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Systems Describing the Tumor Microenvironment:
A Geometric Optimal Control Approach
Urszula LedzewiczSouthern Illinois University Edwardsville, IL, USA and Lodz University of Technology, Lodz, Poland
Biological Systems and Networks, IMA,
November 16-20, 2015
Heinz Schättler
Washington University St. Louis, MO USA
Co-author and Support
DMS 0707404/0707410
DMS 1008209/1008221
DMS 1311729/1311733
Research supported by collaborative NSF grants
Springer 2012
Springer , September 2015
Biomedical/ Pharmaceutical CollaboratorsAlberto d’OnofrioEuropean Institute for Oncology, Milano, Italy& International Prevention Research Institute, Eculy, France
Eddy PasquierChildren Cancer Institute Australia,
University of New South Wales, Sydney
Nicolas AndréChildrens Hospital La Timone, Marseille France
Helen Moore
Tumor stimulating myeloid cell
Surveillance T-cell
Fibroblast
EndotheliaChemo-resistant tumor cell
Chemo-sensitivetumor cell
Tumor Microenvironment
Optimal Drug Treatment Protocols
How to optimize the antitumor, antiangiogenic and pro-immune effects of therapy by modulating dose andadministration schedule?
Eddy Pasquier, Nicholas André
More is Not Necessarily Better: Metronomic Chemotherapy,
Eddy Pasquier and Urszula Ledzewicz,
Newsletter of the Society for Mathematical Biology, Vol. 26, No.2, 2013
ABSTRACT DEADLINE APRIL 25, 2014You may submit your abstract by e-mail to francesco.bertolini@ieo.it
INSTRUCTIONS FOR ABSTRACT SUBMISSIONParticipants with new data relevant to the subjects of the conference may submit one abstract to be considered for e-poster presentation. Only papers whose abstracts have been reviewed and approved by the Scientific Committee will be presented as e-posters or oral presentations. Abstracts must be written in English and the text should not exceed 2000 characters (title included). The title should appear in capital letters. For each author, type surname first followed by given name and initials (presenting author in capital letters). List affiliations after the authors’ list. The full address with telephone, fax number, and e-mail of the corresponding author must be provided. This author will receive all the subsequent communications concerning this abstract.Please include the following sections: Background, Material methods, Results, Conclusions. All the slides must be sent to the Abstract selection committee (francesco.bertolini@ieo.it) Please indicate your approval for publication in case the abstracts will be issued in a journal.
PLEASE NOTEOnly abstracts submitted by registered participants will be accepted. It is the responsibility of the presenting author to ascertain whether all authors are aware of the content of the abstract before submission.
Not only WHAT drugs to give but HOW?
• a model for chemotherapy under tumor-immune interactions
• challenges in modeling metronomic chemotherapy,
• work in progress and medical perspective
Outline – An Optimal Control Approach to…
tumor microenvironment
• a model for antiangiogenic treatment (monotherapy and in combination with chemo- and radiotherapy) also with PK
•a model for chemotherapy for heterogeneous tumors
metronomic chemotherapy
Tumor Antiangiogenesis
• suppress tumor growth by preventing the recruitment of newblood vessels that supply the tumor with nutrients (indirect approach)
• done by inhibiting the growth of the endothelial cells that form the lining of the new blood vessels - therapy “resistant to resistance”
• anti-angiogenic agents are biological drugs (enzyme inhibitors like endostatin) – very expensive and with side effects
http://www.gene.com/gene/research/focusareas/oncology/angiogenesis.html
[Hahnfeldt,Panigrahy,Folkman,Hlatky],Cancer Research, 1999
p,q – volumes in mm3
Lewis lung carcinoma implanted in mice
- tumor growth parameter
- endogenous stimulation (birth)
- endogenous inhibition (death)
- anti-angiogenic inhibition parameter
