Papers
Topics
Authors
Recent
Search
2000 character limit reached

Numerical Approximations and Convergence Analysis of Piecewise Diffusion Markov Processes, with Application to Glioma Cell Migration

Published 24 Jan 2024 in math.NA, cs.NA, and math.PR | (2401.13428v2)

Abstract: In this paper, we focus on numerical approximations of Piecewise Diffusion Markov Processes (PDifMPs), particularly when the explicit flow maps are unavailable. Our approach is based on the thinning method for modelling the jump mechanism and combines the Euler-Maruyama scheme to approximate the underlying flow dynamics. For the proposed approximation schemes, we study both the mean-square and weak convergence. Weak convergence of the algorithms is established by a martingale problem formulation. Moreover, we employ these results to simulate the migration patterns exhibited by moving glioma cells at the microscopic level. Further, we develop and implement a splitting method for this PDifMP model and employ both the Thinned Euler-Maruyama and the splitting scheme in our simulation example, allowing us to compare both methods.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (58)
  1. Markus Ableidinger, Evelyn Buckwar and Harald Hinterleitner “A stochastic version of the Jansen and Rit neural mass model: analysis and numerics” In The Journal of Mathematical Neuroscience 7.1 SpringerOpen, 2017, pp. 1–35
  2. “A Hough transform global probabilistic approach to multiple-subject diffusion MRI tractography” In Medical image analysis 15.4 Elsevier, 2011, pp. 414–425
  3. Alfonso Alamo and Jesús María Sanz-Serna “A technique for studying strong and weak local errors of splitting stochastic integrators” In SIAM Journal on Numerical Analysis 54.6 SIAM, 2016, pp. 3239–3257
  4. “A cellular automaton model for the migration of glioma cells” In Physical biology 3.2 IOP Publishing, 2006, pp. 93
  5. Nicholas A Baran, George Yin and Chao Zhu “Feynman-Kac formula for switching diffusions: connections of systems of partial differential equations and stochastic differential equations” In Advances in Difference Equations 2013.1 SpringerOpen, 2013, pp. 1–13
  6. Julien Bect “Processus de Markov diffusifs par morceaux: outils analytiques et numériques”, 2007
  7. Howard C Berg and Douglas A Brown “Chemotaxis in Escherichia Coli analysed by three-dimensional tracking” In Nature 239.5374 Nature Publishing Group, 1972, pp. 500–504
  8. Andrea Bertazzi, Joris Bierkens and Paul Dobson “Approximations of piecewise deterministic Markov processes and their convergence properties” In Stochastic Processes Appl. 154, 2022, pp. 91–153 DOI: 10.1016/j.spa.2022.09.004
  9. Joris Bierkens, Paul Fearnhead and Gareth Roberts “The zig-zag process and super-efficient sampling for Bayesian analysis of big data”, 2019
  10. “Piecewise deterministic Markov processes for scalable Monte Carlo on restricted domains” In Statistics & Probability Letters 136 Elsevier, 2018, pp. 148–154
  11. HAP Blom “From piecewise deterministic to piecewise diffusion Markov processes” In Proceedings of the 27th IEEE Conference on Decision and Control, 1988, pp. 1978–1983 IEEE
  12. Henk AP Blom, Hao Ma and GJ Bert Bakker “Interacting particle system-based estimation of reach probability for a generalized stochastic hybrid system” In IFAC-PapersOnLine 51.16 Elsevier, 2018, pp. 79–84
  13. Alexandre Bouchard-Côté, Sebastian J Vollmer and Arnaud Doucet “The bouncy particle sampler: A nonreversible rejection-free Markov chain Monte Carlo method” In Journal of the American Statistical Association 113.522 Taylor & Francis, 2018, pp. 855–867
  14. Evelyn Buckwar and Martin G Riedler “An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution” In Journal of mathematical biology 63.6 Springer, 2011, pp. 1051–1093
  15. “A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model” In Applied Numerical Mathematics 179 Elsevier, 2022, pp. 191–220
