Jean Clairambault INRIA Bang project-team, Rocquencourt & INSERM U776, Villejuif, France

Cell proliferation, circadian clocks and molecular Cell proliferation, circadian clocks and molecular pharmacokinetics-pharmacodynamics to optimise cancer pharmacokinetics-pharmacodynamics to optimise cancer

treatmentstreatments

Jean ClairambaultJean Clairambault

INRIA Bang project-team, Rocquencourt & INSERM U776, Villejuif, FranceINRIA Bang project-team, Rocquencourt & INSERM U776, Villejuif, France

http://www-roc.inria.fr/bang/JC/Jean_Clairambault_en.htmlhttp://www-roc.inria.fr/bang/JC/Jean_Clairambault_en.html

European biomathematics Summer school, Dundee, August 2010European biomathematics Summer school, Dundee, August 2010

Outline of the lecturesOutline of the lectures• 0. Introduction (abstract) and general modelling framework0. Introduction (abstract) and general modelling framework

• 1. Modelling the cell cycle in proliferating cell populations1. Modelling the cell cycle in proliferating cell populations

• 2. Circadian rhythm and cell / tissue proliferation

• 3. Molecular pharmacokinetics-pharmacodynamics (PK-PD)

• 4. Optimising anticancer drug delivery: present and future

• 5. More future prospects and challenges

Introduction and general modelling frameworkIntroduction and general modelling framework

A general framework to optimise cancer therapeutics: A general framework to optimise cancer therapeutics: designing mathematical methods along 3 axes designing mathematical methods along 3 axes

• Modelling the growing cell populations on which drugs act: proliferating Modelling the growing cell populations on which drugs act: proliferating tumour tumour and healthy and healthy cells and tissuescells and tissues

• Modelling the control system, i.e., the fate of drugs in the organism, at the Modelling the control system, i.e., the fate of drugs in the organism, at the molecular and whole body levels by molecular and whole body levels by molecular pharmacokinetics-molecular pharmacokinetics-pharmacodynamics:pharmacodynamics: PK-PD, ideally WBPBPKPD PK-PD, ideally WBPBPKPD (Malcolm Rowland)(Malcolm Rowland)

• Optimising the control: Optimising the control: dynamic dynamic optimal control of drug delivery flows optimal control of drug delivery flows using time-dependent objectives+constraintsusing time-dependent objectives+constraints

(JC MMNP 2009)(JC MMNP 2009)

0. Introduction0. Introduction

3 short preliminary questions3 short preliminary questions

• What sort of disease is cancer?What sort of disease is cancer?

• How are anticancer drugs delivered and how do they act?How are anticancer drugs delivered and how do they act?

• How can we improve their efficacy?How can we improve their efficacy?

0. Introduction0. Introduction

To justify the choice of these 3 axes: To justify the choice of these 3 axes:

Cancer: a Cancer: a control diseasecontrol disease, defined as , defined as uncontrolled uncontrolled cell population growth in proliferating tissuescell population growth in proliferating tissues

One cell divides in two: a physiologically controlled process at cell and tissue levelsOne cell divides in two: a physiologically controlled process at cell and tissue levelsin all healthy and fast renewing tissues (gut, bone marrow…) that is in all healthy and fast renewing tissues (gut, bone marrow…) that is disrupted in disrupted in cancercancer

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

0. Introduction: what is cancer?0. Introduction: what is cancer?

• InputInput: an intravenous [multi-]drug infusion flow: an intravenous [multi-]drug infusion flow

• Drug concentrations in bloodDrug concentrations in blood and tissueand tissue compartments (compartments (PKPK))

• Control of targets on the cell cycle Control of targets on the cell cycle in tissuesin tissues (cell population (cell population PDPD))

• OutputOutput: a cell population number -or growth rate- in tumour and healthy : a cell population number -or growth rate- in tumour and healthy tissuestissues

Drugs: from delivery (infusion/ingestion) to targetDrugs: from delivery (infusion/ingestion) to targetMolecular Molecular PK-PD PK-PD modelling in oncologymodelling in oncology

““PharmacokineticsPharmacokinetics is what the organism does to the drug, is what the organism does to the drug, PharmacodynamicsPharmacodynamics is what the drug does to the organism” is what the drug does to the organism”

0. Introduction: how are anticancer drugs delivered?0. Introduction: how are anticancer drugs delivered?

Optimising drug delivery:Optimising drug delivery:optimisation under constraintsoptimisation under constraints

• Optimal control of delivery flow (programmable pumps)Optimal control of delivery flow (programmable pumps)

• Objective: minimising tumour cell numberObjective: minimising tumour cell number

• Constraints: - limiting toxicity to healthy cells Constraints: - limiting toxicity to healthy cells - avoiding drug resistance in cancer cells - avoiding drug resistance in cancer cells

- taking into account individual genetic factors- taking into account individual genetic factors