- natural death
p – tumor volume
q – carrying capacityof the vasculature
u – anti-angiogenic dose rate
For a free terminal time minimize
over all measurable functions that satisfy
subject to the dynamics
Optimal Control Problem [LSch, SICON, 2007; LMSch DCDSB, 2009; LCSch, MBE, 2011]
A Bit of Math
Candidates for Optimal Protocols
• bang-bang controls • singular controls
treatment protocols of maximum dose therapy periods with rest periods in between
continuous infusions of varying lower doses
umax
T T
MTD BOD
Φ(t) > 0
Φ(t) < 0Φ(t) < 0 Φ(t) ≡ 0
switching function Φ(t)
Singular Controls
• is singular on an open interval switching function on
• all time derivatives must vanish as well • “allows” to compute the singular control• order : the control appears for the first time in
the derivative• Legendre-Clebsch condition (minimize)
Dynamics in Vector Form
drift control vector field
Lie bracket:
Switching function
Singular control
order 1 singular control strengthened Legendre-Clebschcondition is satisfied
Singular Control
0 0.5 1 1.5 2 2.5 3-20
0
20
40
60
80
100
120
x
psi
feedback control
0 0.5 1 1.5 2 2.5 3 3.5
x 104
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
carrying capacity of the vasculature, q
tum
or v
olum
e,p
Admissible Singular Arc
q
p
Synthesis of Optimal Controls [LSch, SICON, 2007]
0 2000 4000 6000 8000 10000 12000 14000 16000 180000
2000
4000
6000
8000
10000
12000
14000
16000
18000
endothelial cells
tumor
cells
an optimal trajectorybegin of therapy
final point – minimum of p
end of “therapy”
p
q
u=au=0
typical structure of optimal controls: umax→s→0
An Optimal Controlled Trajectory [LSch, JTB, 2008; LMMSch,MMMB, 2010]
Initial condition: p0 = 12,000 q0 = 15,000, umax=75
0 1 2 3 4 5 6 7
0
10
20
30
40
50
60
70
time
optim
al co
ntro
l u
maximum dose rate
no dose
lower dose rate - singular
Multi-input Control: Antiangiogenic Treatment with Chemotherapy
Minimize subject to
A Model for a Combination Therapy[d’OLMSch, Math. Biosciences, 2009, MBE 2014]
with d’Onofrio and H. Maurer
angiogenic inhibitors
cytotoxic agent or other killing term
Optimal Protocols (sequencing)
4000 6000 8000 10000 12000 14000 16000
7000
8000
9000
10000
11000
12000
13000
carrying capacity of the vasculature, q
tum
or v
olum
e, p
optimal angiogenic monotherapy
Optimal Controls and Corresponding Trajectory
0 1 2 3 4 5 6 7
0
10
20
30
40
50
60
70
time (in days)
dosa
ge a
ngio
4000 6000 8000 10000 12000 14000 16000
7000
8000
9000
10000
11000
12000
13000
carrying capacity of the vasculature, q
tum
or v
olum
e, p
0 1 2 3 4 5 6 7-0.2
0
0.2
0.4
0.6
0.8
1
dosa
ge c
hem
o
time (in days) “Therapeutic window” ( medical)
Medical Connection
Rakesh Jain, Steele Lab, Harvard Medical School,
“there exists a therapeutic window when changes in the tumor in response to anti-angiogenic treatment may allow chemotherapy to be particularly effective”
Justification: Pruning
Tumor Anti-Angiogenesis
And Radiotherapy
and terminal constraints and
For a free terminal time T, minimize the tumor volume p(T)over all measurable functions
andsubject to the dynamics
A Model for Anti-Angiogenic Treatment with Radiotherapy [LSch, JOTA, 2012]
Model based on – Ergun et al., Bull. Math. Biology, 2003
Totally Singular Controls and Surface
• a system of 2 linear equations: totally singular controls
•
•
• the totally singular vector field is only optimal on the hyper-surface
• does not depend on the variables y and z
optimal term for angiogenic monotherapy
Singular Flow on Surface S
-0.2 0 0.2 0.4 0.6 0.8 1 1.20
5
10
15
20
25
30
time
anti-a
ngioge
nic do
se rat
e u
-0.2 0 0.2 0.4 0.6 0.8 1 1.20
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time
radiat
ion do
se w
anti-angiogenic dose rate
radiation schedule
Including Pharmacokinetics
PKdynamics for
p and qdosage
u c
concentration
old system with the control replaced by the output of a first order linear system with control as input
p
To what extent is previous analysis preserved ?
