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Healthy and cancer tissue growth , therapies and the use of mathematical [ cell population dynamic ] models. Jean Clairambault. INRIA, Bang project-team, Paris & Rocquencourt, France Laboratoire Jacques-Louis Lions, UPMC, Paris, France

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Jean clairambault

Healthy and cancer tissue growth, therapies and the use of mathematical [cell population dynamic] models

Jean Clairambault

INRIA, Bang project-team, Paris & Rocquencourt, France

Laboratoire Jacques-Louis Lions, UPMC, Paris, France

INSERM U776, « Biologic rhythms and cancers», Villejuif, France

http://www.rocq.inria.fr/bang/JC/ [email protected]


Jean clairambault

  • Motivation: modelling tissue growth to represent proliferating

  • cell populations as targets, wanted or unwanted, of anticancer

  • drugs, to optimise their delivery at the whole body level.

  • Main pitfalls of cancer therapeutics in the clinic:

  • unwanted toxic side effects on healthy cell populations

  • emergence of drug-resistant clones in cancer cell populations

  • Designing a model of tissue growth must be undertaken after

  • answering the question: Modelling, what for? Resulting models

  • may be very different according to the answers to this question.


Jean clairambault

Recalling a few facts about healthy and cancer tissue growth


Jean clairambault

Tissues thatmayevolvetowardmalignancy

…are the tissues where cells are committed to fast proliferation

(fast renewing tissues):

-epithelial cells+++, i.e., cells belonging to those tissues which

cover the free surfaces of the body (i.e., epithelia):

gut (colorectal cancer), lung, glandular coverings (breast, prostate),skin,…

-cells belonging to the different blood lineages, produced in

the bone marrow: liquid tumours, or malignant haemopathies

-others (rare: sarcomas, neuroblastomas, dysembryomas…)


Natural history of cancers from genes to bedside

Natural history of cancers: from genes to bedside

Gene mutations: an evolutionary process which may give rise to abnormal DNA when a cell duplicates its genome, due to defects in tumour suppressor or DNA

mismatch repair genes (Yashiro et al. Canc Res. 2001; Gatenby & Vincent, Canc. Res. 2003)

  • Entry in the cell cycle for quiescent (=non-proliferating) cells

  • Phase transitions and apoptosis for cycling cells

  • Ability to use anaerobic glycolysis (metabolic selective advantage for cancer cells)

  • Contact inhibition by surrounding cells (cell adhesion, cell density pressure)

  • Ability to stimulate formation of new blood vessels from the neighbourhood

  • Linking to the extracellular matrix (ECM) fibre network and basal membranes

  • Recognition (friend or foe) by the immune system

Resulting genomic instability allows malignant cells to escape proliferation and

growth control at different levels: subcellular, cell, tissue and whole organism:

Cancer invasion is the macroscopic result of breaches in these control mechanisms


Jean clairambault

Evading proliferation and growth control mechanisms

Hanahan & Weinberg, Cell 2011

Hanahan & Weinberg, Cell 2000

…but just what is cell proliferation?


Jean clairambault

Cell population growth in proliferating tissues

(after Lodish et al., Molecular cell biology, Nov. 2003)

One cell divides in two: a controlled process at cell and tissue levels

... that is disrupted in cancer

Populations of cells:

the right observation level to model proliferation, apoptosis, differentiation


Jean clairambault

Cyclin B

+ CDK1

M

G2

G1

S

Cyclin A

+ CDK2

At the origin of proliferation: the cell division cycle

S:=DNA synthesis; G1,G2:=Gap1,2; M:=mitosis

Cyclin D

+ CDK4, 6

(after Lodish et al., Molecular cell biology, Nov. 2003)

  • Physiological or therapeutic control on:

  • - transitions between phases

  • (G1/S, G2/M, M/G1)

  • - death rates (apoptosis or necrosis)

  • inside phases

  • progression speed in phases (G1)

  • exchanges with quiescent phase G0

Cyclin E

+ CDK2

(after F. Lévi, INSERM U776)


Jean clairambault

Proliferating and quiescent cells

after R:

mitogen-independent

progression through G1 to S

(no way back to G0)

