Structured Population Models
for 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/
Outline
Introduction : biological & medical motivation
Quick review of models of hematopoiesis
Short 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
Marie Doumic
Bedlewo, September 14th, 2010
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
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)
Marie Doumic
Bedlewo, September 14th, 2010
Marie Doumic
Bedlewo, September 14th, 2010
What is hematopoiesis ?
Formation of blood components
All derived from Hematopoietic Stem Cells (HSC)
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 overview
on 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
-> 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
Short focus on I. Roeder’s model
IBM model built on the following main ideas
• IBM model built on the following main
ideas:
Marie Doumic
Bedlewo, September 14th, 2010
Short focus on I. Roeder’s model
IBM model built on the following main ideas
• IBM model built on the following main
ideas:
Marie Doumic
Bedlewo, September 14th, 2010
Short focus on I. Roeder’s 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
-> 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:
Marie Doumic
Bedlewo, September 14th, 2010
Short focus on I. Roeder’s model
IBM model built on the following main ideas
• IBM model built on the following main
ideas:
Marie Doumic
Bedlewo, September 14th, 2010
Anna Marciniak – Czochra ‘s
Group « 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
Marie Doumic
Bedlewo, September 14th, 2010
Experimental data
Marie Doumic
Bedlewo, September 14th, 2010
Original model: discrete structure
differentiation
proliferation
Marciniak, Stiehl, W. Jäger, Ho, Wagner, Stem Cells & Dev., 2008.
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:
Marie Doumic
Bedlewo, September 14th, 2010
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
Marie Doumic
Bedlewo, September 14th, 2010
Marie Doumic
Bedlewo, September 14th, 2010
Marie Doumic
Bedlewo, September 14th, 2010
Marie Doumic
Bedlewo, September 14th, 2010
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
Marie Doumic
Bedlewo, September 14th, 2010
Marie Doumic
Bedlewo, September 14th, 2010
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
Self-renewal
u1, a1, p1, d1 discrete
Proliferation
• Maturing cells: u(x), p(x,s), d(x)
Death rate
ui, ai, pi, di discrete
gi-1 ui-1 - gi ui with gi = 2(1-ai(s))pi(s)
Marie Doumic
Bedlewo, September 14th, 2010
PDE model: from discrete to continuous
1 - We formulate the original model as
Marie Doumic
Bedlewo, September 14th, 2010
PDE model: from discrete to continuous
2 – We adimension it by defining characteristic constants:
3 – We introduce a small parameter ε→0, with n=nε → x*
4 – To have sums
differences
Riemann sums
finite differences
integrals
derivatives:
PDE model: from discrete to continuous
2 – We adimension it by defining characteristic constants:
5 – Define
Marie Doumic
Bedlewo, September 14th, 2010
PDE model: from discrete to continuous
2 – We adimension it by defining characteristic constants:
6 – Continuity assumptions:
Marie Doumic
Bedlewo, September 14th, 2010
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
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
Marie Doumic
Bedlewo, September 14th, 2010
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
Marie Doumic
Bedlewo, September 14th, 2010
Numerical simulations
Discrete model
Stem cells
Continuous model
Maturing cells
Marie Doumic
Bedlewo, September 14th, 2010
mature cells
Analysis of the general PDE model
With initial conditions:
Cell number balance law:
Marie Doumic
Bedlewo, September 14th, 2010
Analysis of PDE -Assumptions
Theorem. The unique solution is uniformly bounded
Marie Doumic
Bedlewo, September 14th, 2010
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:
Marie Doumic
Bedlewo, September 14th, 2010
Analysis of PDE - boundedness
From boundedness of z we deduce
1st and 3rd estimate: directly from boundedness of z
2nd estimate: look at
Marie Doumic
Bedlewo, September 14th, 2010
Analysis of PDE - boundedness
Extra estimate, used for non-extinction (see below):
Proof:
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.
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:
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 –
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
Analysis of PDE –
Linearised stability of the non trivial steady state
Eigenvalue problem:
Defining
it gives:
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 >0.
Proof: look for purely imaginary solutions, which are the
places where a bifurcation can occur.
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…)
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 ?
Marie Doumic
Bedlewo, September 14th, 2010