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

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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