Restriction point

(in late G1 phase)

R

before R:

mitogen-dependent

progression through G1

(possible regression to G0)

From Vermeulen et al. Cell Prolif. 2003

Most cells do not proliferate physiologically, even in fast renewing tissues (e.g., gut)

Exchanges between proliferating (G1SG2M) and quiescent (G0) cell compartments

are controlled by mitogens and antimitogenic factors in G1 phase


Jean clairambault

M

G2

G1

S

Repair or apoptosis

Phase transitions, apoptosis and DNA mismatch repair

-Sensor proteins, e.g. p53, detect defects in DNA, arrest the cycle at G1/S and G2/M phase transitions to repair damaged fragments, or lead the whole cell toward controlled death = apoptosis

-p53 expression is known to be down-regulated in about 50% of cancers

-Physiological inputs, such as circadian gene PER2, control p53 expression; circadian clock disruptions (shiftwork) may result in low p53-induced genomic instability and higher incidence of cancer

p53

Repair or apoptosis

p53

(F. Lévi, INSERM U776)

(Fu & Lee, Nature Rev. 2003)


Jean clairambault

Invasion: local, regional and remote

1) Local invasion by tumour cells implies loss of normal cell-cell and cell-ECM (extracellular matrix) contact inhibition of size growth and progression in the cell cycle. ECM (fibronectin) is digested by tumour-secreted matrix degrading enzymes (MDE=PA, MMP) so that tumour cells can move out of it. Until 106 cells (1 mm ) is the tumour in the avascular stage.

2) To overcome the limitations of the avascular stage, local tumour growth is enhanced by tumour-secreted endothelial growth factors which call for new blood vessel sprouts to bring nutrients and oxygen to the insatiable tumour cells (angiogenesis, vasculogenesis)

3) Moving cancer cells can achieveintravasation, i.e., migration in blood and lymph vessels (by diapedesis), andextravasation, i.e. evasion from vessels, through vascular walls, to form new colonies in distant tissues. These new colonies are metastases.

Proliferating rim

Quiescent layer

Necroticcore

(Images thanks to A. Anderson, M. Chaplain, J. Sherratt, and Cl. Verdier)


Jean clairambault

Mathematical models

(A reference: Optimisation of cancer drug treatments using cell population dynamics, by F. Billy, JC, O. Fercoq, to appear soon in “Mathematical Models and Methods in Biomedicine”, A. Friedman, E. Kashdan, U. Ledzewicz, H. Schättler eds. Springer, NY)


Mathematical models of healthy and tumour growth a great variety of models

Mathematicalmodels of healthy and tumourgrowth(a greatvariety of models)

  • In vivo (tumours) or in vitro (cultured cell colonies) growth? In vivo (diffusion in living organisms) or in vitro (constant concentrations) growth control by drugs?

  • Scale of description for the phenomenon of interest: subcellular, cell, tissue or whole organism level? … may depend upon therapeutic description level

  • Is space a relevant variable? [Not necessarily!] Must the cell cycle be represented?

  • Are there surrounding tissue spatial limitations? Limitations by nutrient supply or other metabolic factors?

  • Is loco-regional invasion the main point? Then reaction-diffusion equations (e.g. KPP-Fisher) are widely used, for instance to describe tumour propagation fronts

  • Is cell migration to be considered? Then chemotaxis [=chemically induced cell movement] models (e.g. Keller-Segel) have been used


Jean clairambault

Simple models of tumour growth, to begin with

Macroscopic, non-mechanistic models: the simplest ones:

exponential, logistic, Gompertz

x= tumour weight

or volume, proportional

to the number of cells,

or tumour cell density

x

t

Exponential model: relevant for the early stages of tumour growth only

[Logistic and] Gompertz model: represent growth limitations (S-shaped curves with plateau=maximal growth, or tumour carrying capacity xmax), due to mechanical pressure or nutrient scarcity


Jean clairambault

Models of tumour growth: Gompertz revisited

ODE models a) with 2 cell compartments, proliferating and quiescent,

or b) varying the tumour carrying capacity xmax in the original Gompertz model

(Gyllenberg & Webb, Growth, Dev. & Aging 1989; Kozusko & Bajzer, Math BioSci 2003)

