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

1. Cell proliferation, circadian clocks and molecular pharmacokinetics-pharmacodynamics to optimise cancer treatments Jean Clairambault INRIA Bang project-team, Rocquencourt & INSERM U776, Villejuif, France http://www-roc.inria.fr/bang/JC/Jean_Clairambault_en.html European biomathematics Summer school, Dundee, August 2010

2. Outline of the lectures • 0. Introduction (abstract) and general modelling framework • 1. 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

3. Introduction and general modelling framework

4. 0. Introduction A general framework to optimise cancer therapeutics: designing mathematical methods along 3 axes • Modelling the growing cell populations on which drugs act: proliferating tumour and healthy cells and tissues • Modelling the control system, i.e., the fate of drugs in the organism, at the molecular and whole body levels by molecular pharmacokinetics-pharmacodynamics: PK-PD, ideally WBPBPKPD (Malcolm Rowland) • Optimising the control: dynamic optimal control of drug delivery flows using time-dependent objectives+constraints (JC MMNP 2009)

5. 0. Introduction 3 short preliminary questions To justify the choice of these 3 axes: • What sort of disease is cancer? • How are anticancer drugs delivered and how do they act? • How can we improve their efficacy?

6. 0. Introduction: what is cancer? Cancer: a control disease, defined as uncontrolled cell population growth in proliferating tissues (from Lodish et al., Molecular cell biology, 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

7. 0. Introduction: how are anticancer drugs delivered? Drugs: from delivery (infusion/ingestion) to target Molecular PK-PD modelling in oncology “Pharmacokinetics is what the organism does to the drug, Pharmacodynamics is what the drug does to the organism” • Input: an intravenous [multi-]drug infusion flow • Drug concentrations in bloodand tissuecompartments (PK) • Control of targets on the cell cycle in tissues (cell population PD) • Output: a cell population number -or growth rate- in tumour and healthy tissues

8. 0. Introduction: therapeutic optimisation? Optimising drug delivery:optimisation under constraints • Optimal control of delivery flow (programmable pumps) • Objective: minimising tumour cell number • Constraints: - limiting toxicity to healthy cells - avoiding drug resistance in cancer cells - taking into account individual genetic factors … and optimal circadian delivery times

9. 0. General modelling framework Mathematical modelling to optimise cancer treatments • … hence the choice of these 3 main research directions: • - Proliferation: cell population dynamics in tissues (PDEs) • - Drugs: molecular pharmacokinetics-pharmacodynamic (ODEs) • - Therapeutic optimisation: optimal control of drug delivery (optimal control algorithms) • …and future prospects: even more challenges for modelling!

10. 0. General modelling framework: cell / tissue proliferation Cyclin B M G2 G1 S Cyclin A At the origin of proliferation: the cell division cycle Mitosis=M phase S:=DNA synthesis; G1,G2:=Gap1,2; M:=mitosis Cyclin D (from Lodish et al., Molecular cell biology, 2003) • Physiological or therapeutic control exerted on: • transitions (checkpoints) between • phases (G1/S, G2/M, M/G1) • - death rates (apoptosis or necrosis) • - progression speeds inside phases • exchanges between quiescent (G0) • and proliferative phases (G1 only) Cyclin E

11. 0. General modelling framework: cell / tissue proliferation Modelling the cell division cycle in cell populationsPhysiologically structured PDEs ni:=cell population density in phase i ; vi :=progression speed; di:=death rate; (from B. Basse et al., J Math Biol 2003) In each phase i, a Von Foerster-McKendrick-like equation: Ki-1->i:=transition rate (with a factor 2 if i=1) di , Ki->i+1 constant or periodic w. r. to time t (1≤i≤I, I+1=1) Death rates di: (“loss”), “speeds” vi and phase transitions Ki->i+1are model targets for physiological (e.g. circadian) and therapeutic (drugs) control (t) (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)

12. 0. General modelling framework: cell / tissue proliferation Main result: a growth exponent for the cell population behaviour Proof of the existence ofa unique growth exponent ,the same for all phases i, such that the are asymptotically (i.e., for large times) bounded, and asymptotically periodic if the control is periodic Surfing on the exponential growth curve, example (periodic control case): 2 phases, control on G2/M transition by 24-h-periodic CDK1-Cyclin B (A. Goldbeter’s model) (Normalised cell population number) All cells, surfing on the exponential growth curve time t =CDK1All cells in G1-S-G2 (phasei=1)All cells in M (phase i=2) Entrainment of the cell division cycle by CDK1 at the circadian period

