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Lötstedt, P. & Mathias, S. (2025). Numerical integration of mechanical forces in center-based models for biological cell populations. Applied Mathematics and Computation, 486, Article ID 129069.
Open this publication in new window or tab >>Numerical integration of mechanical forces in center-based models for biological cell populations
2025 (English)In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 486, article id 129069Article in journal (Refereed) Published
Abstract [en]

Center-based models are used to simulate the mechanical behavior of biological cells during embryonic development or cancer growth. To allow for the simulation of biological populations potentially growing from a few individual cells to many thousands or more, these models have to be numerically efficient, while being reasonably accurate on the level of individual cell trajectories. In this work, we increase the robustness, accuracy, and efficiency of the simulation of center-based models by choosing the time steps adaptively in the numerical method and comparing five different integration methods. We investigate the gain in using single rate time stepping based on local estimates of the numerical errors for the forward and backward Euler methods of first order accuracy and a Runge-Kutta method and the trapezoidal method of second order accuracy. Properties of the analytical solution such as convergence to steady state and conservation of the center of gravity are inherited by the numerical solutions. Furthermore, we propose a multirate time stepping scheme that simulates regions with high local force gradients (e.g. as they happen after cell division) with multiple smaller time steps within a larger single time step for regions with smoother forces. These methods are compared for a model system in numerical experiments. We conclude, for example, that the multirate forward Euler method performs better than the Runge-Kutta method for low accuracy requirements but for higher accuracy the latter method is preferred. Only with frequent cell divisions the method with a fixed time step is the best choice.

Place, publisher, year, edition, pages
Elsevier, 2025
Keywords
Center-based model, Time integration, Adaptivity, Gradient system, Numerical methods
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-540948 (URN)10.1016/j.amc.2024.129069 (DOI)001315187300001 ()
Projects
eSSENCE - An eScience Collaboration
Available from: 2024-10-23 Created: 2024-10-23 Last updated: 2025-01-07Bibliographically approved
Lötstedt, P. & Mathias, S. (2022). Adaptive time integration of mechanical forces in center-based models for biological cell populations.
Open this publication in new window or tab >>Adaptive time integration of mechanical forces in center-based models for biological cell populations
2022 (English)Manuscript (preprint) (Other academic)
Abstract [en]

Center-based models are used to simulate the mechanical behavior of biological cells during embryonic development or cancer growth. To allow for the simulation of biological populations potentially growing from a few individual cells to many thousands or more, these models have to be numerically efficient, while being reasonably accurate on the level of individual cell trajectories. In this work, we increase the robustness, accuracy, and efficiency of the simulation of center-based models by choosing the time steps adaptively in the numerical method. We investigate the gain in using single rate time stepping for the forward and backward Euler methods, based on local estimates of the numerical errors and the stability of the method in the case of the explicit forward Euler method. Furthermore,we propose a multirate time stepping scheme that simulates regions with high local force gradients (e.g. as they happen after cell division) with multiple smaller time steps within a larger single time step for regions with smoother forces. These methods are compared for different model systems in numerical experiments. We conclude that the adaptive single rate forward Euler method results in significant gains in terms of reduced wall clock times for the simulation of a linearly growing tissue, while at the same time eliminating the need for manual determination of a suitable time step size.

Keywords
center-based model, time integration, adaptivity, gradient system, numerical methods
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-476430 (URN)10.48550/ARXIV.2206.00339 (DOI)
Funder
eSSENCE - An eScience CollaborationNIH (National Institute of Health), NIH/2R01EB014877-04A1
Available from: 2022-06-13 Created: 2022-06-13 Last updated: 2022-10-31
Lötstedt, P. (2021). Derivation of continuum models from discrete models of mechanical forces in cell populations. Journal of Mathematical Biology, 83(6-7), Article ID 75.
Open this publication in new window or tab >>Derivation of continuum models from discrete models of mechanical forces in cell populations
2021 (English)In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 83, no 6-7, article id 75Article in journal (Refereed) Published
Abstract [en]

In certain discrete models of populations of biological cells, the mechanical forces between the cells are center based or vertex based on the microscopic level where each cell is individually represented. The cells are circular or spherical in a center based model and polygonal or polyhedral in a vertex based model. On a higher, macroscopic level, the time evolution of the density of the cells is described by partial differential equations (PDEs). We derive relations between the modelling on the micro and macro levels in one, two, and three dimensions by regarding the micro model as a discretization of a PDE for conservation of mass on the macro level. The forces in the micro model correspond on the macro level to a gradient of the pressure scaled by quantities depending on the cell geometry. The two levels of modelling are compared in numerical experiments in one and two dimensions.