Avowed aim: to mimic Gompertz growth (a popular model among radiologists)

However, a lot of cell colonies and tumours do not follow Gompertz growth

(Recent refinements: Hahnfeldt et al., Canc. Res 1999; Ergun et al., Bull Math Biol 2003)

Tumour burden

d14

Data

Example: tumour growth

(GOS) in a population of

mice, laboratory data

d12

Gompertz model

d9

d8

time


Jean clairambault

ODE models with two exchanging cell compartments,

proliferating (P) and quiescent (Q)

Cell exchanges

(Gyllenberg & Webb, Growth, Dev. & Aging 1989; Kozusko & Bajzer, Math BioSci 2003)

where, for instance:

r0 representing here the rate of

inactivation of proliferating cells,

and rithe rate of recruitment from

quiescence to proliferation

Initial goal: to mimic Gompertz growth


Jean clairambault

Models of tumourgrowth: stochastic

Physicallawsdescribingmacroscopic spatial dynamics of an avasculartumour

-Fractal-based phenomenological description of growth of cell colonies and tumours,

relying on observations and measures: roughness parameters for the 2D or 3D tumour

Findings: - all proliferation occurs at the outer rim

- cell diffusion along (not from) the tumour border or surface

- linear growth of the tumour radius after a critical time (before: exponential)

(A. Bru et al. Phys Rev Lett 1998, Biophys J 2003)

-Individual-based models (IBMs):

- cell division and motion described by

a stochastic algorithm, then continuous limit

- permanent regime = KPP-Fisher-like

(also linear growth of the tumour radius)

(D. Drasdo, Math Comp Modelling 2003; Phys Biol 2005

H. Byrne & D. Drasdo, J. Math. Biol. 2009)


Jean clairambault

Models of tumour growth: invasion

Macroscopic reaction-diffusion evolution equations

1 variable c = density of tumour cells): KPP-Fisher equation

  • D(x) = diffusion (motility) in brain tissue,

  • (reaction)=growth of tumour cells,1D x spatial variable and c: density of tumour cells, used to represent brain tumour radial propagation from a centre

  • (K. Swanson& J. Murray, Cell Prolif 2000;

  • Br J Cancer 2002; J Neurol Sci 2003)


Jean clairambault

Models of tumour growth: invasion as competition

Macroscopic reaction-diffusion equations to represent invasion (recession) fronts

2 or more variables: ex.: healthy cells N1, tumour cells N2, excess H+ ions L

N2

N1

L

(Gatenby & Gawlinski, Canc. Res. 1996)

Prediction: interstitial cell gap between tumour

propagation and healthy tissue recession fronts


Jean clairambault

Models for moving tumour cells in the ECM

Chemotaxis: chemo-attractant induced cell movements

Keller-Segel model

p = density of cells

w = density of chemical

(Originally designed for movements of bacteria, with w=[cAMP])

(Keller & Segel, J Theoret Biol 1971)

Anderson-Chaplain model for local invasion by tumour cells in the ECM

n = density of cells

f = ECM density

m = MDE (tumour

metalloproteases)

u = MDE inhibitor

(Anderson & Chaplain, Chap 10 in Cancer modelling and simulation, L. Preziosi Ed, Chapman & Hall 2003)


Jean clairambault

Models for angiogenesis

VEGF-induced endothelial cell movements towards tumour

- Biochemical enzyme kinetics

- Chemical transport (capillary and ECM)

- “Reinforced random walks”

- Cell movements in the ECM

Models by Anderson and Chaplain, Levine and Sleeman

(Levine & Sleeman,Chap. 6 in Cancer modelling and simulation, L. Preziosi Ed, Chapman & Hall 2003)


Jean clairambault

A multiscale angiogenesis model

Interacting cell populations

Proliferating cancer cell population

Coupling by oxygen concentration, acting on actual commitment of cells

into the division cycle (passing the restriction point)

