iwr.uni-heidelberg.de/groups/amj/BioStruct

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/

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


• 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 ?


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)

Formation of blood components

All derived from Hematopoietic Stem Cells (HSC)

What is hematopoiesis ?

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

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

(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


• IBM model built on the following main ideas:

Short focus on I. Roeder’s modelIBM model built on the following main ideas

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


Simplest version of I. Roeder’s model:

Short focus on I. Roeder’s model


• IBM model built on the following main ideas:

Short focus on I. Roeder’s modelIBM model built on the following main ideas


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

http://www.iwr.uniheidelberg.de/groups/amj/BioStruct/


Experimental data

Original model: discrete structure

differentiation

proliferation

Marciniak, Stiehl, W. Jäger, Ho, Wagner, Stem Cells & Dev., 2008.


Regulation and signalling

Cytokines• Extracellular signalling molecules (peptides)• Low level under physiological conditions• Augmented in stress conditions

Dynamics :

Quasi steady-state approximation:


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


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


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


PDE model: from discrete to continuous

1 - We formulate the original model as

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:



5 – Define



6 – Continuity assumptions:

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


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


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


Numerical simulations

Stem cells Maturing cells mature cells

Discrete model Continuous model


Analysis of the general PDE model

With initial conditions:

Cell number balance law:


Theorem. The unique solution is uniformly bounded

Analysis of PDE -Assumptions


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


From boundedness of z we deduce

1st and 3rd estimate: directly from boundedness of z

2nd estimate: look at


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

Proof:


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

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

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

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

Eigenvalue problem:

Defining it gives:


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.


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…)



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 ?

iwr.uni-heidelberg.de/groups/amj/BioStruct

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