… … and optimal circadian delivery timesand optimal circadian delivery times

0. Introduction: therapeutic optimisation?0. Introduction: therapeutic optimisation?

Mathematical modelling to optimise cancer treatments Mathematical modelling to optimise cancer treatments

… … hence the choice of these 3 main research directions:hence the choice of these 3 main research directions:

- Proliferation: cell population dynamics in tissues - Proliferation: cell population dynamics in tissues (PDEs)(PDEs)

- Drugs: molecular pharmacokinetics-pharmacodynamic - Drugs: molecular pharmacokinetics-pharmacodynamic (ODEs)(ODEs)

- Therapeutic optimisation: optimal control of drug delivery - Therapeutic optimisation: optimal control of drug delivery (optimal control algorithms)(optimal control algorithms)

… …and future prospects: even more challenges for modelling!and future prospects: even more challenges for modelling!

0. General modelling framework0. General modelling framework

Cyclin DCyclin D

Cyclin ECyclin ECyclin ACyclin A

Cyclin BCyclin B

SG1

G2

M

At the origin of proliferation: the cell division cycleAt the origin of proliferation: the cell division cycle

Physiological or therapeutic control Physiological or therapeutic control exerted on:exerted on:- transitions (checkpoints) betweentransitions (checkpoints) between phases (Gphases (G11/S, G/S, G22/M, M/G/M, M/G11))

- death rates (apoptosis or necrosis) - death rates (apoptosis or necrosis) - progression speeds inside phases- progression speeds inside phases- exchanges between quiescent (Gexchanges between quiescent (G00))

and proliferative phases (Gand proliferative phases (G11 only) only)

S:=DNA synthesis; GS:=DNA synthesis; G11,G,G22:=Gap1,2; M:=mitosis:=Gap1,2; M:=mitosisMitosis=M phaseMitosis=M phase

(from Lodish et al., Molecular cell biology, 2003)(from Lodish et al., Molecular cell biology, 2003)

0. General modelling framework: cell / tissue proliferation0. General modelling framework: cell / tissue proliferation

Modelling the cell division cycle in cell populationsModelling the cell division cycle in cell populationsPhysiologically structured PDEsPhysiologically structured PDEs

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

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

di , Ki->i+1 constant or periodic w. r. to time t(1≤i≤I, I+1=1)

ni:=cell population density in phase i ;vi :=progression speed;di:=death rate;

Ki-1->i:=transition rate(with a factor 2 if i=1)

Death rates Death rates ddii:: (“loss”), “speeds” (“loss”), “speeds” vvii and phase transitions and phase transitions KKi->i+1i->i+1 are model targetsare model targets

for physiological (e.g. circadian) and therapeutic (drugs) control for physiological (e.g. circadian) and therapeutic (drugs) control (t)(t)(t): e.g., clock-controlled Cdk1 or intracellular output of drug infusion flow](t): e.g., clock-controlled Cdk1 or intracellular output of drug infusion flow](Firstly presented in: JC, B. Laroche, S. Mischler, B. Perthame, RR INRIA #4892, 2003)(Firstly presented in: JC, B. Laroche, S. Mischler, B. Perthame, RR INRIA #4892, 2003)


Proof of the existence ofProof of the existence of a unique growth exponent a unique growth exponent ,,the same for all phases the same for all phases ii, , such that the are asymptotically (i.e., for large times) such that the are asymptotically (i.e., for large times) bounded, and asymptotically periodic if the control is periodicbounded, and asymptotically periodic if the control is periodic

Surfing on the exponential growth curveSurfing on the exponential growth curve, example (periodic control case): 2 phases, , example (periodic control case): 2 phases, control on Gcontrol on G22/M transition by 24-h-periodic CDK1-Cyclin B (A. Goldbeter’s model)/M transition by 24-h-periodic CDK1-Cyclin B (A. Goldbeter’s model)

=CDK1=CDK1 All cells in G1-S-G2 (phaseAll cells in G1-S-G2 (phase ii=1)=1) All cells in M (phase All cells in M (phase ii=2)=2)

Entrainment of the cell division cycle by CDK1 at the circadian periodEntrainment of the cell division cycle by CDK1 at the circadian period

All cells, All cells, surfing on the exponential growth curvesurfing on the exponential growth curve

Main result: a growth exponent for the cell population behaviourMain result: a growth exponent for the cell population behaviour

time time tt

(Normalised cell population number)(Normalised cell population number)


Example: molecular pharmacodynamics (PD) of 5FU Example: molecular pharmacodynamics (PD) of 5FU

CompetitiveCompetitiveinhibitioninhibitionby FdUMP of by FdUMP of dUMP binding dUMP binding to target TSto target TS

++[[Stabilisation Stabilisation by CHby CH22-THF of -THF of

binary complex binary complex

dUMP-TSdUMP-TS]]