Aim: assessment of an antiangiogenic treatment by endostatin

F. Billy et al., J. Theor. Biol. 2009


Jean clairambault

Modelling the cell cycle 1: single cell level

Ordinary differential equations to describe progression in the cell cycle

A. Golbeter’s minimal model for the « mitotic oscillator », i.e., G2/M phase transition

C

M

X

C= cyclin B, M = Cyclin-linked cyclin dependent kinase, X = anticyclin protease

Switch-like dynamics of kinase cdk1, M

Adapted to describe G2/M phase transition,

which is controlled by Cyclin B-CDK1

(A. Goldebeter, Biochemical oscillations and cellular rhythms, CUP 1996)


Jean clairambault

Modelling the cell cycle 2: single cell (continued)Detailed ODE models to describe progression in the cell cycle: Tyson & Novak

Phase transitions:

-G1/S

-G2/M

-Metaphase/anaphase

…due to steep variations

of Cyc-CDK concentrations

(bistability with hysteresis)

(Tyson, Chen, Novak, Nature Reviews 2001)

(Novak, Bioinformatics 1999)


Jean clairambault

Modelling the cell cycle 2: single cell (continued)Detailed ODE models to describe progression in the cell cycle: Gérard & Goldbeter

39 variables. Growth factor, rather than cell mass

(as in Tyson & Novak) is the driving parameter

A simplified model has been proposed, with 5 variables

C. Gérard & A. Goldbeter, PNAS 2009; Interface Focus 2011

C. Gérard, D. Gonze & A. Goldbeter, FEBS Journal 2012


Jean clairambault

Modelling the cell cycle 3: cell population levelAge-structured PDE models for proliferating cell populations

Flow cytometry may help quantify

proliferating cell population repartition

according to cell cycle phases

(after B. Basse et al., J Math Biol 2003)

In each phase i , a Von Foerster-McKendrick-like equation:

ni:=cell population density in phase i

di:=death rate

K i->i+1:=transition rate

(with a factor 2for i=1)

di , K i->i+1 constant or periodic w. r. to time t

(1≤i≤I, I+1=1)

Death rates di and phase transitions K i->i+1are targets

for physiological (e.g. circadian) and therapeutic (drugs) control


Jean clairambault

: a growth exponent governing the cell population behaviour

Otherwise said: proof of the existence of a unique growth exponent ,the same for all phases i, such that the are bounded, and asymptotically periodic if the control is periodic.

Example (periodic control case): 2 phases, control on G2/M by 24 h-entrained periodic CDK1-Cyclin B (from A. Goldbeter’s minimal mitotic oscillator model)

“Surfing on the exponential growth curve”

(= the same as adding

an artificial death term to thedi, stabilising the system)

time t

=CDK1-Cyc BCells in G1-S-G2 (phasei=1)Cells in M (phase i=2)

Entrainment of the cell cycle by = CDK1-Cyc B at a 24 h-period


Jean clairambault

Details (1): 2-phase model, no control on transition

Asynchronous cell growth

The total population of cells

insideeach phase follows

asymptotically an exponential

behaviour

Stationary state distribution of cells inside phases according to age a: no control -> exponential decay


Jean clairambault

Details (2): 2 phases, periodic control on G2/M transition

Synchronisation between cell

cycle phases at transitions

The total population of cells

inside each phase follows

asymptotically an exponential

behaviour tuned by a periodic function

Stationary state

distribution of cells

inside phases

according to agea: sharpperiodic control ->sharprise and decay


Jean clairambault

The simplest case: 1-phase model with division

(Here, v(a)=1, a* is the cell cycle duration, and is the time

during which the 1-periodic controlis actually exerted on cell division)

Then it can be shown that the eigenvalue problem:

has a unique positive 1-periodic eigenvector N, with a positive eigenvalue 


Jean clairambault

Experimental measurements to identify transition kernels Ki_i+1

(and simultaneously experimental evaluation of the first eigenvalue l)

In the simplest model with d=0 (one phase, no cell death) and assuming K=K(x)

(instead of indicator functions , experimentally more realistic transitions):

Whence (by integration

along characteristic lines):