13. 0. General modelling framework: drugs Example: molecular pharmacodynamics (PD) of 5FU RNA way DNA way 2 main metabolic pathways: action on RNA and on DNA Competitive inhibition by FdUMP of dUMP binding to target TS + [Stabilisation by CH2-THF of binary complex dUMP-TS] Incorporation of FUTP instead of UTP to RNA Incorporation of FdUTP instead of dTTP to DNA (Longley, Nat Rev Canc 2003)

14. 0. General modelling framework: drugs Inhibition of ThymidylateSynthase (TS) by 5FU and Leucovorin Formyltetrahydrofolate (CHO-THF) = LV Precursor of CH2-THF, coenzyme of TS, that forms with it and FdUMP a stable ternary complex, blocking the normal biochemical reaction TS 5,10-CH2-THF + dUMP + FADH2 dTMP +THF + FAD (TS affinity: FdUMP > dUMP) (Longley, Nat Rev Canc 2003)

15. 0. General modelling framework: drugs ODEs: PK-PD of 5FU [+ drug resistance] + Leucovorin F. Lévi, A. Okyar, S. Dulong, JC, Annu Rev Pharm Toxicol 2010

16. 0. General modelling framework: circadian physiology for chronotherapeutic optimisation CNS, hormones, peptides, mediators Pineal NPV NPY Melatonin RHT Glutamate Arbitrary units Metabolism 11 23 7 23 7 Time (h) Proliferation Circadian clocks The circadian system… Central coordination Supra Chiasmatic Nuclei Entrainment by light Glucocorticoids Food intake rhythm Autonomic nervous system TGF, EGF Prokineticin Rest-activity cycle: open window on SCN central clock Peripheral oscillators Lévi, Lancet Oncol 2001 ; Mormont & Lévi, Cancer 2003

17. 0. General modelling framework: circadian physiology for chronotherapeutic optimisation …is an orchestra of cell clocks with one neuronal conductor in the SCN and molecular circadian clocks in all nucleated cells SCN=suprachiasmatic nuclei (in the hypothalamus) (Hastings, Nature Rev. Neurosci. 2003)

18. 0. General modelling framework: circadian physiology for chronotherapeutic optimisation In eachnucleatedcell: a molecularcircadianclock Inhibition loop Cellular rhythms Clock-controlled genes Metabolism 24 h-rhythmic transcription: 10% of genome, amongwhich: 10% : cell cycle 2% : growthfactors Proliferation Activation loop (after Hastings, Nature Rev. Neurosci. 2003)

19. 0. General modelling framework: actual (clinical) chronotherapeutic optimisation Time-scheduled delivery regimen for metastatic CRC 5-FU 600 - 1100 mg/m2/d Infusion over 5 days every 3rd week OxaliPt 25 mg/m2/d Folinic Acid 300 mg/m2/d 04:00 16:00. Time (local h) Circadian rhythms and cancer chronotherapeutics (Results from Francis Lévi’s INSERM team U 776, Villejuif, France) 3. Drug delivery optimisation Multichannel programmable ambulatory injector for intravenous drug infusion (pompe Mélodie, Aguettant, Lyon, France) Can such therapeutic schedules be improved? F. Lévi, INSERM U 776 Rythmes Biologiques et Cancers

20. 0. General modelling framework: actual (clinical) chronotherapeutic optimisation Results of cancer chronotherapy Metastatic colorectal cancer (FolinicAcid, 5-FU, Oxaliplatin) Infusion flow Constant Chrono Toxicity p Oral mucositis gr 3-4 74% 14% <10-4 Neuropathy gr 2-3 31% 16% <10-2 Responding rate 30% 51% <10-3 Lévi et al. JNCI 1994 ; Lancet 1997 ; Lancet Oncol 2001 How does it work? Impact of circadian clocks on 1) cell drug detoxication enzymes and 2) cell division cycle determinant proteins (cyclins/CDKs) INSERM U 776 Rythmes Biologiques et Cancers

21. 0. General modelling framework: actual (clinical) chronotherapeutic optimisation A working hypothesis that could explain observed 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 peripheral cells from central circadian clock control messages, just as tumour cells evade most physiological controls, cf. Hanahan & Weinberg:

22. 0. General modelling framework: theoretical (future?) chronotherapeutic optimisation under constraints Optimal control of anticancer chronopharmacotherapy Objective functionto be minimised: cell population growth rate or cell population density in tumour tissues Control function: instantaneous [dynamic] intravenous infusion = [multi-]drug delivery flow via external programmable pumps Constraintsto be satisfied: - maintaining healthy cell population over a tolerability threshold - taking into account circadian phases of drug processing systems (model prerequisite) - maintaining normal tissue synchronisation control by circadian clocks - limiting resistances in tumour cells (e.g. controlling induction of nrf2) - others: maximal daily dose, maximal delivery flow,… 4)With adaptationof the designed controlled system model (and hence of the optimised drug delivery flow) to patient-specific parameters: clock phases, enzyme genetic polymorphism, target protein levels, going towards personalised medicine

23. 0. General modelling framework: theoretical (future?) chronotherapeutic optimisation under constraints PK-PD simplified model for cancer chronotherapy (herewithonlytoxicityconstraints; target=death rate) Healthy cells (jejunal mucosa) Tumour cells (PK) (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, Pathol-Biol 2003; Adv Drug Deliv Rev 2007)

24. 0. General modelling framework: theoretical (future?) chronotherapeutic optimisation under constraints Optimal 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) (C. Basdevant, JC, F. Lévi, M2AN 2005; JC Adv Drug Deliv Rev 2007)

25. 0. General modelling framework: some future prospects Individualised treatments in oncology Genetic polymorphism: between-subject variabilityfor pharmacological model parameters • According to subjects, different expression and activity levels of • drug processing enzymes and proteins (uptake, degradation, active efflux, e.g. GST, DPYD, UGT1A1, P-gp,…) • and drug targets (e.g. Thymidylate Synthase, Topoisomerase I) • The same is true of DNA mismatch repair enzyme gene expression (e.g., ERCC1, ERCC2) • More generally, pharmacotherapeutics should be guided more by molecular 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 oncology in the future (see e.g. G. Milano & J. Robert in Oncologie 2005) • …Which leads, using actively searched-for biomarkers, to populational PK-PD

26. 0. General modelling framework: some future prospects Other frontiers in cancer therapeutics Immunotherapy: 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 system (Review by L. Zitvogel in Nature Rev. Immunol. 2008) 2. The various facets of (innate/acquired/(ir)reversible) drug resistance: - 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) - Microenvironment, interactions with stromal cells: competition/cooperativity models - Mutations of the targets: evolutionary game theory, evolutionary dynamics models 3.Developing non-cell-killing therapeutic means: - Associations of cytotoxics and redifferentiating agents (e.g. retinoic acid in AML3) - Modifying local metabolic parameters? (pH) to foster proliferation of healthy cells

27. Modelling the cell division cycle in proliferating cell populations

28. 1. Modelling the cell cycle in proliferating cell populations Why model the cell division cycle? • Need for detailed models of cell proliferation to represent the action of anticancer drugs in cell populations with: 1) Cell cycle phase specificity 2) Different pharmacological targets on cell cycle control 3) Action with same targets on tumour cells and on healthy cells (toxic side effects of anticancer drugs) • Hence, even independently of therapeutics, need for models with: 1) Cell cycle phases and age-in-phase, possibly cyclin, structure 2) Transitions between cell division cycle phases (G1/S, G2/M) 3) Exchanges between quiescent and proliferative phases (G0/G1) 4) Targets for control of cell proliferation (physiological / by drugs)

29. 1. Modelling the cell cycle in proliferating cell populations Cyclin B / CDK1 M G2 G1 S Cyclin A / CDK2 At the origin of proliferation: the cell division cycle in proliferating cell populations Mitosis=M phase S:=DNA synthesis; G1,G2:=Gap1,2; M:=mitosis (one cell divides in two) Cyclin D / CDK2 Mitotic human HeLa cell (from LBCMCP-Toulouse) • Physiological and therapeutic control exerted on: • transitions (checkpoints) between • phases (G1/S, G2/M, M/G1) • - death rates (apoptosis or necrosis) • and progression speeds inside phases • exchanges between quiescent (G0) • and proliferative phases (G1 only) Cyclin E / CDK 4or6

30. 1. Modelling the cell cycle in proliferating cell populations Age-structured PDE models Flow cytometry ni:=cell population density in phase i ; vi :=progression speed; di:=death rate; (from B. Basse et al., J Math Biol 2003) In each phase i, a Von Foerster-McKendrick-like linear model: Ki-1->i:=transition rate (with a factor 2for i=1) di , Ki->i+1 constant or periodic w. r. to time t (1≤i≤I, I+1=1) Death rates di: (“loss”), “speeds” vi and phase transitions Ki->i+1are model targets for physiological (e.g., circadian) or therapeutic (drug) control (t) (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)