Incorporation of Incorporation of FUTP instead of FUTP instead of

UTP to RNAUTP to RNA

Incorporation of Incorporation of FdUTP instead of FdUTP instead of dTTP to DNA dTTP to DNA (Longley, Nat Rev Canc 2003)(Longley, Nat Rev Canc 2003)

RNA wayRNA way DNA wayDNA way2 main metabolic pathways:2 main metabolic pathways: action on RNA and on DNAaction on RNA and on DNA

0. General modelling framework: drugs0. General modelling framework: drugs

Formyltetrahydrofolate (CHO-THF) = LVFormyltetrahydrofolate (CHO-THF) = LV

Precursor of CHPrecursor of CH22-THF, coenzyme of TS, that forms with it and FdUMP -THF, coenzyme of TS, that forms with it and FdUMP

a stable ternary complex, blocking the normal biochemical reactiona stable ternary complex, blocking the normal biochemical reaction

(Longley, Nat Rev Canc 2003)(Longley, Nat Rev Canc 2003)

5,10-CH5,10-CH22-THF + dUMP + FADH-THF + dUMP + FADH22 dTMP +THF + FAD dTMP +THF + FADTSTS

Inhibition of Thymidylate Synthase (TS) by 5FU and LeucovorinInhibition of Thymidylate Synthase (TS) by 5FU and Leucovorin

((TS affinity:TS affinity:

FdUMP > dUMPFdUMP > dUMP))


ODEs: PK-PD of 5FU [+ drug resistance] + ODEs: PK-PD of 5FU [+ drug resistance] + LeucovorinLeucovorin

F. Lévi, A. Okyar, S. Dulong, JC, Annu Rev Pharm Toxicol 2010F. Lévi, A. Okyar, S. Dulong, JC, Annu Rev Pharm Toxicol 2010


0. General modelling framework: circadian physiology for chronotherapeutic optimisation0. General modelling framework: circadian physiology for chronotherapeutic optimisation

Peripheral oscillatorsPeripheral oscillatorsProliferationProliferation

MetabolismMetabolism

Rest-activity cycle: open window on SCN central clockRest-activity cycle: open window on SCN central clock

23 23 7711

Arb

itra

ry u

nits

Time (h)

RHTRHT

MelatoninMelatonin

CNS, hormones,CNS, hormones,peptides, mediatorspeptides, mediators

NPV

GlutamateGlutamate

PinealPineal

NPYNPY

Central coordinationCentral coordination

TGFTGF, , EGFEGFProkineticinProkineticin

GlucocorticoidsGlucocorticoids Food intake rhythmFood intake rhythm

Autonomic nervous systemAutonomic nervous system

CircCircadian clocksadian clocks

The circadian system…The circadian system…

Lévi, Lancet Oncol 2001 ; Mormont & Lévi, Cancer 2003

Entrainment by lightEntrainment by light SupraSupraChiasmaticChiasmaticNucleiNuclei

……is an orchestra of cell clocks with one neuronal conductor in is an orchestra of cell clocks with one neuronal conductor in the SCN and molecular circadian clocks in all nucleated cellsthe SCN and molecular circadian clocks in all nucleated cells

(Hastings, Nature Rev.Neurosci. 2003)

SCN=suprachiasmatic nuclei SCN=suprachiasmatic nuclei (in the hypothalamus)(in the hypothalamus)


(after Hastings, Nature Rev. Neurosci. 2003)

Activation loop

Inhibition loop

Clock-controlled

genes

In each In each nucleated cell: a molecular circadian clocknucleated cell: a molecular circadian clock

Proliferation

Metabolism

Cellular rhythmsCellular rhythms

24 h-rhythmic transcription:24 h-rhythmic transcription: 10% of genome, among which:10% of genome, among which: 10% : cell cycle10% : cell cycle 2% : growth factors2% : growth factors


Time-scheduled delivery regimen for metastatic CRCTime-scheduled delivery regimen for metastatic CRCInfusion over 5 days every 3rd week Infusion over 5 days every 3rd week

Folinic Acid300 mg/m2/d

Time (local h) 04:0016:00.

5-FU600 - 1100 mg/m2/dOxaliPt

25 mg/m2/d

Multichannel programmable ambulatoryMultichannel programmable ambulatoryinjector for intravenous drug infusion injector for intravenous drug infusion (pompe Mélodie, Aguettant, Lyon, France)(pompe Mélodie, Aguettant, Lyon, France)

F. Lévi, INSERM U 776 Rythmes Biologiques et Cancers

Circadian rhythms and cancer chronotherapeuticsCircadian rhythms and cancer chronotherapeutics(Results from Francis Lévi’s INSERM team U 776, Villejuif, France)(Results from Francis Lévi’s INSERM team U 776, Villejuif, France)

Can such therapeutic schedules be improvedCan such therapeutic schedules be improved??