  16. Manuela L Bujorianu and John Lygeros “Reachability questions in piecewise deterministic Markov processes” In International Workshop on Hybrid Systems: Computation and Control, 2003, pp. 126–140 Springer
  17. Manuela L Bujorianu and John Lygeros “Toward a general theory of stochastic hybrid systems” In Stochastic hybrid systems Springer, 2006, pp. 3–30
  18. SERGIO BLANES1 FERNANDO CASAS and ANDER MURUA “Splitting and composition methods in the numerical integration of differential equations” In BOLETÍN NUMERO 45 Diciembre 2008, 2008, pp. 89
  19. Zhengdao Chen, Baranidharan Raman and Ari Stern “Structure-Preserving Numerical Integrators for Hodgkin–Huxley-Type Systems” In SIAM Journal on Scientific Computing 42.1 SIAM, 2020, pp. B273–B298
  20. “Assessment of brain tumor cell motility in vivo and in vitro” In Journal of Neurosurgery 82.4 Journal of Neurosurgery Publishing Group, 1995, pp. 615–622 DOI: 10.3171/jns.1995.82.4.0615
  21. “Probabilistic and piecewise deterministic models in biology” In ESAIM: Proceedings and Surveys 60 EDP Sciences, 2017, pp. 225–245
  22. Martina Conte, Luca Gerardo-Giorda and Maria Groppi “Glioma invasion and its interplay with nervous tissue and therapy: A multiscale model” In Journal of theoretical biology 486 Elsevier, 2020, pp. 110088
  23. Mark HA Davis “Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models” In Journal of the Royal Statistical Society: Series B (Methodological) 46.3 Wiley Online Library, 1984, pp. 353–376
  24. Alain Durmus, Arnaud Guillin and Pierre Monmarché “Piecewise deterministic Markov processes and their invariant measures” In Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 57.3, 2021, pp. 1442–1475 Institut Henri Poincaré
  25. Christian Engwer, Alexander Hunt and Christina Surulescu “Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings” In Mathematical medicine and biology: a journal of the IMA 33.4 Oxford University Press, 2016, pp. 435–459
  26. Christian Engwer, Markus Knappitsch and Christina Surulescu “A multiscale model for glioma spread including cell-tissue interactions and proliferation” In Mathematical Biosciences & Engineering 13.2 American Institute of Mathematical Sciences, 2016, pp. 443
  27. “Glioma follow white matter tracts: a multiscale DTI-based model” In Journal of mathematical biology 71.3 Springer, 2015, pp. 551–582
  28. “Stochastic simulation and Monte Carlo methods: mathematical foundations of stochastic simulation” Springer Science & Business Media, 2013
  29. Hana LP Harpold, Ellsworth C Alvord Jr and Kristin R Swanson “The evolution of mathematical modeling of glioma proliferation and invasion” In Journal of Neuropathology & Experimental Neurology 66.1 American Association of Neuropathologists, Inc., 2007, pp. 1–9
  30. Thomas Hillen “On the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-moment closure of transport equations: The Cattaneo approximation” In Discrete & Continuous Dynamical Systems-B 4.4 American Institute of Mathematical Sciences, 2004, pp. 961
  31. Thomas Hillen and Kevin J Painter “Transport and anisotropic diffusion models for movement in oriented habitats” In Dispersal, individual movement and spatial ecology Springer, 2013, pp. 177–222
  32. Alexander Hunt “DTI-based multiscale models for glioma invasion”, 2018
  33. “The regime switching portfolios” In Asia-Pacific Financial Markets 18 Springer, 2011, pp. 167–189
  34. “Approximation methods for piecewise deterministic Markov processes and their costs” In Scandinavian Actuarial Journal 2019.4 Taylor & Francis, 2019, pp. 308–335
  35. Douglas A Lauffenburger and Jennifer Linderman “Receptors: models for binding, trafficking, and signaling” Oxford University Press, 1996
  36. Vincent Lemaire, Michèle Thieullen and Nicolas Thomas “Exact simulation of the jump times of a class of piecewise deterministic Markov processes” In Journal of Scientific Computing 75.3 Springer, 2018, pp. 1776–1807
