Journal of Mathematical Biology

, Volume 67, Issue 1, pp 69–103 | Cite as

Estimating the division rate for the growth-fragmentation equation

  • M. DoumicEmail author
  • Léon M. Tine


Growth-fragmentation equations arise in many different contexts, ranging from cell division, protein polymerization, neurosciences etc. Direct observation of temporal dynamics being often difficult, it is of main interest to develop theoretical and numerical methods to recover reaction rates and parameters of the equation from indirect observation of the solution. Following the work done in Perthame and Zubelli (Inverse Probl 23:1037–1052, 2007) and Doumic et al. (2009) for the specific case of the cell division equation, we address here the general question of recovering the fragmentation rate of the equation from the observation of the time-asymptotic solution, when the fragmentation kernel and the growth rates are fully general. We give both theoretical results and numerical methods, and discuss the remaining issues.


Growth-fragmentation equation Cell division equation General fragmentation kernels Inverse problem Eigenvalue problem 

Mathematics Subject Classification

35Q92 35R30 45Q05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Brézis H (1983) Functional analysis. Masson, ParisGoogle Scholar
  2. Doumic M, Perthame B, Zubelli JP (2009) Numerical solution of an inverse problem in size-structured population dynamics. Inverse Probl 25(4). doi: 10.1088/0266-5611/25/4/045008
  3. Doumic M, Maia P, Zubelli JP (2010) On the calibration of a size-structured population model from experimental data. Acta Biotheor 58(4): 405–413CrossRefGoogle Scholar
  4. Doumic M, Gabriel P (2010) Eigenelements of a general aggregation-fragmentation modelGoogle Scholar
  5. Doumic M, Hoffmann M, Reynaud-Bouret P, Rivoirard V Nonparametric estimation of the division rate of a size-structured population. (submitted)Google Scholar
  6. Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems, volume 375 of mathematics and its applications. Kluwer, DordrechtCrossRefGoogle Scholar
  7. Greer ML, Pujo-Menjouet L, Webb GF (2006) A mathematical analysis of the dynamics of prion proliferation. J Theor Biol 242: 598–606MathSciNetCrossRefGoogle Scholar
  8. Groh A, Krebs J, Wagner M (2011) Efficient solution of an inverse problem in cell population dynamics. Inverse Probl 27Google Scholar
  9. Gyllenberg M, Osipov A, Päivärinta L (2002) The inverse problem of age-structured population dynamics. J Evol Equ 2: 222–239CrossRefGoogle Scholar
  10. Hardy GH, Littlewood JE, Polya G (1988) Inequalities. Cambride Mathematical Library, CambridgezbMATHGoogle Scholar
  11. Heijmans HJAM (1984) On the stable size distribution of populations reproducing by fission into two inequal parts. Math Biosci 72(1): 19–50MathSciNetzbMATHCrossRefGoogle Scholar
  12. Metz JAJ, Diekmann O (1986) The dynamics of physiologically structured populations, Lecture Notes in Biomathematics 68. Springer, BerlinGoogle Scholar
  13. Michel P, Mischler , Perthame B (2005) General entropy equations for structured population models and scattering. C R Math Acad Sci Paris 338(9): 697–702MathSciNetCrossRefGoogle Scholar
  14. Perthame B (2007) Transport equations in biology. Frontiers in mathematics. Birkhäuser Verlag, BaselGoogle Scholar
  15. Perthame B, Ryzhik L (2005) Exponential decay for the fragmentation or cell-division equation. J Differ Equ 210(1): 155–177MathSciNetzbMATHCrossRefGoogle Scholar
  16. Perthame B, Zubelli JP (2007) On the inverse problem for a size structured population model. Inverse Probl 23: 1037–1052MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.INRIA Paris-Rocquencourt, EPI BANGLe Chesnay CedexFrance
  2. 2.Laboratoire d’Analyse Numérique et d’Informatique (LANI)Université Gaston BergerSaint-LouisSenegal
  3. 3.Labo P. Painlevé UMR 8524 CNRS, Université des Scienceset TechnologiesLille 1France
  4. 4.Project-Team SIMPAF, INRIA Lille Nord Europe Research CenterVilleneuve d’Ascq CedexFrance

Personalised recommendations