31. 1. Modelling the cell cycle in proliferating cell populations 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  and an explicitformula can be found for  when K0+∞(T. Lepoutre)

32. 1. Modelling the cell cycle in proliferating cell populations General case: I phases (last = mitosis, or M phase) (Note that exchanges between G0 and G1 are not considered in this linear model, i.e., all cells are assumed to proliferate) Then, provided that reasonable assumptions on death and transition rates are satisfied: (thusensuring positive growth), one canestablishthat:

33. 1. Modelling the cell cycle in proliferating cell populations According to the Krein-Rutman theorem (infinite-dimensional form of the Perron-Frobenius theorem), there exists a nonnegative first eigenvalue such that, if , then there exist Ni, bounded solutions to the problem: with a numbersuchthat for all i: . (the weightsi ≥0 being solutions to the dual problem); this can be proved by using a generalised entropy principle (GRE). Moreover, if the control (diorKi->i+1) is periodic, so are the eigenvectors Niand weights i, with the same period. Ph. 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 2005

34. 1. Modelling the cell cycle in proliferating cell populations : a growth exponent governing the cell population behaviour 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 of control (periodic control case): 2 phases, control on G2/M transition by 24-h-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) time t =CDK1All cells in G1-S-G2 (phasei=1)All cells in M (phase i=2) Entrainment of the cell division cycle by = CDK1 at the circadian period

35. 1. Modelling the cell cycle in proliferating cell populations Details (1): 2-phase model, no control on transition The total population of cells insideeach phase follows asymptotically an exponential behaviour Stationary (=asymptotic) state distribution of cells inside phases according to age a: no control -> exponential decay

36. 1. Modelling the cell cycle in proliferating cell populations Details (2): 2 phases, periodic control on G2/M transition The total population of cells inside each phase follows asymptotically an exponential behaviour tuned by a periodic function Stationary (=asymptotic) state distribution of cells inside phases according to age a: sharp periodic control ->sharp rise and decay

37. 1. Modelling the cell cycle in proliferating cell populations Desynchronisation of cells within populations w. r. to cell cycle timing = phase overlapping at transition: Possible experimental measurements to identify transition kernel K Starting from the simplest model with d=0 (one phase with division): whence Interpreted as: if tis the age at division, or transition: (remark by Th. Lepoutre) with where the probability density (experimentally identifiable) is: i.e.,

38. 1. Modelling the cell cycle in proliferating cell populations Modelling cell proliferation and quiescence

39. 1. Modelling the cell cycle in proliferating cell populations Nonlinear models: introducing exchanges between proliferating (G1/S/G2/M) and quiescent (G0) cells after R: mitogen-independent progression through G1 to S (no way back to G0) Restriction point (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 proliferatephysiologically, even in fastrenewing tissues (e.g. gut) Exchanges between proliferating (G1/S/G2/M) and quiescent (G0) cell compartments are controlled by mitogens and antimitogenic factors in G1 phase

40. 1. Modelling the cell cycle in proliferating cell populations ODE modelswithtwoexchangingcellcompartments, 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 justify Gompertz growth (a popular model among radiologists)

41. 1. Modelling the cell cycle in proliferating cell populations Simple PDE models, age-structuredwith exchanges betweenproliferation 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

42. 1. Modelling the cell cycle in proliferating cell populations Delay differentialmodels with two cell compartments, proliferating (P)/quiescent (Q): Haematopoiesis models (obtained from the previous model with additional hypotheses and integration in x along characteristics) (delay  cell division cycle duration) (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)

43. 1. Modelling the cell cycle in proliferating cell populations Modelling haematopoiesis for Acute 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: (see Adimy et al. JBS 2008 for more details)

44. 1. Modelling the cell cycle in proliferating cell populations 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

45. 1. Modelling the cell cycle in proliferating cell populations Next step: integrating the two models (Von Foerster-McKendrick-like linear and nonlinear proliferation/quiescence) in a complete cell cycle model with phases G0-G1-S-G2-M Keeping the same control targets, adding control by growth factors, hence on cyclin D, on the recruitment function G from G0 to G1 and on progression speed in G1 …Work in progress…