Diapositiva 1

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Identification of parameters in tumour growth
models based on empirical eigenfunctions
Damiano Lombardi
In collaboration with
Angelo Iollo
Thierry Colin
Olivier Saut
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Overview
Introduction
A simplified 2-D Darcy-type model
Empirical eigenfunction basis obtained by
POD-Sirovich method
Inverse problems direct approach and hybrid
regularized approach
Results
Conclusions and perspectives.
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Introduction
Tumour growth is a complex phenomenon involving
a very large number of degree of freedom, so that
a direct approach based on first principles is
not feasible.
Tumour growth models are parametric models
Due to the phenomenological nature of these
models the parameters are not known and they
often can not be measured
We need to perform inverse problems in order to
recover the unknown parameters from the data
coming from medical analysis.
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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A simplified PDE model
Simplified 2-D Darcy-type model, with 2 cellular
species
Balance equations
Constitutive equations
Boundary conditions
We can find non-dimensional similarity parameters.
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Empirical eigenfunction basis obtained by
POD-Sirovich method (1)
In order to extract coherent structures that
capture the information of our system we have to
build a data base of direct numerical simulations
(DNS).
We have varied the non-dimensional ratios
describing the behaviour of cancerous cells with
respect to healthy cells.
Because of the instrinsic different physical
nature of the variables, we have decided to build
an eigenmode set for each variable.
In these first cases we have decided to keep the
geometry of blood vessels fixed, with a single
blood vessel (on the left of the tumour).
Reconstruction of the proliferating cells density
with 20 modes
Direct numerical simulation of a Darcy-2D
simplified flow dynamic of the proliferating
cells density
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Empirical eigenfunction basis obtained by
POD-Sirovich method (2)
Analysis of the approximation properties
L2 norm of proliferating cells residue
L2 norm of pressure residue
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problems (1)
We have performed several inverse problems in
order to evaluate the minimum quantity of
information that allow us to identify a complex
fluid system.
Furthermore we have tried to identify complex
systems with more simple ones, trying to
reproduce what we would do in real applications.
We have solved 4 different inverse problems,
increasing complexity
1) Identification of porosity ratio (k2/k1) given
P, Q, hypoxia function (?)
2) Identification of porosity ratio (k2/k1) and
hypoxia function (?), supposed to be constant,
given P,Q
3) Identification of porosity ratio (k2/k1) and
hypoxia function (?), given PQ only
4) Identification of a more complex Stokes-type
flow with a Darcy-type flow, given PQ only.
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem III only PQ is given
3) Identification of porosity ratio (k2/k1) and
hypoxia function (?), given PQ only
We make the following assumption
In inverse problems I and II, this assumption
allows us to derive all the quantities we need
from the PDE system, directly, using least square
and Tikhonov regularisation method.
On the contrary, in inverse problems III and IV
no approach based on the PDE equations is
possible since the resulting system would be
under-determined, infinite solutions would be
possible.
We introduce a new hybrid method, based on POD
eigenfuncitons we add information to our system
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem III only PQ is given
The system of balance equations we have to deal
with become
Curl of Darcy-law
We write the velocity field as expansion of POD
modes and substitute it into our system
We get the velocity field from the transport
equations and then we try to identify porosity
using the curl of the Darcy-type law
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem III only PQ is given
We solve for the porosity field using a least
square method with a Tikhonov regularisation.
We can derive our functional setting
We can make simulations varying several
parameters
Porosity ratio
Time of the avascular growth at which we take two
subsequent snapshots
Number of POD eigenfunctions we use.
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem III results
Errors in velocity reconstruction
Kinetic energy reconstructed using Nv30
eigenmodes
Kinetic energy of direct numerical simulations
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem III results
Analysis of porosity error when we vary all the
simulation parameters
Good identification of near potential flows, at
T20
Good identification with N30 POD modes
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem III results
Analysis of the error of the mitosis function
Good identification for Tlt20
Good results with N50 POD modes
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem IV
4) Identification of a more complex Stokes-type
flow with a Darcy-type flow, given PQ only.
We identify porosity, hypoxia functions,
proliferating cells density, diffusivity and
oxygen consuption as if the snapshots were part
of a Darcy-type flow evolution.
We introduce the parameters in the DNS code, with
the reference snapshot as initial condition, and
we evaluate the differences between Stokes and
Darcy.
While in inverse problem II is possible to
regularize the simple logistic model using
eigenfunctions of Darcy-type model, in this case
the procedure is more realistic, since we
regularize our fields using eigenfunctions
obtained by POD method applied to the more simple
model.
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem IV the complete identification
We have Y and we identify the mitosis function
We want to use these informations to identify all
the remaining parameters and scalar fields P,Q,C
and Dm,K,?,a,R,Chyp
We use the equation for the oxygen
We write
Equations for the inverse problem.
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem IV the complete identification
We can write the equations in a more compact
manner
The have to find the minimum of the following
functional
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem IV Results
We can look at the behaviour of our solutions
Reconstruction of the proliferating cells density
and oxygen with a Darcy-type model
Direct numerical simulation of a Stokes-2D
simplified flow dynamic of the proliferating
cells density and oxygen
We want to evaluate quantitavely how good is our
reconstruction as function of reference time at
wich we take two subsequent snapshots.
We will introduce an error measure on the PQY
variable in order to compare the solutions
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Inverse problem IV Results
Error in PQ field as function of time, for
different identifications
At the very beginning we are not able to identify
well the behaviour
At 30 of avascular growth we can identify the
transport phenomenon quite reasonably.
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Conclusions

• We have started doing inverse problems with
  increasing complexity
• Model complexity we have to identify a complex
  phenomenon using a model that is more simple
• Information we have only partial informations
  coming from data and we have to deal with
  underdetermined systems.

We have proposed a hybrid regularized method
based on empirical eigenfunctions, that have
shown great stability and good results
We have assumed that the data are sufficiently
close in time, that is to say, the time interval
between two subsequent snaphots is small compared
to the time scale of the process.
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi
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Perspectives
We want to go toward real applications in order
to do so we increase both
1) Model complexity aged structured
Navier-Stokes models, that account for more
realistic phenomena
2) Geometrical complexity 3-D real geometries,
non-isotropic tissues
In order to relax the hypotesis on the time
derivative we want to apply optimal mass
transport theory and Monge-Kantorovich equation.
Identification of parameters in tumour growth
models based on empirical eigenfunctions Damiano
Lombardi