1 A Population Model Structured by Age and Molecular Content of the Cells Marie Doumic Jauffret Work with Jean CLAIRAMBAULT and Benoît PERTHAME Workshop on mathematical methods and modeling of biophysical phenomena – IMPA - Rio de Janeiro, Brazil 30th, August 2007

2 Outline Introduction: models of population growth I. Presentation of our model: A. Biological motivation B. Simplification & link with other models II. Resolution of the eigenvalue problem A. A priori estimates B. Existence and unicity III. Asymptotic behaviour

3 Introduction: Models of population growth Malthus parameter: Exponential growth Logistic growth (Verhulst): 1. Historical models of population growth -> various ways to complexify this equation: Cf. B. Perthame, Transport Equations in Biology, Birkhäuser 2006.

4 Introduction: Models of population growth 2. The age variable McKendrick-Von Foerster equation: Birth rate (division rate) P. Michel, General Relative Entropy in a Non Linear McKendrick Model, AMS proceeding, 2006.

5 I.Presentation of our Model: an Age and Molecular-Content Structured Model for the Cell Cycle A. Two Compartments Model P Q Proliferating cellsQuiescent cells L G d1d1 d2d2 B 3 variables: time t, age a, cyclin-content x

6 I.A. Presentation of our model – 2 compartments model a) 2 equations : proliferating and quiescent Cf. F. Bekkal-Brikci, J. Clairambault, B. Perthame, Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle, Math. And Comp. Modelling, available on line, july quiescent cells Proliferating cells=1 Demobilisation DIVISION (=birth) RATE Death rate Recruitment with N(t) =« total population » Death rate

7 I.A. Presentation of our model – 2 compartments model b) Initial conditions: for t=0 and a=0 Initial conditions at t=0: Birth condition for a=0: with daughter mother

8 Conservation of the number of cells: Conservation of the cyclin-content of the mother: shared in 2 daughter cells: I.A. Presentation of our model – 2 compartments model c) Properties of the birth rates b and B

9 I.A. Presentation of our model – 2 compartments model c) Properties of the birth rates b and B Examples: - Uniform division: - Equal division in 2 daughter cells:

10 Goal: find out the asymptotic behaviour of the model : Way to do it: Look for a « Malthus parameter » λ such that there exists a solution of type p(t,a,x)=e λt P(a,x),q(t,a,x)=e λt Q(a,x) Goal of our study and steps of the work Eigenvalue linearised problem

11 Goal: find out the asymptotic behaviour of the model : the « Malthus parameter » resolution of the eigenvalue linearised problem part II: A. a priori estimates B. Existence and unicity theorems Back to the time-dependent problem part III: A. General Relative Entropy Method Cf. Michel P., Mischler S., Perthame B., General relative entropy inequality: an illustration on growth models, J. Math. Pur. Appl. (2005). B. Back to the non-linear problem C. Numerical validation Goal of our study and steps of the work

12 I. Presentation of our model B. Eigenvalue Linearised Model Non-linearity : G(N(t)) simplified in : Simplified in:

13 I.B. Presentation of our model – Eigenvalue Linearised Problem a) Link with other models If Γ=Γ(a) and B=B(a) independent of x Integration in x gives for : = Linear McKendrick – Von Foerster equation

14 I.B. Presentation of our model – Eigenvalue Linearised Problem a) Link with other models If Γ=Γ(x)>0 and B=B(x) independent of age a Integration in a gives for : Cf. works by P. Michel, B. Perthame, L. Ryzhik, J. Zubelli…

15 I.B. Presentation of our model – Eigenvalue Linearised Problem b) Form of Γ a x Γ=0 Γ<0 Γ>0 Ass. 2: Γ(a,0)=0 or N(a,0)=0 Ass. 1: xMxM

16 II. Study of the Eigenvalue Linearised Problem Question to solve: Exists a unique (λ 0, N) solution ? A.Estimates – a) Conservation of the number of cells : integrating the equation in a and x gives:

17 II.A. Study of the Eigenvalue Linearised Problem - Estimates b) Conservation of the cyclin-content of the mother: integrating the equation multiplied by x gives:

18 II.A. Study of the Eigenvalue Linearised Problem - Estimates c ) Limitation of growth according to age a Integrating the equation multiplied by a gives: multiplying by and integrating we find:

19 a x XMXM X0X0 II. Resolution of the Eigenvalue Problem B. Method of characteristics Γ=0 Γ<0 Γ>0 Assumption: N=0

20 II.B.Resolution of the Eigenvalue Problem – Method of Characteristics Step 1: Reformulation of the problem (b continuous in x) Formula of characteristics gives: Introducing this formula in the boundary condition a=0:

21 Step 2: study of the operator : With For ε>0 and λ>0, is positive and compact on C (0,x M ) Apply Krein-Rutman theorem (=Perron-Frobenius in inf. dim.) : Lemma: there exists a unique N λ,ε 0 >0, s.t. Moreover, for λ=0, =2 and for λ=, =0 and is a continuous decreasing function. II.B.Resolution of the Eigenvalue Problem – Method of Characteristics

22 we choose the unique λ s.t. =1. Following steps : Step 3. Passage to the limit when ε tends to zero Step 4. N(a,x) is given by N(a=0,x) by the formula of characteristics and must be in L 1 Key assumption: Which can also be formulated as :

23 Following steps Step 5. Resolution of the adjoint problem (Fredholm alternative) Step 6. Proof of unicity and of λ 0 >0 (lost when ε 0)

24 Theorem: under the preceding assumption (+ some other more technical…), there exists a unique λ 0 >0 and a unique solution N, with for all, of the problem: II.B.Resolution of the Eigenvalue Problem – Method of Characteristics

25 Some remarks -B(a,x=0)=0 makes unicity more difficult to prove: supplementary assumptions on b and B are necessary. -The result generalizes easily to the case x in : possibility to model various phenomena influencing the cell cycle: different proteins, DNA content, size… - The proof can be used to solve the cases of pure age- structured or pure size-structured models. II.B.Resolution of the Eigenvalue Problem – Method of Characteristics

26 Some remarks -The preceding theorem is only for b(a,x,y) continuous in x. e.g. in the important case of equal mitosis: the proof has to be adapted : reformulation gives: compacity is more difficult to obtain but the main steps remain. II.B.Resolution of the Eigenvalue Problem – Method of Characteristics

27 Theorem: Under the same assumptions than for existence and unicity in the eigenvalue problem, we have III. Asymptotic behaviour of the time- dependent problem A. Linearised problem: based on the « General Relative Entropy » principle

28 II. Asymptotic Behaviour of the Time-Dependent Problem B. Back to the 2 compartments eigenvalue problem Theorem. For L constant there exists a unique solution (λ, P, Q) and we have the following relation between λ and the eigenvalue λ 0 >0 of the 1-equation model:

29 Since G=G(N(t)) we have p=Pe λ[G(N(t))].t Study of the linearised problem in different values of G(N) F. Bekkal-Brikci, J. Clairambault, B. Perthame, Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle, Math. And Comp. Modelling, available on line, july II. Asymptotic Behaviour of the Time-Dependent Problem B. Back to the 2 compartments problem

30 III.B. Asymptotic Behaviour – Two Compartment Problem a) Healthy tissues: (H1) forwe have λ=λ G=0 >0 non-extinction (H2) for we have λ=λ lim <0 no blow-up ; convergence towards a steady state ? P=e λ[G(N(t))].t

31 b) Tumour growth: (H3) for we have λ=λ inf >0 unlimited exponential growth (H4) for we have λ=λ inf =0 subpolynomial growth (not robust) P=e λ[G(N(t))].t III.B. Asymptotic Behaviour – Two Compartment Problem Polynomial growth, Log-Log scale Exponential growth, Log scale

32 Recall : link between λ and λ 0 : If d 2 =0 and α 2 =0 in the formula we can obtain (H4) and unlimited subpolynomial growth in a « robust »way: III.B. Asymptotic Behaviour – Two Compartment Problem c) Robust subpolynomial growth Robust polynomial growth, Log scale (not affected by small changes in the coefficients)

33 Perspectives -compare the model with data and study the inverse problem… cf. B. Perthame and J. Zubelli, On the Inverse Problem for a Size-Structured Population Model, IOP Publishing (2007). -Use and adapt the method to similar models: e.g. to model leukaemia, genetic mutations, several phases models…