Numerical Solution of an Inverse Problem in Size-Structured Population Dynamics Marie Doumic (projet...
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![Page 1: Numerical Solution of an Inverse Problem in Size-Structured Population Dynamics Marie Doumic (projet BANG – INRIA & ENS) Torino – May 19th, 2008 with B.](https://reader035.fdocuments.in/reader035/viewer/2022070306/55164519550346b2068b54bd/html5/thumbnails/1.jpg)
Numerical Solution of an Inverse Problemin Size-Structured Population Dynamics
Marie Doumic (projet BANG – INRIA & ENS)
Torino – May 19th, 2008
with B. PERTHAME (L. J-L. Lions-Paris) J. ZUBELLI (IMPA-Rio de Janeiro)
and Pedro MAIA (UFRJ - master student)
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Marie DOUMIC - CancerSim2008 May ,19th
Outline
Structured Population ModelsMotivation to structured population modelsThe model under consideration
The Inverse ProblemRegularisation by quasi-reversibility methodRegularisation by filtering approach
Numerical SolutionChoice of a convenient schemeSome results and application to experimental data
Conclusion and perspectives
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Marie DOUMIC - CancerSim2008 May ,19th
Structured Populations
Recent Réf: B. Perthame, Transport Equations in Biology, Birkhäuser (2006)
Population density Examples of x:
Unicellular organisms: the mass of the cell DNA content of the cell Cell age (age-structured populations) Protein content: cyclin, cyclin-dependent kinases,
complexes…
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From B. Basse et al, Modelling the flow of cytometric data obtained from unperturbed human tumour cell lines: parameter fitting and comparison,B. of Math. Bio., 2005
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Cell volumes distribution for E. Coli THU in a glucose minimal medium at a doubling time of 2 hrs. H.E. Kubitschek, Biophysical J. 9:792-809 (1969)
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Marie DOUMIC - CancerSim2008 May ,19th
Our Structured Population Model: Model of Cell Division under Mitosis
Density of cells:
Size of the cell:
Birth rate:
1 cell of size gives birth to 2 cells of size
The growth of the cell size by nutrient uptake is given by a rate g(x):
here for the sake of simplicity, g(x)≡1.
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Model obtained by a mass conservation law:
Applications of this model (or adaptations of this model): cell division cycle, prion replication, fragmentation equation…
Our Structured Population Model: Model of Cell Division under Mitosis
Density of the cells
Growth by nutrient
Division of cells of size x
Division of cells of size 2x
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Marie DOUMIC - CancerSim2008 May ,19th
(Some) Related Works
J.A.J Metz and O. Diekmann,
Physiologically Structured Models (1986)
Engl, Rundell, Scherzer,
Regularisation Scheme for an Inverse Problem in Age-Structured Populations (1994)
Gyllenberg, Osipov, Päivärinta,
The Inverse Problem of Linear Age-Structured Population Dynamics (2002)
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The Question: find the birth rate B
What is really observed ? Recall the figures:
We do not observe B ; not even n(t,x):
but a DOUBLING TIME and a STEADY PROFILE N(x)
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Marie DOUMIC - CancerSim2008 May ,19th
Our approach: Use the stable size distribution
This uses recent results on Generalised Relative Entropy: refer for instance to
B. Perthame, L. Ryzhik, Exponential Decay for the fragmentation or cell-division Equation J. Diff. Equ. (2005)
P. Michel, S. Mischler, B. Perthame, General Relative Entropy for Structured Population Models and Scattering, C.R. Math. Acad. Sci. Paris (2004)
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Dynamics of this equationIf you look at solutions n(t,x) under the form n(t,x)=eλtN(x):
Theorem (above ref.): There is a unique solution for a unique of:
And under fairly general conditions we have (in weighted Lp topologies) :
(Assumptions on B(x) quite general – exponential decay can also been proved)
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Marie DOUMIC - CancerSim2008 May ,19th
Dynamics of the equation
2 Major & fundamental & useful relations:
1. Integration of the equation:
Interpretation: number of cells increases by division
2. Integration of the equation multiplied by x:
Interpretation: biomass increases by nutrient uptake
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Marie DOUMIC - CancerSim2008 May ,19th
1st question on our inverse problem:is it well-posed ?
Hadamard’s definition of well-posedness:1- For all admissible data, a solution exists2- For all admissible data, the solution is unique3- The solution depends continuously on the data.
Our problem becomes: Knowing N ≥ 0, with N(x=0)=0, find B such that :
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An ill-posed problem
Now N is the parameter, B the unknown. The equation can be written as:
With .
If N is regular, e.g. if L is in L2 : H=BN is in L2 (see prop. below) what we really know is not but a noisy data with
L is not in L2
B is not well-defined in L2
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How to regularize this problem: « quasi-reversibility » method
1st method (in Perthame-Zubelli, Inverse Problems (2006)):
Add a (small) derivative for B: we obtain the following well-posed problem:
Theorem (Perthame-Zubelli): we have the error estimate:
is optimal
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How to regularize this problem – Filtering approach
2nd method: regularize in order to make L be regular:
With and .
