iwr.uni-heidelberg.de/groups/amj/BioStruct
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Structured Population Modelsfor Hematopoiesis
Marie Doumic
with Anna MARCINIAK-CZOCHRA, Benoît PERTHAME and Jorge ZUBELLI
part of A. Marciniak group « BIOSTRUCT » aims
http://www.iwr.uni-heidelberg.de/groups/amj/BioStruct/
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Marie Doumic Bedlewo, September 14th, 2010
Outline
Introduction : biological & medical motivationQuick review of models of hematopoiesisShort focus on I. Roeder’s model
The original model: a discrete compartment model
A continuous model: link with the discrete model boundedness steady states stability and instability
Perspectives
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Marie Doumic Bedlewo, September 14th, 2010
• Functionally undifferentiated• Able to proliferate• Give rise to a large number of more differentiated
progenitor cells• Maintain their population by dividing to undifferentiated
cells• Heterogeneous in respect to morphological and
biochemical properties
What are stem cells ?
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Marie Doumic Bedlewo, September 14th, 2010
Role of (adult) stem cells
• Found in lots of different tissues
• Govern regeneration processes: importance in– Bone marrow transplantation (leukemia), liver
regeneration…– Cancerogenesis (cancer stem cells)
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Marie Doumic Bedlewo, September 14th, 2010
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Formation of blood components
All derived from Hematopoietic Stem Cells (HSC)
What is hematopoiesis ?
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Open questions
• How is cell differentiation and self-renewal regulated ?
• Which factors influence repopulation kinetics ?
• How cancer cells and healthy cells interact ?
• How drug resistance of cancer cells can appear ?
• How acts a drug therapy (e.g. Imatinib for leukemia) ? Can it cure the patient completely ?
• … and many others
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Models of hematopoiesis• Compartments / quiescence and proliferation
• Maturation : discrete or continuous process?
• IBM or PDE/ODE/DDE models
• Modelling the Cell cycle (or simplifications)
• Nonlinearities to regulate the system:– Feedback-loops (A. Marciniak’s model)– competition for space (stem cells niches – I. Roeder)
Choice of a model depends on which aim is pursued
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(very partial) short overviewon models of hematopoiesisA good review: Adimy et al., Hemato., 2008
First models: MacKey, 1978 ; Loeffler, 1985
• F. Michor et al (Nature 2005, …): linear ODE and stochastic
• I. Roeder et al (Nature 2006,…): IBM model
Nonlinearity + reversible maturation process
-> Kim, Lee, Levy (PloS Comp Biol 2007, …):
PDE model based on Roeder IBM model
• Adimy, Crauste, Pujo-Menjouet et al.: DDE and application to chronic leukemia
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Marie Doumic Bedlewo, September 14th, 2010
• IBM model built on the following main ideas:
Short focus on I. Roeder’s modelIBM model built on the following main ideas
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Marie Doumic Bedlewo, September 14th, 2010
• IBM model built on the following main ideas:
Short focus on I. Roeder’s modelIBM model built on the following main ideas
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Goal: to model leukemia & Imatinib treatment. 2 Main ideas: 1. Reversible maturation process2. Competition for room in « stem cell niches »: this nonlinearity
controls the system
Work of Kim, Lee, Levy:• Write a full PDE model mimicking the IBM model• Show strictly equivalent (quantitatively & qualitatively) behaviours-> very efficient numerical simulations
Work of MD, Kim, Perthame:• Write successive simplified PDE models, keeping ideas 1. & 2.• Show equivalent qualitative behaviours (stability or instability)-> analytical analysis explaining these behaviours
Short focus on I. Roeder’s model
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Marie Doumic Bedlewo, September 14th, 2010
Simplest version of I. Roeder’s model:
Short focus on I. Roeder’s model
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Marie Doumic Bedlewo, September 14th, 2010
• IBM model built on the following main ideas:
Short focus on I. Roeder’s modelIBM model built on the following main ideas
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Marie Doumic Bedlewo, September 14th, 2010
Anna Marciniak – Czochra ‘sGroup « BioStruct » aim
See http://www.iwr.uniheidelberg.de/groups/amj/BioStruct/
To model hematopoietic reconstitution –> model Cytokin control (feedback loop)
Medical applications
• Stress conditions (chemotherapy)• Bone marrow transplantation• Blood regeneration
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Marie Doumic Bedlewo, September 14th, 2010
Experimental data
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Original model: discrete structure
differentiation
proliferation
Marciniak, Stiehl, W. Jäger, Ho, Wagner, Stem Cells & Dev., 2008.
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Marie Doumic Bedlewo, September 14th, 2010
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Marie Doumic Bedlewo, September 14th, 2010
Regulation and signalling
Cytokines• Extracellular signalling molecules (peptides)• Low level under physiological conditions• Augmented in stress conditions
Dynamics :
Quasi steady-state approximation:
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Marie Doumic Bedlewo, September 14th, 2010
ModelsWhat is regulated?• Evidence of cell cycle regulation• Evidence of high self-renewal capacity in HSC
Regulation modes• Regulation of proliferation:
• Regulation of self renewal versus differentiation
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Marie Doumic Bedlewo, September 14th, 2010
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Marie Doumic Bedlewo, September 14th, 2010
Model analysis
Steady states• Trivial: stable iff it is the only equilibrium• Semi-trivial: linearly unstable iff there exists a steady
state with more positive components• Positive steady: unique if it exists – (globally) stable ?