Interpreted as: if t is the age in phase at division, or transition:

with

With probability density (experimentally identifiable):

i.e.,


Jean clairambault

Experimental identification of basic model parameters

with FUCCI reporters on a 2-phase model G1 / S-G2-M

FUCCI=Fluorescent Ubiquitination-based Cell Cycle Indicator


Jean clairambault

FUCCI: a movie (Sakaue-Sawano 2008) on HeLa cells


Jean clairambault

Another FUCCI movie (C. Feillet, IBDC Nice), NIH3T3 cells


Jean clairambault

FUCCI reporters

Measuring time intervals: G1 and total division cycle durations

Data from G. van der Horst’s lab, Erasmus University, Rotterdam, processed by Frédérique Billy at INRIA

(NIH3T3 cells)(Billy et al., Math. Comp. Simul. 2012)


Jean clairambault

Phase transitions w.r.t. age x

Gamma p.d.f.s f(x) fitted from data on 55 NIH 3T3 cells

(Billy et al., Math. Comp. Simul. 2012)


Jean clairambault

Phase transitions w.r.t. age x

Transition rates K(x)from p.d.f.s f(x)on NIH 3T3 cells

and resulting population evolution on one complete cycle

K1 (G1 to S)

K2 (M to G1)

G1

SG2M

(Billy et al., Math. Comp. Simul. 2012)


Jean clairambault

A possible application to the investigation of

synchronisation betweencell cycle phases

(fromLodish et al., Molecularcellbiology, Nov. 2003)

One cell divides in two: a physiologically controlled process at cell and tissue levels

in all healthy and fast renewing tissues (gut, bone marrow) that is disrupted in cancer:

Is cell cycle phase synchronisation a mark of health in tissues?


Jean clairambault

A working hypothesis that could explain differences in responses to drug treatments between healthy and cancer tissues

Healthy tissues, i.e., cell populations, would be well synchronised

w. r. to proliferation rhythms and w. r. to circadian clocks, whereas…

...tumour cell populations would be desynchronised w. r. to both, and such

proliferation desynchronisation would be a consequence of an escape

by tumour cells from central circadian clock control messages, just as

they evade most physiological controls, cf. e.g., Hanahan & Weinberg:

Question:

iscell cycle phase

desynchronisationanotherhallmark of cancer in cell populations?


Jean clairambault

A mathematical result: l increases with desynchronisation

(expected, and shown in Th. Ouillon’s internship report, INRIA 2010)

where desynchronisation is defined as a measure of phase overlapping at transition

i.e.,for a given family (fi) of p.d.f.s with second moment si, l is increasing with each si

(also shown in Billy et al., Math. Comp. Simul., 2012)


Jean clairambault

Computing the growth exponent, fitting data to p.d.f.s:

Gamma p.d.f.s were best fits and yielded simple computations:

2-phase Lotka’s equation simply reads:

... which yields here l= 0.039 h-1

(and yields mean doubling time Td =17.77 h, and mean cell cycle time Tc =17.95 h)

Now, what if we add periodic (e.g., circadian) control??

i.e., what if we put K(x,t) = k(x).y(t)

with k = FUCCI-identified and y = a cosine?

[cosine: in the absence of a better identified clock thus far]


Jean clairambault

Phases: asynchronous cell growth

Global: sheer exponential cell growth

[Agreement between

model and data on

the first division]

F. Billy


Jean clairambault

Steep synchronisation within the cell cycle

Stepwise cell population growth

F. Billy


Jean clairambault

Soft synchronisation within the cell cycle

Stepwise cell population growth

F. Billy


Jean clairambault

F. Billy


Jean clairambault

Other cell population dynamic models relying on the cell cycle


Jean clairambault

A model of tissue growth with proliferation/quiescence

An age[a]-and-cyclin[x]-structured PDE model with proliferating and quiescent cells

(exchanges between (p) and (q), healthy and tumour tissue cases: G0 to G1 recruitment differs)

p: proliferating cells

q: quiescent cells

N: all cells (p+q)

Tumour recruitment:

exponential (possibly

polynomial) growth

Healthy tissue recruitment: homeostasis

F. Bekkal Brikci, JC, B. Ribba,

B. Perthame

JMB 2008; MCM 2008

M. Doumic-Jauffret, MMNP 2008

0

for small

0

for large N

0

for all


Jean clairambault

P-Q again: Simple PDE models, age-structured with

exchanges between proliferation and quiescence

p=density of proliferating cells; q=density of quiescent cells; ,=death terms;

