Analysis and Calibration of a Linear Model for Structured Cell Populations with Unidirectional Motion: Application to the Morphogenesis of Ovarian Follicles

Abstract

We analyze a multitype age-dependent model for cell populations subject to unidirectional motion in both a stochastic and deterministic framework. Cells are distributed into successive layers; they may divide and move irreversibly from one layer to the next. We adapt results on the large-time convergence of PDE systems and branching processes to our context, where the Perron--Frobenius or Krein--Rutman theorem cannot be applied. We derive explicit analytical formulas for the asymptotic cell number moments and the stable age distribution. We illustrate these results numerically and apply them to the study of the morphodynamics of ovarian follicles. We prove the structural parameter identifiability of our model in the case of age independent division rates. Using a set of experimental biological data, we estimate the model parameters to fit the changes in the cell numbers in each layer during the early stages of follicle development.

Keywords

  1. structured cell populations
  2. multitype age dependent branching processes
  3. renewal equations
  4. McKendrick--Von Foerster model
  5. parameter calibration
  6. structural identifiability

MSC codes

  1. 35L65
  2. 60K15
  3. 60J80
  4. 92D25

Get full access to this article

View all available purchase options and get full access to this article.

Supplementary Material


PLEASE NOTE: These supplementary files have not been peer-reviewed.


Index of Supplementary Materials

Title of paper: Analysis and Calibration of a Linear Model for Structured Cell Populations with Unidirectional Motion: Application to the Morphogenesis of Ovarian Follicles

Authors: Frédérique Clément, Fréderique Robin, and Romain Yvinec

File: SM_M116133.pdf

Type: PDF

Contents: supplemental proofs

References

1.
F. Clément, P. Michel, D. Monniaux, and T. Stiehl, Coupled somatic cell kinetics and germ cell growth: Multiscale model-based insight on ovarian follicular development, Multiscale Model. Simul., 11 (2013), pp. 719--746, https://doi.org/10.1137/120897249.
2.
T. E. Harris, The Theory of Branching Processes, Springer-Verlag, Berlin 1963.
3.
P. Jagers and F. C. Klebaner, Population-size-dependent and age-dependent branching processes, Stochastic Process. Appl., 87 (2000), pp. 235--254, https://doi.org/10.1016/S0304-4149(99)00111-8.
4.
F. C. Klebaner, Introduction to Stochastic Calculus with Applications, 3rd ed., Imperial College Press, London, UK, 2012.
5.
T. Lundy, P. Smith, A. O'connell, N. L. Hudson, and K. P. McNatty, Populations of granulosa cells in small follicles of the sheep ovary, J. Reprod. Fertil., 115 (1999), pp. 251--262.
6.
J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath. 68, Springer-Verlag, Berlin, 1986.
7.
P. Michel, S. Mischler, and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl. (9), 84 (2005), pp. 1235--1260, https://doi.org/10.1016/j.matpur.2005.04.001.
8.
A. Perasso and U. Razafison, Identifiability problem for recovering the mortality rate in an age-structured population dynamics model, Inverse Probl. Sci. Eng., 24 (2016), pp. 711--728, https://doi.org/10.1080/17415977.2015.1061522.
9.
B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 2007.
10.
P. E. Protter, Stochastic Integration and Differential Equations, 2nd ed., Appl. Math. (N. Y.) 21, Springer-Verlag, Berlin, 2004.
11.
A. Raue, C. Kreutz, T. Maiwald, J. Bachmann, M. Schilling, U. Klingmüller, and J. Timmer, Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood, Bioinformatics, 25 (2009), pp. 1923--1929, https://doi.org/10.1093/bioinformatics/btp358.
12.
A. Raue, B. Steiert, M. Schelker, C. Kreutz, T. Maiwald, H. Hass, J. Vanlier, C. Tönsing, L. Adlung, R. Engesser, W. Mader, T. Heinemann, J. Hasenauer, M. Schilling, T. Höfer, E. Klipp, F. Theis, U. Klingmüller, B. Schöberl, and J. Timmer, Data2dynamics: A modeling environment tailored to parameter estimation in dynamical systems, Bioinformatics, 31 (2015), pp. 3558--3560, https://doi.org/10.1093/bioinformatics/btv405.
13.
P. Smith, R. Braw-Tal, K. Corrigan, N. L. Hudson, D. A. Heath, and K. P. McNatty, Ontogeny of ovarian follicle development in Booroola sheep fetuses that are homozygous carriers or non-carriers of the FecB gene, J. Reprod. Fertil., 100 (1994), pp. 485--490, https://doi.org/10.1530/jrf.0.1000485.
14.
P. Smith, W.-S. O, N. L. Hudson, L. Shaw, D. A. Heath, L. Condell, D. J. Phillips, and K. P. McNatty, Effects of the Booroola gene (FecB) on body weight, ovarian development and hormone concentrations during fetal life, J. Reprod. Fertil., 98 (1993), pp. 41--54, https://doi.org/10.1530/jrf.0.0980041.
15.
V. C. Tran, Large population limit and time behaviour of a stochastic particle model describing an age-structured population, ESAIM Probab. Stat., 12 (2008), pp. 345--386, https://doi.org/10.1051/ps:2007052.
16.
G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 207 - 229
ISSN (online): 1095-712X

History

Submitted: 14 December 2017
Accepted: 22 October 2018
Published online: 5 February 2019

Keywords

  1. structured cell populations
  2. multitype age dependent branching processes
  3. renewal equations
  4. McKendrick--Von Foerster model
  5. parameter calibration
  6. structural identifiability

MSC codes

  1. 35L65
  2. 60K15
  3. 60J80
  4. 92D25

Authors

Affiliations

Frédérique Clément

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media