Singular Controls: Chattering
Comparison of Optimal and Suboptimal Controls [LSch,M, Springer 2010]
“optimal” chattering control and corresponding concentration
sub-optimal control and corresponding concentration
Heterogeneous Tumor Cell Populations
2-Compartment Model with Linear PK
for simplicity, just consider two populations of different chemotherapeutic sensitivity and call them ‘sensitive’ and ‘resistant’
S – sensitive cell population
R – resistant cell population
α1 – growth rate of sensitive population
α2 – growth rate of resistant population
γ1 – transfer rate from sensitive to resistant population
γ2 – transfer rate from resistant to sensitive population
φ1 – linear log-kill parameter for sensitive population
φ2 – linear log-kill parameter for resistant population
β – pharmacokinetic parameter related to half-life of chemotherapeutic agent
S
R
c
[Hahnfeldt, Folkman and Hlatky, JTB, 2003]
For a fixed therapy horizon minimize
over all functions subject to the dynamics
where
As Optimal Control Problem[LSch, JBS, 2014]
• in the region MTD (Maximum Tolerable Dose)
- no singular arcs exist
- optimal controls are bang-bang with one switching from u=umax to u=0
- in particular, this holds if the number of sensitive cells lies above the following threshold:
Bang-bang vs. Singular Solutions
• outside the region MTD,
- the Legendre-Clebsch condition is satisfied,
- singular controls are of order k=2
-difficulty: concatenations with bang controls are through chattering arcs
Bang-bang vs. Singular Solutions
Chemo-Switch Protocols
(bang-singular)
0 20 40 60 80 100 120 140 1600
1
2
3
4
5
6
7
8
9
10x 104
time
popu
lation
s R an
d S
Chemo-Switch Protocols
0
200
400
600
800
1000
1200
1400
1600
1800
2000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
Sensibles A549
Résistantes Epo40
Fluo
resc
ence
(%)
D17
Resistant cells (Epo40-EGFP)Sensitive cells (A549-mtDsRed)Fl
uore
scen
ce (%
)
Days
Chemotherapy (epo 2-5 nM)IC50 - IC90
IN VITRO: Effects of MTD Treatment on Tumor Heterogeneity
Manon Carre and Nicholas André, Childrens’ Hospital La Timone, Marseille
inhibitory concentration (IC) - a measure of the effectiveness of a substance inhibiting a specific biological or biochemical function
Fluo
resc
ence
(%)
days
IN VITRO: Effects of Metronomic Chemo on Tumor Heterogeneity
Manon Carre and Nicholas André, Childrens’ Hospital La Timone, Marseille
daysFluo
resc
ence
(%)
020406080
100120
1 2 3 4
Metronomic Chemotherapy
<IC10MTD: black arrows10% MTD: red + green
Tumor Immune Interactions
Scientific American
Tumor Immune System Interactions [Stepanova]
- tumor growth parameter
- rate at which cancer cells are eliminated through the activity of T-cells
- constant rate of influx of T-cells generated by primary organs
- natural death of T-cells
- calibrate the interactions between immune system and tumor
- threshold beyond which immune reaction becomes suppressed by the tumor
- tumor volume
- immunocompetent cell density
Stepanova, 1980 Kuznetsov, Makalkin, Taylor and Perelson, 1994
de Vladar and Gonzalez, 2004 d’Onofrio, 2005
Phaseportrait for Gompertzian Model
[Kuznetsov et al., 1994
de Vladar et al., 2004]
asymptotically stable focus – “good”, benign equilibrium
saddle point
asymptotically stable node – “bad”, malignant equilibrium
bi- stability
0 100 200 300 400 500 600 700 8000
0.5
1
1.5
2
2.5
tumor volume, x
imm
uno c
om
pete
nt c
ell
den
sity
, y
(xs,ys)
(xb,yb) *
*
* (xm,ym)
Phaseportrait of uncontrolled dynamics
• we want to move the state of the system into the region of attraction of the benign equilibrium
minimize
For a free terminal time T minimize
over all measurable functions andsubject to the dynamics
Optimal Control Formulation[LNSch, JMB, 2011; LMSch, DCDSB 2013]
Chemotherapy – log-kill hypothesis
Immune boost
0 200 400 600 8000
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10 12
0
0.2
0.4
0.6
0.8
1
Chemotherapy with Immune Boost [DCDSB, 2013]
• chemo: bang-singular-bang-bang
(MTD/metronomic, chemo-switch)
• immuno: bang-bang
**
*
“free pass”
1s01 010
- chemo
- immune boost
**
Tumor Microenvironment and Metronomic Chemotherapy
with Eddy Pasquier, CCIA, University of New South Wales
Metronomic Chemotherapy: modeling challenge
• treatment at lower doses
( between 10% and 50% of MTD)
• constant ? varying in time ? short rest periods ?