K=term describing cells leaving proliferation to quiescence, due to mitosis;

=term describing “reintroduction” (or recruitment) from quiescence to proliferation


Jean clairambault

Delay differentialmodels with two cell compartments,

proliferating (P)/quiescent (Q): Haematopoiesis models

(obtained from the previous model with additional hypotheses and integration in age x along characteristics)

(delay  cell division cycle time)

(from Mackey, Blood 1978)

Properties of this model: depending on the parameters, one can have positive

stability, extinction, explosion, or sustained oscillations of both populations

(Hayes stability criteria, see Hayes, J London Math Soc 1950)

Oscillatory behaviour is observed in periodic Chronic Myelogenous Leukaemia

(CML) where oscillations with limited amplitude are compatible with survival,

whereas explosion (blast crisis, alias acutisation) leads to AML and death

(Mackey and Bélair in Montréal; Adimy, Bernard, Crauste, Pujo-Menjouet, Volpert in Lyon)


Jean clairambault

From Adimy, Crauste, ElAbdllaoui J Biol Syst 2008 (see also: Özbay, Bonnet, Benjelloun, JC MMNP 2012)

Modelling haematopoiesis

forAcute Myelogenous Leukaemia (AML)

…aiming at non-cell-killing therapeutics

by inducing re-differentiation of cells using

molecules (e.g. ATRA)enhancing differentiation

rates represented by Ki terms

where ri and pi represent resting and proliferating

cells, respectively, with reintroduction term i=i(xi) positive decaying to zero,

with population argument:

and boundary conditions:


Modelling leukaemic haematopoiesis mackey adimy proliferation advantage

TK (flt3-ITD) mutation

Modellingleukaemichaematopoiesis: (Mackey/Adimy) proliferationadvantage?

Stem cells CD34+/CD38-

Committed cells CD34+/CD38+

Blood/ bonemarrowsampling

in AML patients

Cellsorting (magneticbeads)

FACS for cell cycle phases

Self-renewal

Measuringapoptosis and cell

division in each population

Model adaptation (in charge: A. Ballesta, Hosp. St Antoine)


Optimisation of cancer therapy by cytotoxic drugs

Optimisation of cancer therapy by cytotoxic drugs

  • Optimal control strategies to overcome the development of drug resistant cell populations, using different drugs (Kimmel & Swierniak, 2006)

  • Pulsed chemotherapies aiming at synchronising drug injections with cell cycle events to enhance the effect of drugs on tumours: e.g. optimal control of IL21 injection times and doses ui (t-ti) using variational methods (Z. Agur, IMBM, Israel)

  • Chronotherapy = continuous infusion time regimens taking advantage

    of optimal circadian anti-tumour efficacy and healthy tissue tolerability

    for each particular drug: has been in use for the last 15 years, with particular achievements for colorectal cancer(F. Lévi, INSERM U776, M.-C. Mormont & F. Lévi, Cancer 2003)


Jean clairambault

Examples of (still theoretical) applications to therapeutics:

Controlling unwanted toxic side effects on healthy tissues


Jean clairambault

PK-PD simplified model for cancer chronotherapy

(here with only toxicity constraints; target=death rate)

Healthy cells (jejunal mucosa)

Tumour cells (Glasgow osteosarcoma)

(PK)

(pop.dyn.)

(tumour growth=Gompertz model)

(homeostasis=damped harmonic oscillator)

(« chrono-PD »)

f(C,t)=F.C/(C50+C).{1+cos 2(t-S)/T}

g(D,t)=H.D/(D50+D).{1+cos 2(t-T)/T}

Aim: balancing IV delivereddruganti-tumourefficacy by healthy tissue toxicity

(JC et al. Pathol-Biol 2003; Basdevant, JC, Lévi M2AN 2005; JC ADDR 2007)


Optimal control step 1 deriving an objective function from the tumoral cell population model

Optimal control, step 1: deriving an objective function from the tumoral cell population model

( characteristic function of the allowed infusion periods)

