ)
o LQ-modified with oxygenation effect -hypoxic cells less vulnerable to radiation: Need OER (oxygen enhancement ratio):standard is 2.5 to 3
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Radiation Treatment
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50x50 grid. Cycles 80 to 140. Radiation: AlternateDays (cycles 100-120), 3 Grays per dose.
No treatment Radiation cycles 100-120
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• The three cell types within the model are:– proliferating cells: alive, can divide and grow– quiescent cells: alive, but dormant– necrotic cells: dead
HMC Mathematics Clinic: 5 Undergraduates and LANL
HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
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• Grid site
• Tumor cell
Simulated Model Cross-Section
HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
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Cell Types Growth Factor Concentration
Matlab Imaging – 2D Slices
HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
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Linear Vasculature - Tumor Growth
2D slice taken over several time steps
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HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
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Lattice Vasculature – Tumor Growth: 2D Slice of 3D Computation
Series of fixed-depth 2D slices taken over several time steps
HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
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Hex Lattice Vasculature –Tumor Growth
2D slice taken over several time steps
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HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
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3D Vasculature – Fly-Through
Series of fixed-depth 2D slices taken over several time steps
HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
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Chemotherapy Treatments
No Treatment
0 Grid Sites 200
Grid
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0 Grid Sites 200
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Low Dose Chemotherapy
High Dose Chemotherapy
37 MCS
HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
Red: Quiescent; Dark Yellow: Proliferating; Blue: Apoptotic;Light yellow: Nutrient Medium; Light Blue Line: Blood Vessel
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Chemotherapy Treatments
No Treatment
0 Grid Sites 200
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0 Grid Sites 200
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0 Grid Sites 200
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Low Dose Chemotherapy
High Dose Chemotherapy
40 MCS
HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
Red: Quiescent; Dark Yellow: Proliferating; Blue: Apoptotic;Light yellow: Nutrient Medium; Light Blue Line: Blood Vessel
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Chemotherapy Treatments
No Treatment
0 Grid Sites 200
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0 Grid Sites 200
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0 Grid Sites 200
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Low Dose Chemotherapy
High Dose Chemotherapy
50 MCS
HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
Red: Quiescent; Dark Yellow: Proliferating; Blue: Apoptotic;Light yellow: Nutrient Medium; Light Blue Line: Blood Vessel
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Chemotherapy Treatments
No Treatment
0 Grid Sites 200
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00
0 Grid Sites 200
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0 Grid Sites 200
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Low Dose Chemotherapy
High Dose Chemotherapy
60 MCS
HMC STUDENTS: Tiffany Head, Cris Cecka, Alan Davidson, Liam Robinson, Dana MohamedLANL Liason: Yi Jiang. Faculty Supervisor: L.G. de Pillis
Red: Quiescent; Dark Yellow: Proliferating; Blue: Apoptotic;Light yellow: Nutrient Medium; Light Blue Line: Blood Vessel
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3D Tumor GrowthHybrid ODE-PDE Approach
withSpherical Harmonics
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Spherical Harmonics to Model 3D Tumor Growth
• Use knowledge from ODE models• Incorporate spatial components to allow
visualization of 3D tumor• Spherical harmonics: Motivated by
medical imaging techniques
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ODE/PDE Equations
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PDE Types in Problem
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Truncated Spherical Coordinates
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Evolving Tumor Simulations
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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
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• The more we cooperate, the more rapid progress we can make.
• The more we cooperate, the more interesting problems we can solve.
• The more we cooperate, the more relevant our contributions.
Final Thoughts on Cooperation
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Sample References:De Pillis, Radunskaya, “A Mathematical Tumor Model with Immune Resistance and Drug Therapy: an Optimal Control Approach”, J. Theor. Medicine, 2001De Pillis, Radunskaya, “The Dynamics of an Optially Controlled Tumor Model: A Case Study”, Math. Comp. Model., 2003De Pillis, Radunskaya, Wiseman “A Validated Mathematical Model of Cell Mediated Response to Tumor Growth”, Cancer Res., 2005De Pillis, Gu, Radunskaya, “Mixed Immunotherapy and Chemotherapy of Tumors: Modeling, Applications and Biological Interpretations”, J. Theor. Biol., 2006Mallet, De Pillis, “A Cellular Automata Model of Tumor-immune System Interactions”, J. Theor. Biol., 2006
De Pillis, Fister, Gu, Head, Maples, Neal, Murugan, Yoshida, “Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls ”, Math. Biosci., 2007
De Pillis, Gu, Fister, Head, Maples, Neal, Murugan, Yoshida, “Optimal Control of Mixed Immunotherapy and Chemotherapy of Tumors ”, J. Biol. Systems 2008
De Pillis, Mallet, Radunskaya, “Spatial Tumor-immune Modeling ”, J. Comp. & Math. Methods in Med., 2006
De Pillis, Fister, Gu, Collins, Daub, Gross, Moore, Preskill, “Seeking Bang-Bang Solutions of Mixed Chemo-Immunotherapy of Tumors ”, EJDE., 2007
De Pillis, Gu, Fister, Collins, Daub, Gross, Moore, Preskill, “Mathematical model creation for Cancer Chemo-Immunotherapy ”, Comp. & Math. Methods in Med., 2009
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Dept. of Mathematics
Claremont, CA, 91711
USA
Thanks for listening!
Prof. L.G. de Prof. L.G. de PillisPillishttp://www.math.http://www.math.hmchmc..edu/~depillisedu/~depillis