Theorem (D-Perthame-Zubelli): we have the error estimate:
is optimal
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Generic form of the problem for any regularization:
With 1.Naive method:
2. « Quasi-reversibility » method:
3. Filtering method:
Numerical Implementation
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Several requirements:A. Avoid instabilityB. Conserve main properties of the continuous model:
laws for the increase - of biomass:
- of number of cells, e.g. for the quasi-reversibility method:
C. 1 question: begin from the left, deducing B(2x) from B(x)or from the right, deducing B(x) from B(2x) ?
Numerical Implementation - strategy
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Numerical Implementation:a result to choose the right scheme
3
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Marie DOUMIC - CancerSim2008 May ,19th
Numerical Implementation:choice of the scheme
H0: deduce H(x) from larger x scheme departs from infinity
H1: deduce H(x) from smaller x scheme departs from 0.
H1 « more regular » choice: scheme departing from 0.
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Numerical Scheme – filtering approach
-> departs from zero (mimics H1)
-> mass and number of cells balance laws preserved:
-> stability: 4H(2x) is approximated by 4 H2i
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Numerical Scheme – Quasi-Reversibility
-> departs from zero (mimics H1)
-> mass and number of cells balance laws preserved:
-> stability: 4H(2x) is approximated by 4 H2i
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Marie DOUMIC - CancerSim2008 May ,19th
Numerical Scheme: steps1. solve the direct problem for a given B(x)
Method: use of the exponential convergence of n(t,x) to N(x): Finite volume scheme to solve the time-dependent problem:
Then renormalization at each time-step
2. Add a noise to N(x) to get a noisy data Nε(x)
1. Run the numerical scheme for the inverse problemto get a birth rate Bε,α (x) Nε(x) and compare it with the initial data B(x) – look for the best α for a given error ε
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First step: direct problem
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Results with ε=0: no noise for the entry data
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Marie DOUMIC - CancerSim2008 May ,19th
Є=0, α=0.01Results with ε=0: no noise for the entry data
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Marie DOUMIC - CancerSim2008 May ,19th
Results with ε=0: no noise for the entry data
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Marie DOUMIC - CancerSim2008 May ,19th
Results with ε=0: no noise for the entry data
Measures of error for the different methods
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Marie DOUMIC - CancerSim2008 May ,19th
Results with ε=0.01 and 0.1: measure of error
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Marie DOUMIC - CancerSim2008 May ,19th
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Marie DOUMIC - CancerSim2008 May ,19th
Application to experimental data – work of Pedro MAIA
Main difficulty: find data to which the equation can be applied !
- No death (in vitro experiment) or constant death rate
- Symetrical division
- Known growth speed (assumption is needed)
Bacteria: E. Coli
Reference: H.E. Kubitschek, Growth during the bacterial cell cycle: analysis of cell size distribution, Biophys. J., (1969)
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Application to Kubitschek’s data
4 kinds of growth environment for E. Coli:
Den
sity
N(x
)
Relative size (x=2: mean doubling size)
First curve:
Doubling time = 20 mns
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Application to Kubitschek’s data
4 kinds of growth environment for E. Coli:
Den
sity
N(x
)
Relative size (x=2: mean doubling size)
Second curve:
Doubling time = 54 mns
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Application to Kubitschek’s data
4 kinds of growth environment for E. Coli:
Den
sity
N(x
)
Relative size (x=2: mean doubling size)
Third curve:
Doubling time = 120 mns
![Page 36: Numerical Solution of an Inverse Problem in Size-Structured Population Dynamics Marie Doumic (projet BANG – INRIA & ENS) Torino – May 19th, 2008 with B.](https://reader035.fdocuments.in/reader035/viewer/2022070306/55164519550346b2068b54bd/html5/thumbnails/36.jpg)
Application to Kubitschek’s data
4 kinds of growth environment for E. Coli:
Den
sity
N(x
)
Relative size (x=2: mean doubling size)
Fourth curve:
Doubling time = 720 mns
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Application to Kubitschek’s data
Assumption: Linear growth, constant growth speed calculated from the knowledge of the doubling time.
B(x
)N(x
)
Relative size (x=2: mean doubling size)
α= 0.2, 2 different methods, doubling time=20 mns
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Application to Kubitschek’s data
Assumption: Linear growth, constant growth speed calculated from the knowledge of the doubling time.
B(x
)
Relative size (x=2: mean doubling size)
α= 0.2, 2 different methods, doubling time=20 mns
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Marie DOUMIC - CancerSim2008 May ,19th
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Marie DOUMIC - CancerSim2008 May ,19th
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Marie DOUMIC - CancerSim2008 May ,19th
Doubling time: 120 minutes
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Marie DOUMIC - CancerSim2008 May ,19th
Doubling time: 720 minutes
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-> A birth pattern ?
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Marie DOUMIC - CancerSim2008 May ,19th
Perspectives-> Compare with the assumption of exponential growth (work
with Pedro Maia) ; precise what is the noise-> Adaptation to recent data on plankton (M. Felipe)-> apply the method to adaptations of this model: e.g. for non-
symetrical division equation, for a cyclin-structured model (cf. work of F. Bekkal-Brikci, J. Clairambault, B. Perthame, B. Ribba), for tumor growth (early stage),…
-> improve the method: compare it with other regularizations like Tikhonov understand why the combination of both methods seems better recover B globally, even where N vanishes, by getting a priori information on B.
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Marie DOUMIC - CancerSim2008 May ,19th
Grazie mille per la Sua attenzione !