  37. Vincent Lemaire, Michèle Thieullen and Nicolas Thomas “Thinning and Multilevel Monte Carlo for Piecewise Deterministic (Markov) Processes. Application to a stochastic Morris-Lecar model” In Advances in Applied Probability 52.1 Cambridge University Press, 2020, pp. 138–172
  38. PA W Lewis and Gerald S Shedler “Simulation of nonhomogeneous Poisson processes by thinning” In Naval Research Logistics Quarterly 26.3 Wiley Online Library, 1979, pp. 403–413
  39. Roland Malhamé “A jump-driven Markovian electric load model” In Advances in Applied Probability 22.3 Cambridge University Press, 1990, pp. 564–586
  40. Xuerong Mao “Stochastic differential equations and applications” Elsevier, 2007
  41. Robert I McLachlan and G Reinout W Quispel “Splitting methods” In Acta Numerica 11 Cambridge University Press, 2002, pp. 341–434
  42. Amira Meddah, Martina Conte and Evelyn Buckwar “A stochastic hierarchical model for low grade glioma evolution” In Journal of Mathematical Biology, 2023
  43. Hiroko Ohgaki “Epidemiology of brain tumors” In Cancer Epidemiology: Modifiable Factors Springer, 2009, pp. 323–342
  44. Gilles Pagès “Numerical probability” In Universitext, Springer Springer, 2018
  45. “Mathematical modelling of glioma growth: the use of diffusion tensor imaging (DTI) data to predict the anisotropic pathways of cancer invasion” In Journal of theoretical biology 323 Elsevier, 2013, pp. 25–39
  46. Khashayar Pakdaman, Michele Thieullen and Gilles Wainrib “Fluid limit theorems for stochastic hybrid systems with application to neuron models” In Advances in Applied Probability 42.3 Cambridge University Press, 2010, pp. 761–794
  47. WP Petersen “A general implicit splitting for stabilizing numerical simulations of Itô stochastic differential equations” In SIAM journal on numerical analysis 35.4 SIAM, 1998, pp. 1439–1451
  48. “Jump-Adapted Strong Approximations” In Numerical Solution of Stochastic Differential Equations with Jumps in Finance Springer, 2010, pp. 347–388
  49. Martin G Riedler “Almost sure convergence of numerical approximations for piecewise deterministic Markov processes” In Journal of Computational and Applied Mathematics 239 Elsevier, 2013, pp. 50–71
  50. Tony Shardlow “Splitting for dissipative particle dynamics” In SIAM Journal on Scientific computing 24.4 SIAM, 2003, pp. 1267–1282
  51. “Cofilin determines the migration behavior and turning frequency of metastatic cancer cells” In The Journal of Cell Biology 179.4 Rockefeller University Press, 2007, pp. 777–791 DOI: 10.1083/jcb.200707009
  52. Abhyudai Singh and Joao P Hespanha “Stochastic hybrid systems for studying biochemical processes” In Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368.1930 The Royal Society Publishing, 2010, pp. 4995–5011
  53. Kristin R Swanson, Ellsworth C Alvord Jr and JD Murray “A quantitative model for differential motility of gliomas in grey and white matter” In Cell proliferation 33.5 Wiley Online Library, 2000, pp. 317–329
  54. “Expansion of the global error for numerical schemes solving stochastic differential equations” In Stochastic analysis and applications 8.4 Taylor & Francis, 1990, pp. 483–509
  55. Romain Veltz “A new twist for the simulation of hybrid systems using the true jump method” In arXiv preprint arXiv:1504.06873, 2015
  56. “Properties of solutions of stochastic differential equations with continuous-state-dependent switching” In Journal of Differential Equations 249.10 Elsevier, 2010, pp. 2409–2439
  57. G George Yin and Chao Zhu “Hybrid switching diffusions: properties and applications” Springer Science & Business Media, 2009
  58. “Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching” In Mathematics and Computers in Simulation 64.2 Elsevier, 2004, pp. 223–235
Citations (1)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Authors (2)

  1. Evelyn Buckwar 
  2. Amira Meddah 

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 1 tweet with 4 likes about this paper.

Sign Up for Free