3. Drug delivery optimisation3. Drug delivery optimisation

0. General modelling framework: actual (clinical) chronotherapeutic optimisation0. General modelling framework: actual (clinical) chronotherapeutic optimisation

Results of cancer chronotherapyResults of cancer chronotherapy

Infusion flowInfusion flowMetastatic colorectal cancerMetastatic colorectal cancer

(Folinic Acid, 5-FU, Oxaliplatin)(Folinic Acid, 5-FU, Oxaliplatin) ChronoConstant

16%16%31%31% Neuropathy gr 2-3Neuropathy gr 2-3

51%51%30%30% Responding rateResponding rate

14%14%74%74% Oral mucositis gr 3-4Oral mucositis gr 3-4

ToxicityToxicity

<10-3

<10-2

<10-4

p

Lévi et al. Lévi et al. JNCI 1994 ;JNCI 1994 ;Lancet 1997 ;Lancet 1997 ;Lancet Oncol 2001Lancet Oncol 2001

INSERM U 776 Rythmes Biologiques et Cancers

How does it work? Impact of circadian clocks on 1) cell drug How does it work? Impact of circadian clocks on 1) cell drug detoxication enzymes and 2) cell division cycle determinant proteins detoxication enzymes and 2) cell division cycle determinant proteins (cyclins/CDKs)(cyclins/CDKs)

0. General modelling framework: actual 0. General modelling framework: actual (clinical) chronotherapeutic optimisation(clinical) chronotherapeutic optimisation

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

Healthy tissues, i.e., cell populations, would be well synchronised w. r. to Healthy tissues, i.e., cell populations, would be well synchronised w. r. to proliferation rhythms and w. r. to circadian clocks, whereas…proliferation rhythms and w. r. to circadian clocks, whereas…

Tumour cell populations would be desynchronised w. r. to both, and suchTumour cell populations would be desynchronised w. r. to both, and suchproliferation desynchronisation would be a consequence of an escapeproliferation desynchronisation would be a consequence of an escapeby peripheral cells from central circadian clock control messages, just asby peripheral cells from central circadian clock control messages, just astumour cells evade most physiological controls, cf. Hanahan & Weinberg:tumour cells evade most physiological controls, cf. Hanahan & Weinberg:

0. General modelling framework: actual (clinical) chronotherapeutic optimisation0. General modelling framework: actual (clinical) chronotherapeutic optimisation

Optimal control of anticancer chronopharmacotherapyOptimal control of anticancer chronopharmacotherapy

1)1) Objective functionObjective function to be minimised: cell population growth rate or cell population to be minimised: cell population growth rate or cell population density in tumour tissuesdensity in tumour tissues

2)2) Control functionControl function: : instantaneous [dynamic] intravenous infusion = [multi-]drug instantaneous [dynamic] intravenous infusion = [multi-]drug delivery flow via external programmable pumpsdelivery flow via external programmable pumps

3)3) ConstraintsConstraints to be satisfied: to be satisfied: - maintaining healthy cell population over a tolerability threshold- maintaining healthy cell population over a tolerability threshold - taking into account circadian phases of drug processing systems (model prerequisite)- taking into account circadian phases of drug processing systems (model prerequisite) - - maintaining normal tissue synchronisation control by circadian clocksmaintaining normal tissue synchronisation control by circadian clocks - limiting resistances in tumour cells (- limiting resistances in tumour cells (e.g. controlling induction of nrf2)e.g. controlling induction of nrf2) - others: maximal daily dose, maximal delivery flow,…- others: maximal daily dose, maximal delivery flow,…

4)4) With adaptationWith adaptation of the designed controlled system model (and hence of the optimised of the designed controlled system model (and hence of the optimised drug delivery flow) to drug delivery flow) to patient-specific parameterspatient-specific parameters: clock phases, enzyme genetic : clock phases, enzyme genetic polymorphism, target protein levels, going polymorphism, target protein levels, going towards personalised medicinetowards personalised medicine

0. General modelling framework: theoretical (future?) chronotherapeutic optimisation under constraints0. General modelling framework: theoretical (future?) chronotherapeutic optimisation under constraints

PK-PD simplified model for cancer chronotherapyPK-PD simplified model for cancer chronotherapy(here with only toxicity constraints; target=death rate)(here with only toxicity constraints; target=death rate)

Healthy cells (jejunal mucosa)Healthy cells (jejunal mucosa) Tumour cellsTumour cells

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}

(PK)(PK)

(« chrono-PD »)(« chrono-PD »)

(homeostasis=damped harmonic oscillator)(homeostasis=damped harmonic oscillator) (tumour growth=Gompertz model)(tumour growth=Gompertz model)

(JC, Pathol-Biol 2003; Adv Drug Deliv Rev 2007)(JC, Pathol-Biol 2003; Adv Drug Deliv Rev 2007)

Aim: balancing IV delivered drug anti-tumour efficacy by healthy tissue toxicity Aim: balancing IV delivered drug anti-tumour efficacy by healthy tissue toxicity


