Transcript Document

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
[email protected]
Work with Jean CLAIRAMBAULT and Benoît PERTHAME
30th, August 2007
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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
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Introduction: Models of population growth
1. Historical models of population growth
Malthus parameter: Logistic growth (Verhulst):
Exponential growth
-> various ways to complexify this equation:
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Cf. B. Perthame, Transport Equations in Biology, Birkhäuser 2006.
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
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Model, AMS proceeding, 2006.
I. Presentation of our Model:
an Age and Molecular-Content Structured Model
for the Cell Cycle
A. Two Compartments Model
d1
B
P
L
d2
Q
G
Proliferating cells
Quiescent cells
3 variables: time t, age a, cyclin-content x 5
I.A. Presentation of our model – 2 compartments model
a) 2 equations : proliferating and quiescent
Proliferating cells
quiescent cells
Demobilisation
=1
DIVISION (=birth) RATE
Death rate
Cf. F. Bekkal-Brikci, J. Clairambault, B. Perthame,
Death rate
Recruitment with N(t)
=« total population »
Analysis of a molecular structured population model with possible polynomial growth
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for the cell division cycle, Math. And Comp. Modelling, available on line, july 2007.
I.A. Presentation of our model – 2 compartments model
b) Initial conditions: for t=0 and a=0
Initial conditions at t=0:
daughter
Birth condition for a=0:
mother
with
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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 » 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
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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 Γ=Γ(a) and B=B(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
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…
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I.B. Presentation of our model – Eigenvalue Linearised Problem
b) Form of Γ
x
xM
Ass. 1:
Γ<0
Γ=0
Γ>0
Ass. 2: Γ(a,0)=0 or
N(a,0)=0
a
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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:
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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
N=0
x
XM
Γ<0
Γ=0
Γ>0
Assumption:
X0
a
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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 a=0:
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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.
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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 :
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Following steps
Step 5. Resolution of the adjoint problem
(Fredholm alternative)
Step 6. Proof of unicity and of λ0>0 (lost when ε
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 λ0>0 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,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 agestructured 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 timedependent 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 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 λ0 >0 of the 1-equation model:
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II. Asymptotic Behaviour of the Time-Dependent Problem
B. 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. 29
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 ?
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III.B. Asymptotic Behaviour – Two Compartment Problem
P=eλ[G(N(t))].t
b) Tumour growth:
(H3) for
(H4) for
we have λ=λinf >0
unlimited exponential growth
we have λ=λinf =0
subpolynomial growth (not robust)
Polynomial growth,
Log-Log scale
Exponential
growth, Log scale
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III.B. Asymptotic Behaviour – Two Compartment Problem
c) Robust subpolynomial growth
Recall : link between λ and 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)
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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…
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