Eradication strategy: minimise GB(i), where:

or else:

Stabilisation strategy: minimise GB(i), where:

(t1 < tf being some fixed observation time

after t0,beginning of infusion interval)


Jean clairambault

Optimal control, step 2: deriving a constraint

function from the enterocyte population model

Minimal toxicity constraint, for 0<A<1(e.g. A =50%):

±other possible constraints:


Jean clairambault

Optimal control problem: defining a lagrangian:

then:

If GB and FA were convex, then a necessary and sufficient condition would be:

…i.e. the minimum would be obtained at a saddle point of the lagrangian, reachable by a Uzawa-like algorithm


Jean clairambault

Optimised control: results of a tumour stabilisation strategy using this simple one-drug PK-PD model

Objective: minimising the maximumof the tumour cell population

Constraint : preserving the jejunal mucosa according to the patient’s state of health

Result : optimal infusion flow i(t) adaptable to the patient’s state of health (according to a tunable parameterA: here preserving A=50% of enterocytes)

(Basdevant, JC, Lévi M2AN 2005; JC ADDR 2007; Lévi, Okyar, Dulong, JC Annu Rev Pharm Toxicol 2010)


Jean clairambault

Another way to take into account a toxicity constraint (with Frédérique Billy and Olivier Fercoq):

Theoretical chronotherapeutic optimisation

of a 1st eigenvalue (cancer growth) under the constraint

of preserving another 1st eigenvalue (healthy tissue growth)

  • McKendrick’s model of cell population proliferation

  • Control of proliferation by blockingKi_i+1 usingtheoreticperiodicdrugdelivery:

  • K(t,x)=[1-g(t)].y(t).k(x) where: g(t) is a periodicexternal control (chronotherapy) y(t) is a circadianclock control on the cell cycle k(x) is an [only] age-dependent transition rate

  • Objective function to beminimised: l1, 1st eigenvalue of cancer cell population

  • Constraintfunction to bepreserved: l2 [≥L], 1st eigenvalue of healthycell population

  • Design of an augmentedLagrangian by combiningl1and l2-L (with penalty)

  • -Arrow-Hurwitz (or Uzawa) algorithm to tracksaddle points of the Lagrangian

  • …thusobtainingonlysuboptimality (necessary to obtaincritical points) conditions


Jean clairambault

Circadian + pharmacological control on transitions

K(x,t) = k(x).y(t).[1-g(t)]: k FUCCI-identified, y clock, g drug effect to be optimised

green and red: y ,

gating at transitions

blue: (1-g).y

(therapy g blocks y)


Jean clairambault

Evolution of the two populations of cancer and healthy cells

Cancer

Circadian control,

no drug infusion

Healthy

Circadian control,

added drug infusion

Healthy

Cancer

(F. Billy et al. 2012)


Jean clairambault

Numerical solution to the optimal infusion problem

and effect on eigenvalues, healthy and cancer

Infusion scheme g(t)

In favour of this approach:

- characterises long-term

trends with one number,

- easily accessible

target for control

- fits to physiologically

structured growth models

Target eigenvalues:

Cancer (blue)

Healthy (green)

  • Its drawbacks:

  • deals with asymptotics,

  • not with transients

  • assumes a linear model

  • for proliferation

  • - assumes periodic control

  • by drugs (but the period

  • can be infinitely long)


Jean clairambault

...within the “DarEvCan” French research consortiumDarwinian Evolution and Cancerhttp://www.darevcan.univ-montp2.fr/

Now tackling the problem of drug resistance


Jean clairambault

A soaring theme on the international scene:

Carlo Maley

Robert Gatenby, MD*

First international

Evolution and cancer conference SF, June 3-5, 2011, next one in 2013

* RG advocates ‘adaptive therapy’, cf. Gatenby Nature 2009, Gatenby et al. Cancer Research 2009


Jean clairambault

Gatenby’s new paradigm: rational management of cancer burden by ‘adaptive therapy’

See also review on evolution and cancer by Aktipis et al. PLoS One, Nov. 2011


Jean clairambault

Tackling this other issue in cancer pharmacotherapeutics:

Emergence of drug resistance in cancer cell populations

Instead of controlling drug resistance at the individual cell level (ABC transporters),

representation of the possible emergence of resistant cell clones due to mutation

occurring at mitoses or adaptation to the environment in a cell Darwinism perspective.

