Structured Population Models for Hematopoiesis Marie Doumic with Anna MARCINIAK-CZOCHRA, Beno - PowerPoint PPT Presentation

Title: Structured Population Models for Hematopoiesis Marie Doumic with Anna MARCINIAK-CZOCHRA, Beno


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Structured Population Modelsfor
HematopoiesisMarie 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/BioStr
uct/
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Outline

• Introduction biological medical motivation
• Quick review of models of hematopoiesis
• Short focus on I. Roeders 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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What are stem cells ?

• 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

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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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What is hematopoiesis ?

• Formation of blood components
• All derived from Hematopoietic Stem Cells (HSC)

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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. Marciniaks 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
hematopoiesis

• A 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
• -gt 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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Short focus on I. Roeders modelIBM model built
on the following main ideas

• IBM model built on the following main ideas

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Short focus on I. Roeders modelIBM model built
on the following main ideas

• IBM model built on the following main ideas

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Short focus on I. Roeders model

• Goal to model leukemia Imatinib treatment. 2
  Main ideas
• 1. Reversible maturation process
• 2. 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
• -gt 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)
• -gt analytical analysis explaining these
  behaviours

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Short focus on I. Roeders model

• Simplest version of I. Roeders model

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Short focus on I. Roeders modelIBM model built
on the following main ideas

• IBM model built on the following main ideas

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Anna Marciniak Czochra sGroup  BioStruct 
aim

• See http//www.iwr.uniheidelberg.de/groups/amj/Bio
  Struct/
• To model hematopoietic reconstitution
• gt model Cytokin control (feedback loop)
• Medical applications
• Stress conditions (chemotherapy)
• Bone marrow transplantation
• Blood regeneration

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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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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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Models

• What 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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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 ?
• -gt Stiehl, Marciniak (2010) T. Stiehls talk on
  friday

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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)

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PDE model from discrete to continuous

• 1 - We formulate the original model as

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PDE model from discrete to continuous

• 2 We adimension it by defining characteristic
  constants
• 3 We introduce a small parameter e?0, with nne
  ? x
• 4 To have sums Riemann sums integrals
• differences finite differences
  derivatives

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PDE model from discrete to continuous

• 2 We adimension it by defining characteristic
  constants
• 5 Define

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PDE model from discrete to continuous

• 2 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(u1e) solution of
• we get
• If moreover u is continuous in x and un-1e
  converges to u(t,x)
• Then une converges to v solution of

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Analysis of the PDE model

• Remark decorrelation between differentiation and
• proliferation is needed, else due to orders of
  magnitude
• transport becomes a corrective term and we get

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Analysis of the PDE model

• Remark decorrelation between differentiation and
• proliferation is needed, else due to orders of
  magnitude
• transport becomes a corrective term and we get
• see Grzegorz Jamrozs talk for more insight

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Numerical simulations
Discrete model
Continuous model
Stem cells
Maturing cells
mature cells
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Analysis of the general PDE model
With initial conditions
Cell number balance law
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Analysis of PDE -Assumptions

• Theorem. The unique solution is uniformly bounded

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Analysis of PDE - boundedness

• Main difficulty feed-back loop involves a delay
• Main tool the following lemma
• Sketch of the proof deriving the equation
  divided by u

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Analysis of PDE - boundedness

• From boundedness of z we deduce
• 1st and 3rd estimate directly from boundedness
  of z
• 2nd estimate look at

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Analysis of PDE - boundedness

• Extra estimate, used for non-extinction (see
  below)
• Proof

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Analysis of PDE steady states

• Solution of
• With .
• Proposition. There exists a steady state iff
• In this case, it is unique.
• Remark similar assumption for the discrete
  system
• BUT here no semi-trivial steady state.

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Analysis of PDE extinction or persistance

• Theorem.
• extinction with exponential rate
• bounded away from zero
• Proof for extinction uses entropy by calculating
• Proof for positivity

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Analysis of PDE extinction or persistance

• 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)

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Analysis of PDE Linearised stability of the
non trivial steady state

• Linearised equation around the steady state
• Method look for the sign of the real part of the
  eigenvalues

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Analysis of PDE Linearised stability of the
non trivial steady state

• Eigenvalue problem
• Defining it gives

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Analysis of PDE Linearised stability of the
non trivial steady state

• 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
  gt0.
• Proof look for purely imaginary solutions, which
  are the
• places where a bifurcation can occur.

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Analysis of PDE Linearised stability of the
non trivial steady state

• 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)

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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 ?