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

1. 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/BioStruct/

2. 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

3. 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

4. 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

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6. What is hematopoiesis ? Formation of blood components All derived from Hematopoietic Stem Cells (HSC)

7. 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

8. 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

9. (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 -> 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

10. Short focus on I. Roeder’s modelIBM model built on the following main ideas • IBM model built on the following main ideas: Marie Doumic Bedlewo, September 14th, 2010

11. Short focus on I. Roeder’s modelIBM model built on the following main ideas • IBM model built on the following main ideas: Marie Doumic Bedlewo, September 14th, 2010

12. 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

13. Short focus on I. Roeder’s model Simplest version of I. Roeder’s model: Marie Doumic Bedlewo, September 14th, 2010

14. Short focus on I. Roeder’s modelIBM model built on the following main ideas • IBM model built on the following main ideas: Marie Doumic Bedlewo, September 14th, 2010

15. 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 Marie Doumic Bedlewo, September 14th, 2010

16. Experimental data Marie Doumic Bedlewo, September 14th, 2010

17. Original model: discrete structure differentiation proliferation Marciniak, Stiehl, W. Jäger, Ho, Wagner, Stem Cells & Dev., 2008.

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19. 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

20. 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

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25. 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

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28. Self-renewal Proliferation Death rate PDE model derived from the discrete one(MD, Marciniak, Zubelli, Perthame,in progress) • Stem cells: w, aw, pw, dwu1, a1, p1, d1 discrete • Maturing cells: u(x), p(x,s), d(x) ui, ai, pi, di discrete gi-1 ui-1 - gi uiwith gi = 2(1-ai(s))pi(s) Marie Doumic Bedlewo, September 14th, 2010

29. PDE model: from discrete to continuous 1 - We formulate the original model as Marie Doumic Bedlewo, September 14th, 2010

30. 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 Riemann sums integrals differences finite differences derivatives:

31. PDE model: from discrete to continuous 2 – We adimension it by defining characteristic constants: 5 – Define Marie Doumic Bedlewo, September 14th, 2010

32. PDE model: from discrete to continuous 2 – We adimension it by defining characteristic constants: 6 – Continuity assumptions: Marie Doumic Bedlewo, September 14th, 2010

33. 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-1εconverges to u(t,x*) Then unεconverges to v solution of

34. 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

35. 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

36. Numerical simulations Discrete model Continuous model Stem cells Maturing cells mature cells Marie Doumic Bedlewo, September 14th, 2010

37. Analysis of the general PDE model With initial conditions: Cell number balance law: Marie Doumic Bedlewo, September 14th, 2010

38. Analysis of PDE -Assumptions Theorem. The unique solution is uniformly bounded Marie Doumic Bedlewo, September 14th, 2010

39. 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

40. 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

41. Analysis of PDE - boundedness Extra estimate, used for non-extinction (see below): Proof:

42. 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.

43. 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:

44. 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…)

45. 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

46. Analysis of PDE – Linearised stability of the non trivial steady state Eigenvalue problem: Defining it gives:

47. 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.

48. 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…)

49. 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