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A Population Model Structured by Age and Molecular Content of the Cells

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

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A Population Model Structured by Age and Molecular Content of the Cells

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

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

  3. Introduction: Models of population growth 1. Historical models of population growth Malthus parameter: Exponential growth Logistic growth (Verhulst): -> 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. Presentation of our Model: • an Age and Molecular-Content Structured Model for the Cell Cycle • A. Two Compartments Model d2 d1 B L P Q G Proliferating cells Quiescent cells 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 Proliferating cells =1 DIVISION (=birth) RATE quiescent cells Death rate Death rate Recruitment with N(t) =« total population » Demobilisation 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 2007.

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

  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 of our study and steps of the work • 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)=eλt P(a,x), q(t,a,x)=eλt Q(a,x) Eigenvalue linearised problem

  11. Goal of our study and steps of the work • 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

  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 • 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 x Γ<0 Γ=0 Γ>0 a Ass. 1: xM Ass. 2:Γ(a,0)=0 or N(a,0)=0

  16. II. Study of the Eigenvalue Linearised Problem Question to solve: Exists a unique (λ0, 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. II. Resolution of the Eigenvalue Problem B. Method of characteristics N=0 x XM Γ<0 Γ=0 Assumption: Γ>0 X0 a

  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. II.B.Resolution of the Eigenvalue Problem – Method of Characteristics Step 2: study of the operator : With For ε>0 and λ>0, is positive and compact on C(0,xM) Apply Krein-Rutman theorem (=Perron-Frobenius in inf. dim.): Lemma: there exists a unique Nλ,ε0 >0, s.t. Moreover, for λ=0, =2 and for λ= , =0 and is a continuous decreasing function.

  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 L1 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 λ0>0 (lost when ε 0)

  24. II.B.Resolution of the Eigenvalue Problem – Method of Characteristics Theorem:under the preceding  assumption (+ some other more technical…), there exists a unique λ0>0 and a unique solution N, withfor all , of the problem:

  25. II.B.Resolution of the Eigenvalue Problem – Method of Characteristics 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.

  26. II.B.Resolution of the Eigenvalue Problem – Method of Characteristics 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.

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

  28. II. Asymptotic Behaviour of the Time-Dependent ProblemB. 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 λ0 >0 of the 1-equation model:

  29. II. Asymptotic Behaviour of the Time-Dependent ProblemB. Back to the 2 compartments problem Since G=G(N(t)) we have p=Peλ[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 2007.

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

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

  32. III.B. Asymptotic Behaviour – Two Compartment Problem c) Robust subpolynomial growth Recall : link between λand λ0: If d2=0 and α2=0 in the formula we can obtain (H4) and unlimited subpolynomial growth in a « robust »way: 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…

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