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

Objective: Objective: minimising the maximumminimising the maximum of the tumour cell populationof the tumour cell population

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

((C. BasdevantC. Basdevant, JC, F. Lévi, M2AN 2005; JC Adv Drug Deliv Rev 2007), JC, F. Lévi, M2AN 2005; JC Adv Drug Deliv Rev 2007)

Result : optimal infusion flow i(t) adaptable 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(according to a tunable parameter AA: : here preserving here preserving AA=50% of enterocytes=50% of enterocytes))


Individualised treatments in oncology Individualised treatments in oncology Genetic polymorphismGenetic polymorphism: between-subject variability: between-subject variability

for pharmacological model parametersfor pharmacological model parameters

• According to subjects, different expression and activity levels of According to subjects, different expression and activity levels of drug processing enzymes and proteins (uptake, degradation, active efflux, e.g. drug processing enzymes and proteins (uptake, degradation, active efflux, e.g.

GST, DPYD, UGT1A1, P-gp,…) GST, DPYD, UGT1A1, P-gp,…) and drug targets (e.g. Thymidylate Synthase, Topoisomerase I)and drug targets (e.g. Thymidylate Synthase, Topoisomerase I)

• The same is true of DNA mismatch repair enzyme gene expression (e.g., The same is true of DNA mismatch repair enzyme gene expression (e.g., ERCC1, ERCC2)ERCC1, ERCC2)

• More generally, pharmacotherapeutics should be guided more by molecular More generally, pharmacotherapeutics should be guided more by molecular alterations of the DNA than by location of tumours in the organism: genotyping alterations of the DNA than by location of tumours in the organism: genotyping patients with respect to anticancer drug processing may become the rule in patients with respect to anticancer drug processing may become the rule in oncology in the future oncology in the future ((see e.g. G. Milano & J. Robert in Oncologie 2005)see e.g. G. Milano & J. Robert in Oncologie 2005)

• ……Which leads, using actively searched-for biomarkers, to Which leads, using actively searched-for biomarkers, to populational PK-PDpopulational PK-PD

0. General modelling framework: some future prospects0. General modelling framework: some future prospects

Other frontiers in cancer therapeuticsOther frontiers in cancer therapeutics

1.1. ImmunotherapyImmunotherapy:: Not only using cytokines and actual anticancer vaccines, but also examining Not only using cytokines and actual anticancer vaccines, but also examining

delivery of cytotoxics from the point of view of their action on the immune systemdelivery of cytotoxics from the point of view of their action on the immune system (Review by L. Zitvogel in Nature Rev. Immunol. 2008)(Review by L. Zitvogel in Nature Rev. Immunol. 2008)

2. 2. The various facets of (innate/acquired/(ir)reversible) The various facets of (innate/acquired/(ir)reversible) drug resistancedrug resistance::- Repair enzymes, mutated p53: cell cycle models with by-pass of DNA damage control- Repair enzymes, mutated p53: cell cycle models with by-pass of DNA damage control- ABC transporters, cellular drug metabolism: molecular PK-PD ODEs (or PDEs)- ABC transporters, cellular drug metabolism: molecular PK-PD ODEs (or PDEs)- Microenvironment, interactions with stromal cells: competition/cooperativity models- Microenvironment, interactions with stromal cells: competition/cooperativity models- Mutations of the targets: evolutionary game theory, evolutionary dynamics models- Mutations of the targets: evolutionary game theory, evolutionary dynamics models

3.3. Developing Developing non-cell-killing therapeutic meansnon-cell-killing therapeutic means:: - Associations of cytotoxics and redifferentiating agents (e.g. retinoic acid in AML3)- Associations of cytotoxics and redifferentiating agents (e.g. retinoic acid in AML3) - Modifying local metabolic parameters? (pH) to foster proliferation of healthy cells- Modifying local metabolic parameters? (pH) to foster proliferation of healthy cells

0. General modelling framework: some future prospects0. General modelling framework: some future prospects

Modelling the cell division cycle inModelling the cell division cycle inproliferating cell populationsproliferating cell populations

Why model the cell division cycle?Why model the cell division cycle?

• Need for detailed models of cell proliferation to represent the action of anticancer Need for detailed models of cell proliferation to represent the action of anticancer drugs drugs in cell populationsin cell populations with: with:

1) Cell cycle phase specificity1) Cell cycle phase specificity 2) Different pharmacological targets on cell cycle control2) Different pharmacological targets on cell cycle control 3) Action with same targets on tumour cells 3) Action with same targets on tumour cells and on healthy cellsand on healthy cells (toxic side effects of anticancer drugs)(toxic side effects of anticancer drugs)

• Hence, even independently of therapeutics, need for models with:Hence, even independently of therapeutics, need for models with: 1) Cell cycle phases and age-in-phase, possibly cyclin, structure1) Cell cycle phases and age-in-phase, possibly cyclin, structure