-> Stiehl, Marciniak (2010) & T. Stiehl’s talk on friday
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Marie Doumic Bedlewo, September 14th, 2010
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Marie Doumic Bedlewo, September 14th, 2010
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Marie Doumic Bedlewo, September 14th, 2010
PDE model derived from the discrete one(MD, Marciniak, Zubelli, Perthame,in progress)
• Stem cells: w, aw, pw, dw u1, a1, p1, d1 discrete
• Maturing cells: u(x), p(x,s), d(x) ui, ai, pi, di discrete
gi-1 ui-1 - gi ui with gi = 2(1-ai(s))pi(s)
Self-renewal Proliferation Death rate
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Marie Doumic Bedlewo, September 14th, 2010
PDE model: from discrete to continuous
1 - We formulate the original model as
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PDE model: from discrete to continuous2 – We adimension it by defining characteristic constants:
3 – We introduce a small parameter ε→0, with n=nε → x*
4 – To have sums Riemann sums integrals
differences finite differences derivatives:
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Marie Doumic Bedlewo, September 14th, 2010
PDE model: from discrete to continuous2 – We adimension it by defining characteristic constants:
5 – Define
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Marie Doumic Bedlewo, September 14th, 2010
PDE model: from discrete to continuous2 – We adimension it by defining characteristic constants:
6 – Continuity assumptions:
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7 – Proposition: under the continuity assumptions, the
Solution to the discrete system converges, up to a
subsequence, to with
if moreover the convergence is strong in
for w = lim(u1ε) solution of
we get
If moreover u is continuous in x* and un-1ε converges to u(t,x*)
Then unε converges to v solution of
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Marie Doumic Bedlewo, September 14th, 2010
Remark: decorrelation between differentiation and
proliferation is needed, else due to orders of magnitude
transport becomes a corrective term and we get
Analysis of the PDE model
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Marie Doumic Bedlewo, September 14th, 2010
Remark: decorrelation between differentiation and
proliferation is needed, else due to orders of magnitude
transport becomes a corrective term and we get
see Grzegorz Jamroz’s talk for more insight
Analysis of the PDE model
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Marie Doumic Bedlewo, September 14th, 2010
Numerical simulations
Stem cells Maturing cells mature cells
Discrete model Continuous model
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Marie Doumic Bedlewo, September 14th, 2010
Analysis of the general PDE model
With initial conditions:
Cell number balance law:
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Marie Doumic Bedlewo, September 14th, 2010
Theorem. The unique solution is uniformly bounded
Analysis of PDE -Assumptions
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Marie Doumic Bedlewo, September 14th, 2010
Main difficulty: feed-back loop involves a delay
Main tool: the following lemma:
Sketch of the proof: deriving the equation divided by u:
Analysis of PDE - boundedness
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Marie Doumic Bedlewo, September 14th, 2010
From boundedness of z we deduce
1st and 3rd estimate: directly from boundedness of z
2nd estimate: look at
Analysis of PDE - boundedness
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Extra estimate, used for non-extinction (see below):
Proof:
Analysis of PDE - boundedness
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Solution of:
With .
Proposition. There exists a steady state iffIn this case, it is unique.
Remark: similar assumption for the discrete system BUT here: no semi-trivial steady state.
Analysis of PDE – steady states
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Theorem.
extinction with exponential rate
bounded away from zero
Proof for extinction: uses entropy by calculating
Proof for positivity:
Analysis of PDE – extinction or persistance
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Theorem.
extinction with exponential rate
bounded away from zero
Remark: a similar alternative is found
in many other nonlinear structured models
(see D, Kim, Perthame for CML ;
Calvez, Lenuzza et al. for prion equations;
Bekkal Brikci, Clairambault, Perthame for cell cycle…)
Analysis of PDE – extinction or persistance
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Linearised equation around the steady state:
Method: look for the sign of the real part of the eigenvalues
Analysis of PDE – Linearised stability of the non trivial steady state
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Eigenvalue problem:
Defining it gives:
Analysis of PDE – Linearised stability of the non trivial steady state
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Simplest case: no feed-back on the maturation process.The characteristic equation becomes
Proposition.
If
There is a Hopf bifurcation for one value of μ >0.
Proof: look for purely imaginary solutions, which are theplaces where a bifurcation can occur.
Analysis of PDE – Linearised stability of the non trivial steady state
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Case derived from the discrete model:
Proposition.
If the maturation and the proliferation rates are independent of maturity: linear stability.
If proliferation rate varies: instability may appear.
Proof: same ideas (but longer calculations…)
Analysis of PDE – Linearised stability of the non trivial steady state
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Marie Doumic Bedlewo, September 14th, 2010
perspectives
• Comparison discrete & continuous :– biological interpretation of analytical constraints– What could give a measure of differentiation ? – Opportunity of the discrete vs continuous modelling ?
• Inverse problems: recover g from data of differentiated cells ?
• Mathematical challenge: prove nonlinear (in)stability by the use of entropy-type arguments ?