Assumption: cancer cell populations, under the pressure of a drug-enriched

environment, may develop (costly) mutations or adaptations yielding resistant cell

clones, less fit in a drug-free environment, but better survivors in a hostile environment.

A therapeutic objective, under these circumstances, may be not to eradicate all

cancer cells (in fact only all drug-sensitive cells), but instead to let some of them

live so as to limit the growth of an emergent resistant cell clone (‘adaptive therapy’).

Mathematical modelling: design of cell population models structured according to a

phenotypic trait (representing drug resistance, cell division cycle not considered here)


Jean clairambault

First point of view: ‘mutations only’, one cytotoxic drug

a) Healthy cells


Jean clairambault

First point of view: ‘mutations only’, one cytotoxic drug

b) Cancer cells


Jean clairambault

Point of view ‘mutations only’: monomorphism in the healthy cell population


Jean clairambault

Point of view ‘mutations only’: monomorphism also in the cancer cell population

Simulations by Alexander Lorz(JC, T. Lorenzi, A. Lorz, B. Perthame, in revision)


Jean clairambault

2nd point of view : two different drugs, cytotoxic and cytostatic, two resistance traits x and y,‘no mutations, exchanges with the environment instead’


Jean clairambault

Monomorphism in the healthy cell populationNo mutations: non-resistant (‘healthy’) cells: starting from a common medium phenotype (cytotoxic res.=.5, cytostatic res.= .5), evolution towards the non-resistant (0,0) phenotype

Model, simulations and figures by TommasoLorenzi


Jean clairambault

Dimorphism in the cancer cell populationNo mutations: Resistant (‘cancer’) cells: starting from the same common medium phenotype (.5,.5), evolution towards 2 different resistant phenotypes: (1,0) and (0,1)

Model, simulations and figures by TommasoLorenzi


Jean clairambault

What more now? Need for multiscale models!!Drugs act at the molecular and cell level, produce measurable effectsat the cell population level, and are delivered at the whole body level

  • Modelling challenge: integrating

  • these levels in whole body models

  • to optimise drug control on tissue

  • proliferation, healthy and cancer,

  • in a systems biology perspective:

  • intracellular signalling networks,

  • actual drug targets

  • cell populations: proliferation,

  • apoptosis, differentiation

  • whole body ensemble of tissues

  • as interacting cell populations

  • under drug control

(JC, Personalized Medicine, 2011)


Jean clairambault

Collaborators

Bang & LJLL: Annabelle Ballesta,Fadia Bekkal Brikci, Frédérique Billy, Luna Dimitrio, Marie Doumic, Pierre Gabriel, Herbert Gayrard, Erwan Hingant, Thomas Lepoutre, Tommaso Lorenzi, Alexander Lorz, Thomas Ouillon, Benoît Perthame, Emilio Seijo Solis

Other INRIA teams: Catherine Bonnet (Disco), Stéphane Gaubert, Olivier Fercoq (Maxplus), Jean-Charles Gilbert (Estime), Mostafa Adimy, Samuel Bernard, Fabien Crauste, Stéphane Génieys, Laurent Pujo-Menjouet, Vitaly Volpert (Dracula)

INSERM U776 “Biological Rhythms and Cancers” (Francis Lévi, Paul-Brousse hospital, Villejuif): Solid tumours of Mice and Men, chronotherapeutics of colorectal cancer

UMRs UPMC- INSERM U872 Team 18 “Resistance and survival of tumour cells”

(J.-P. Marie, Cordeliers Research Centre and St Antoine Hospital, Paris): Leukaemias

Université Paris-Nord (Claude Basdevant):Optimal control theory and algorithms

FP7ERASysBio C5Sys: http://www.erasysbio.net/index.php?index=272

ANR Bimod (Lyon, Paris, Bordeaux)

GDR DarEvCan (Montpellier, Paris, Lyon,...): http://www.darevcan.univ-montp2.fr/


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