2) Transitions between cell division cycle phases (G2) Transitions between cell division cycle phases (G11/S, G/S, G22/M)/M)

3) Exchanges between quiescent and proliferative phases (G3) Exchanges between quiescent and proliferative phases (G00/G/G11))

4) Targets for control of cell proliferation (physiological / by drugs)4) Targets for control of cell proliferation (physiological / by drugs)

1. Modelling the cell cycle in proliferating cell populations1. Modelling the cell cycle in proliferating cell populations

Cyclin DCyclin D/ CDK2/ CDK2

Cyclin ECyclin E/ CDK 4or6/ CDK 4or6

Cyclin A Cyclin A / CDK2/ CDK2

Cyclin B Cyclin B / CDK1/ CDK1

SG1

G2

M

At the origin of proliferation: the cell division cycleAt the origin of proliferation: the cell division cyclein proliferating cell populationsin proliferating cell populations

Physiological and therapeutic control Physiological and therapeutic control exerted on:exerted on:- transitions (checkpoints) betweentransitions (checkpoints) between phases (Gphases (G11/S, G/S, G22/M, M/G/M, M/G11))

- death rates (apoptosis or necrosis) - death rates (apoptosis or necrosis) and progression speeds inside phasesand progression speeds inside phases- exchanges between quiescent (Gexchanges between quiescent (G00))

and proliferative phases (Gand proliferative phases (G11 only) only)

S:=DNA synthesis; GS:=DNA synthesis; G11,G,G22:=Gap1,2; M:=mitosis:=Gap1,2; M:=mitosis (one cell divides in two)(one cell divides in two)

Mitosis=M phaseMitosis=M phase

Mitotic human HeLa cell (from LBCMCP-Toulouse)


Age-structured PDE modelsAge-structured PDE models

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

In each phase In each phase ii, a Von Foerster-McKendrick-like linear model:, a Von Foerster-McKendrick-like linear model:

di , Ki->i+1 constant or periodic w. r. to time t(1≤i≤I, I+1=1)

ni:=cell population density in phase i ;vi :=progression speed;di:=death rate;

Ki-1->i:=transition rate(with a factor 2for i=1)

Death rates Death rates ddii:: (“loss”), “speeds” (“loss”), “speeds” vvii and phase transitions and phase transitions KKi->i+i->i+11 are model targetsare model targets

for physiological (e.g., circadian) or therapeutic (drug) control for physiological (e.g., circadian) or therapeutic (drug) control (t)(t)(t)(t): e.g., clock-controlled CDK1 or intracellular output of drug infusion flow]: e.g., clock-controlled CDK1 or intracellular output of drug infusion flow](Firstly presented in: JC, B. Laroche, S. Mischler, B. Perthame, RR INRIA #4892, 2003)(Firstly presented in: JC, B. Laroche, S. Mischler, B. Perthame, RR INRIA #4892, 2003)

Flow cytometry


The simplest case: 1-phase model with divisionThe simplest case: 1-phase model with division

(Here(Here, , vv(a)(a)=1, =1, a* a* is the cell cycle duration, and is the cell cycle duration, and is the timeis the timeduring which the 1-during which the 1-periodic controlperiodic control is actually exerted on cell division)is actually exerted on cell division)

Then it can be shown that the eigenvalue problem:Then it can be shown that the eigenvalue problem:

has a unique positive 1has a unique positive 1-periodic-periodic eigenvector eigenvector NN, with a positive eigenvalue , with a positive eigenvalue and an explicitformula can be found for and an explicitformula can be found for when when KK00 ++∞∞ (T. Lepoutre)(T. Lepoutre)


General case: General case: II phases (last = mitosis, or phases (last = mitosis, or MM phase) phase) (Note that exchanges between G(Note that exchanges between G00 and G and G11 are not considered in this linear model, i.e., are not considered in this linear model, i.e.,

all cells are assumed to proliferate)all cells are assumed to proliferate)

Then, provided that reasonable assumptions on death and transition rates are satisfied:Then, provided that reasonable assumptions on death and transition rates are satisfied:

(thus ensuring positive growth), one can establish that:(thus ensuring positive growth), one can establish that:


According to the Krein-Rutman theorem (infinite-dimensional form of theAccording to the Krein-Rutman theorem (infinite-dimensional form of the

Perron-Frobenius theorem), there exists a nonnegative first eigenvalue Perron-Frobenius theorem), there exists a nonnegative first eigenvalue such that,such that,if , then there exist if , then there exist NNii,, bounded solutions to the problem: bounded solutions to the problem:

(the weights(the weightsi i ≥0 being solutions to the dual problem); this can be proved by using≥0 being solutions to the dual problem); this can be proved by using

a generalised entropy principle (GRE). Moreover, if the control (a generalised entropy principle (GRE). Moreover, if the control (ddii oror KKi->i+1i->i+1) is ) is

periodic, so are the eigenvectors periodic, so are the eigenvectors NNii and weights and weights ii, with the same period., with the same period.