How is it administered?
Advantages:
1. lower cytotoxic effects on tumor cells
• lower toxicity (in many cases, none)
• lower drug resistance and even resensitization effect
2. antiangiogenic effects
3. boost to the immune system
Adapted from Pasquier et al., Nature Reviews Clinical Oncology, 2010
A Combined Model for Low Dose Chemotherapy
p(t) – primary tumor volume
q(t) – carrying capacity of the tumor vasculature
r(t) – immunocompetent cell density
u(t) – concentration of a chemotherapeutic agent
Ledzewicz, Schättler, Amini,
JMB 2015, MBE 2015
effectiveness (PD)
Projections into (p,q)- and (p,r)-space
asymptotically stable focus – “good”, benign equilibrium
saddle point and stability boundary
asymptotically stable node – “bad”, malignant equilibrium
0 100 200 300 400 500 600 7000
100
200
300
400
500
600
700
800
p ( Tumor Volume )
q ( C
arry
ing
Vas
cula
ture
)
γ=0.05
q (c
arry
ing
capa
city
)
p (tumor volume)
300 350 400 450 500 550 600 650 7000
0.005
0.01
0.015
0.02
0.025
0.03
p ( Tumor Volume )
r ( Im
mun
e S
yste
m )
γ = 0.05
r (im
mun
ocom
pete
nt c
ell d
ensi
ty)
p (tumor volume)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
100
200
300
400
500
600
tumor growth ξ
p*(2)
p*(1)
p*(3)
tum
or v
olum
e p * a
t equ
ilibriu
m p
oint
Bifurcation diagram in Tumor Growth Rate
Optimal Control Problem
“move an initial condition that lies in the malignant region through chemotherapy into the benign region”
minimize
over all Lebesgue measurable functions u: [0,T] → [0,umax] subject to the dynamics
where (A,B,-C) (A,B and C are positive) is the tangent vector to the unstable manifold of the saddle point, oriented to point from the benign into the malignant region.
0 100 200 300 400 500 600 7000
100
200
300
400
500
600
700
800
p ( Tumor Volume )q
( Car
ryin
g Va
scul
atur
e )
γ=0.05q
p
Legendre-Clebsch Condition and Singular Controls
slices for constant value of r
Legendre-Clebsch condition
singular control
using
Chemo-Switch Protocols
Metronomics and Other Alternatives to MTD:
Medical Evidence
with Eddy Pasquier, CCIA, University of New South Wales
How to optimize the anti-tumor, anti-angiogenic and pro-immuneeffects of chemotherapy by modulating dose and administrationschedule? Different therapeutic approaches:
- “Pure” metronomic / Metronomics (R. Kerbel, D. Hanahan)
J Clin Oncol 2010
-Weekly VLB-Daily CPA-2x weekly MTX-Daily CLX
How to optimize the anti-tumour, anti-angiogenic and pro-immuneeffects of chemotherapy by modulating dose and administrationschedule?
Different therapeutic approaches:- MTD/Metronomic: Chemo-Switch strategies (D. Hanahan)
J Clin Oncol 2005
Lancet Oncol 2010
How to optimize the anti-tumour, anti-angiogenic and pro-immuneeffects of chemotherapy by modulating dose and administrationschedule?
Different therapeutic approaches:- chaotic therapy (Nicolas André and Eddy Pasquier)
Nature 2009
Cancer Research 2009
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