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

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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

• Marie Doumic Jauffret
• doumic_at_dma.ens.fr
• Work with Jean CLAIRAMBAULT and Benoît PERTHAME

30th, August 2007
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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

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Introduction Models of population growth
1. Historical models of population growth
Malthus parameter Exponential growth
Logistic growth (Verhulst)
-gt various ways to complexify this equation Cf.
B. Perthame, Transport Equations in Biology,
Birkhäuser 2006.
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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.
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• 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
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I.A. Presentation of our model 2 compartments
model a) 2 equations proliferating and quiescent
Proliferating cells
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DIVISION (birth) RATE
quiescent cells
Death rate
Recruitment with N(t)  total population 
Death rate
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.
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I.A. Presentation of our model 2 compartments
model b) Initial conditions for t0 and a0
Initial conditions at t0 Birth condition for
a0 with
daughter
mother
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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

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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

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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 ? such that
  there exists a solution of type
• p(t,a,x)e?t P(a,x), q(t,a,x)e?t Q(a,x)

Eigenvalue linearised problem
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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

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I. Presentation of our model B. Eigenvalue
Linearised Model Non-linearity G(N(t))
simplified in
Simplified in
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• I.B. Presentation of our model Eigenvalue
  Linearised Problem
• a) Link with other models
• If GG(a) and BB(a) independent of x
• Integration in x gives for
  

Linear McKendrick Von Foerster equation
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• I.B. Presentation of our model Eigenvalue
  Linearised Problem
• Link with other models
• If GG(x)gt0 and BB(x) independent of age a
• Integration in a gives for
  
• Cf. works by P. Michel, B. Perthame, L. Ryzhik,
  J. Zubelli

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I.B. Presentation of our model Eigenvalue
Linearised Problem b) Form of G
Ass. 1
xM
Ass. 2 G(a,0)0 or N(a,0)0
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II. Study of the Eigenvalue Linearised
Problem Question to solve Exists a unique
(?0, N) solution ? A.Estimates a) Conservation
of the number of cells integrating the
equation in a and x gives
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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
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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
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II. Resolution of the Eigenvalue Problem B.
Method of characteristics
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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 a0
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II.B.Resolution of the Eigenvalue Problem
Method of Characteristics

• Step 2 study of the operator
• With
• For egt0 and ?gt0, is positive and compact on
  C(0,xM)
• Apply Krein-Rutman theorem (Perron-Frobenius in
  inf. dim.)
• Lemma there exists a unique N?,e0 gt0,
  s.t.
• Moreover, for ?0, 2 and for ? ,
  0 and
• is a continuous decreasing
  function.

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• we choose the unique ? s.t. 1.
• Following steps
• Step 3. Passage to the limit when e tends to zero
• Step 4. N(a,x) is given by N(a0,x) by the
  formula of characteristics and must be in L1
• Key assumption
• Which can also be formulated as

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• Following steps
• Step 5. Resolution of the adjoint problem
• (Fredholm alternative)
• Step 6. Proof of unicity and of ?0gt0 (lost when e
  0)

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II.B.Resolution of the Eigenvalue Problem
Method of Characteristics
Theorem under the preceding  assumption (
some other more technical), there exists a
unique ?0gt0 and a unique solution N, with
for all , of the problem
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II.B.Resolution of the Eigenvalue Problem
Method of Characteristics

• Some remarks
• B(a,x0)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.

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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.

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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

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II. Asymptotic Behaviour of the Time-Dependent
ProblemB. Back to the 2 compartments eigenvalue
problem
Theorem. For L constant there exists a unique
solution (?, P, Q) and we have the following
relation between ? and the eigenvalue ?0 gt0 of
the 1-equation model
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II. Asymptotic Behaviour of the Time-Dependent
ProblemB. Back to the 2 compartments problem
Since GG(N(t)) we have pPe?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.
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III.B. Asymptotic Behaviour Two Compartment
Problem
 Pe?G(N(t)) .t

• a) Healthy tissues
• (H1) for we have ??G0 gt0
• non-extinction
• (H2) for we have ??lim lt0
• no blow-up
• convergence towards a steady state ?

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III.B. Asymptotic Behaviour Two Compartment
Problem
Pe?G(N(t)).t

• b) Tumour growth
• (H3) for we have ??inf gt0
• unlimited exponential growth
• (H4) for we have ??inf 0
• subpolynomial growth (not robust)

Exponential growth, Log scale
Polynomial growth, Log-Log scale
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III.B. Asymptotic Behaviour Two Compartment
Problem c) Robust subpolynomial growth

• Recall link between ? and ?0
• If d20 and a20 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)
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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