with a number with a number such that for all such that for all ii::

Ph. Michel, S. Mischler, B. Perthame, C. R. Acad. Sci. Paris Ser. I (Math.) 2004; J Math Pures Appl 2005Ph. Michel, S. Mischler, B. Perthame, C. R. Acad. Sci. Paris Ser. I (Math.) 2004; J Math Pures Appl 2005

JC, Michel, Perthame C. R. Acad. Sci. Paris Series I (Math.) 2006; Proceedings ECMTB Dresden 2005JC, Michel, Perthame C. R. Acad. Sci. Paris Series I (Math.) 2006; Proceedings ECMTB Dresden 2005

..


Proof of the existence of a Proof of the existence of a unique growth exponent unique growth exponent ,,the same for all phases the same for all phases ii, , such that the are bounded, and asymptotically periodic if such that the are bounded, and asymptotically periodic if the control is periodicthe control is periodic

Example of control (periodic control case): 2 phases, control on GExample of control (periodic control case): 2 phases, control on G22/M transition by /M transition by

24-h-periodic CDK1-Cyclin B (from A. Goldbeter’s minimal mitotic oscillator 24-h-periodic CDK1-Cyclin B (from A. Goldbeter’s minimal mitotic oscillator model)model)

=CDK1 All cells in G1-S-G2 (phase i=1) All cells in M (phase i=2)

Entrainment of the cell division cycle by = CDK1 at the circadian period

: a growth exponent governing the cell population behaviour: a growth exponent governing the cell population behaviour

time t

““Surfing on the Surfing on the exponential growth exponential growth curve”curve”

(= the same as adding(= the same as addingan artificial death terman artificial death termto theto theddii))


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

The total population of cellsThe total population of cells

inside each phase followsinside each phase followsasymptotically an exponentialasymptotically an exponentialbehaviourbehaviour

Stationary (=asymptotic) Stationary (=asymptotic) state distribution of cells state distribution of cells inside phases according inside phases according to ageto age a: no control -> a: no control -> exponential decayexponential decay


Details (2): 2 phases, periodic control Details (2): 2 phases, periodic control on Gon G22/M transition/M transition

The total population of cellsThe total population of cells

inside each phase followsinside each phase followsasymptotically an exponentialasymptotically an exponentialbehaviour behaviour tuned by a periodic tuned by a periodic functionfunction

Stationary (=asymptotic) Stationary (=asymptotic) state distribution of cells state distribution of cells inside phases according inside phases according to age to age a: sharp periodic a: sharp periodic control ->sharp rise and control ->sharp rise and decay decay


Desynchronisation of cells within populations w. r. toDesynchronisation of cells within populations w. r. tocell cycle timing = phase overlapping at transition:cell cycle timing = phase overlapping at transition:Possible experimental measurements to identify transition kernel Possible experimental measurements to identify transition kernel KK

Starting from the simplest model with Starting from the simplest model with dd=0 (one phase with division):=0 (one phase with division):

Interpreted as: if Interpreted as: if is the age at division, or transition: (remark by Th. Lepoutre)is the age at division, or transition: (remark by Th. Lepoutre)

where the probability density (experimentally identifiable) is:where the probability density (experimentally identifiable) is:

withwith

whencewhence

i.e.,i.e.,


Modelling cell proliferation Modelling cell proliferation and quiescenceand quiescence


Exchanges between proliferating (GExchanges between proliferating (G11/S/G/S/G22/M) and quiescent (G/M) and quiescent (G00) cell compartments) cell compartments

are controlled by mitogens and antimitogenic factors in Gare controlled by mitogens and antimitogenic factors in G11 phase phase

From Vermeulen et al. Cell Prolif. 2003From Vermeulen et al. Cell Prolif. 2003

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


RRRestriction pointRestriction point(late G(late G11 phase) phase)

before R:before R:mitogen-dependentmitogen-dependentprogression through Gprogression through G11

(possible regression to G(possible regression to G00))

after R:after R:mitogen-independentmitogen-independentprogression through Gprogression through G11 to S to S

(no way back to G(no way back to G00))

Nonlinear models: introducing exchanges between Nonlinear models: introducing exchanges between proliferating (Gproliferating (G11/S/G/S/G22/M) and quiescent (G/M) and quiescent (G00) cells) cells

ODE models with two exchanging cell compartments, ODE models with two exchanging cell compartments, proliferating (P) and quiescent (Q)proliferating (P) and quiescent (Q)

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

where, for instance:where, for instance:rr0 0 representing here the rate ofrepresenting here the rate of

inactivation of proliferating cells,inactivation of proliferating cells,and and rrii the rate of recruitment fromthe rate of recruitment from

quiescence to proliferationquiescence to proliferation

Initial goal: to justify Gompertz growthInitial goal: to justify Gompertz growth(a popular model among radiologists) (a popular model among radiologists)

Cell exchangesCell exchanges


Simple PDE models, age-structured withSimple PDE models, age-structured withexchanges between proliferation and quiescenceexchanges between proliferation and quiescence

pp=density of proliferating cells; =density of proliferating cells; qq=density of quiescent cells; =density of quiescent cells; =death terms;=death terms;KK=term describing cells leaving proliferation to quiescence, due to mitosis;=term describing cells leaving proliferation to quiescence, due to mitosis;=term describing “reintroduction” (or recruitment) from quiescence to proliferation=term describing “reintroduction” (or recruitment) from quiescence to proliferation


DelayDelay differential differential models with two cell compartments,models with two cell compartments, proliferating (P)/quiescent (Q): proliferating (P)/quiescent (Q): Haematopoiesis models Haematopoiesis models

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

(from Mackey, Blood 1978)(from Mackey, Blood 1978)

Properties of this model: depending on the parameters, one can have positiveProperties of this model: depending on the parameters, one can have positivestability, extinction, explosion, or sustained oscillations of both populationsstability, extinction, explosion, or sustained oscillations of both populations

(Hayes stability criteria, see (Hayes stability criteria, see Hayes, J London Math Soc 1950Hayes, J London Math Soc 1950))Oscillatory behaviour is observed in Oscillatory behaviour is observed in periodic Chronic Myelogenous Leukaemiaperiodic Chronic Myelogenous Leukaemia((CMLCML) where oscillations with limited amplitude are compatible with survival, ) where oscillations with limited amplitude are compatible with survival, whereas explosion (blast crisis, alias acutisation) leads to whereas explosion (blast crisis, alias acutisation) leads to AMLAML and death and death (Mackey and Bélair in Montréal; Adimy, Bernard, Crauste, Pujo-Menjouet, Volpert in Lyon)(Mackey and Bélair in Montréal; Adimy, Bernard, Crauste, Pujo-Menjouet, Volpert in Lyon)

(delay (delay cell division cycle cell division cycle duration)duration)


Modelling haematopoiesis Modelling haematopoiesis forfor

Acute Myelogenous LeukaemiaAcute Myelogenous Leukaemia (AML) (AML)……aiming at aiming at non-cell-killing therapeuticsnon-cell-killing therapeuticsby inducing by inducing re-differentiation of cells usingre-differentiation of cells usingmolecules (e.g. ATRA) enhancing differentiationmolecules (e.g. ATRA) enhancing differentiationrates represented by Krates represented by Kii terms terms

(see (see Adimy et al. JBS 2008Adimy et al. JBS 2008 for more details) for more details)

where where rrii and and ppii represent resting and proliferating represent resting and proliferating

cells, respectively, with reintroduction term cells, respectively, with reintroduction term ii==ii(x(xii) )

positive decaying to zero, positive decaying to zero,

with population argument:with population argument:

and boundary conditions:and boundary conditions:


A model of tissue growth with proliferation/quiescenceA model of tissue growth with proliferation/quiescenceAn age[An age[aa]-and-cyclin[]-and-cyclin[xx]-structured PDE model with proliferating and quiescent cells]-structured PDE model with proliferating and quiescent cells(exchanges between (exchanges between (p)(p) and and (q)(q), healthy and tumour tissue cases: G, healthy and tumour tissue cases: G00 to G to G1 1 recruitment differs)recruitment differs)

Healthy tissue Healthy tissue recruitment: recruitment: homeostasishomeostasis

Tumour recruitment:Tumour recruitment:exponential (exponential (possiblypossiblypolynomialpolynomial) growth) growth

F. Bekkal Brikci, JC, B. Ribba, B. PerthameJMB 2008; MCM 2008

M. Doumic-Jauffret, MMNP 2008

p: p: proliferating proliferating cellscells

q: quiescent q: quiescent cellscells

N: all cells N: all cells (p+q)(p+q)

00ffor smallor small00for large Nfor large N

00ffor allor all


Next step: integrating the two models (Next step: integrating the two models (Von Foerster-Von Foerster-McKendrick-like linear and nonlinear proliferation/quiescence) McKendrick-like linear and nonlinear proliferation/quiescence)

in a complete cell cycle model with phases Gin a complete cell cycle model with phases G00-G-G11-S-G-S-G22-M-M

Keeping the same control targets, adding control by growth factors,Keeping the same control targets, adding control by growth factors,hence on cyclin D, on the recruitment function G from Ghence on cyclin D, on the recruitment function G from G00 to G to G11

and on progression speed in Gand on progression speed in G11

……Work in progress…Work in progress…


Jean Clairambault INRIA Bang project-team, Rocquencourt & INSERM U776, Villejuif, France

Documents

Transcript of Jean Clairambault INRIA Bang project-team, Rocquencourt & INSERM U776, Villejuif, France