Place, publisher, year, edition, pages
Springer NatureSpringer Nature, 2021
Keywords
Biomechanics, Cell forces, Coarse-graining, Macroscale, Microscale
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-462440 (URN)10.1007/s00285-021-01697-w (DOI)000728902000001 ()34878601 (PubMedID)
Funder
eSSENCE - An eScience Collaboration
Available from: 2021-12-23 Created: 2021-12-23 Last updated: 2024-01-15Bibliographically approved
Cheng, G., Kirchner, N. & Lötstedt, P. (2021). Sensitivity of ice sheet surface velocity and elevation to variations in basal friction and topography in the full Stokes and shallow-shelf approximation frameworks using adjoint equations. The Cryosphere, 15(2), 715-742
Open this publication in new window or tab >>Sensitivity of ice sheet surface velocity and elevation to variations in basal friction and topography in the full Stokes and shallow-shelf approximation frameworks using adjoint equations
2021 (English)In: The Cryosphere, ISSN 1994-0416, E-ISSN 1994-0424, Vol. 15, no 2, p. 715-742Article in journal (Refereed) Published
Abstract [en]

Predictions of future mass loss from ice sheets are afflicted with uncertainty, caused, among others, by insufficient understanding of spatiotemporally variable processes at the inaccessible base of ice sheets for which few direct observations exist and of which basal friction is a prime example. Here, we present a general numerical framework for studying the relationship between bed and surface properties of ice sheets and glaciers. Specifically, we use an inverse modeling approach and the associated time-dependent adjoint equations, derived in the framework of a full Stokes model and a shallow-shelf/shelfy-stream approximation model, respectively, to determine the sensitivity of grounded ice sheet surface velocities and elevation to time-dependent perturbations in basal friction and basal topography. Analytical and numerical examples are presented showing the importance of including the time-dependent kinematic free surface equation for the elevation and its adjoint, in particular for observations of the elevation. A closed form of the analytical solutions to the adjoint equations is given for a two-dimensional vertical ice in steady state under the shallow-shelf approximation. There is a delay in time between a seasonal perturbation at the ice base and the observation of the change in elevation. A perturbation at the base in the topography has a direct effect in space at the surface above the perturbation, and a perturbation in the friction is propagated directly to the surface in time.

Place, publisher, year, edition, pages
Copernicus PublicationsCOPERNICUS GESELLSCHAFT MBH, 2021
National Category
Physical Geography Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-440412 (URN)10.5194/tc-15-715-2021 (DOI)000619837600002 ()
Funder
Swedish Research Council Formas, 2017-00665
Available from: 2021-04-21 Created: 2021-04-21 Last updated: 2024-01-15Bibliographically approved
Cheng, G., Lötstedt, P. & von Sydow, L. (2020). A full Stokes subgrid scheme in two dimensions for simulation of grounding line migration in ice sheets using Elmer/ICE (v8.3). Geoscientific Model Development, 13, 2245-2258
Open this publication in new window or tab >>A full Stokes subgrid scheme in two dimensions for simulation of grounding line migration in ice sheets using Elmer/ICE (v8.3)
2020 (English)In: Geoscientific Model Development, ISSN 1991-959X, E-ISSN 1991-9603, Vol. 13, p. 2245-2258Article in journal (Refereed) Published
National Category
Computational Mathematics Geosciences, Multidisciplinary
Identifiers
urn:nbn:se:uu:diva-392197 (URN)10.5194/gmd-13-2245-2020 (DOI)000535190100002 ()
Projects
eSSENCE
Available from: 2020-05-13 Created: 2019-08-30 Last updated: 2021-01-08Bibliographically approved
Cheng, G. & Lötstedt, P. (2020). Parameter sensitivity analysis of dynamic ice sheet models — numerical computations. The Cryosphere, 14, 673-691
Open this publication in new window or tab >>Parameter sensitivity analysis of dynamic ice sheet models — numerical computations
2020 (English)In: The Cryosphere, ISSN 1994-0416, E-ISSN 1994-0424, Vol. 14, p. 673-691Article in journal (Refereed) Published
National Category
Computational Mathematics Geosciences, Multidisciplinary
Identifiers
urn:nbn:se:uu:diva-392146 (URN)10.5194/tc-14-673-2020 (DOI)000515168000001 ()
Projects
eSSENCE
Available from: 2020-02-17 Created: 2019-08-29 Last updated: 2021-01-08Bibliographically approved
Cheng, G. & Lötstedt, P. (2019). Parameter sensitivity analysis of dynamic ice sheet models.
Open this publication in new window or tab >>Parameter sensitivity analysis of dynamic ice sheet models
2019 (English)Manuscript (preprint) (Other academic)
National Category
Computational Mathematics Geosciences, Multidisciplinary
Identifiers
urn:nbn:se:uu:diva-392198 (URN)
Available from: 2019-06-19 Created: 2019-08-30 Last updated: 2023-10-26Bibliographically approved
Lötstedt, P. (2019). The linear noise approximation for spatially dependent biochemical networks. Bulletin of Mathematical Biology, 81, 2873-2901
Open this publication in new window or tab >>The linear noise approximation for spatially dependent biochemical networks
2019 (English)In: Bulletin of Mathematical Biology, ISSN 0092-8240, E-ISSN 1522-9602, Vol. 81, p. 2873-2901Article in journal (Refereed) Published
National Category
Computational Mathematics Biochemistry Molecular Biology
Identifiers
urn:nbn:se:uu:diva-351228 (URN)10.1007/s11538-018-0428-0 (DOI)000478772200004 ()29644520 (PubMedID)
Projects
eSSENCE
Available from: 2018-04-11 Created: 2018-05-21 Last updated: 2025-02-20Bibliographically approved
Marchenko, S., Cheng, G., Lötstedt, P., Pohjola, V., Pettersson, R., van Pelt, W. & Reijmer, C. (2019). Thermal conductivity of firn at Lomonosovfonna, Svalbard, derived from subsurface temperature measurements. The Cryosphere, 13, 1843-1859
Open this publication in new window or tab >>Thermal conductivity of firn at Lomonosovfonna, Svalbard, derived from subsurface temperature measurements
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2019 (English)In: The Cryosphere, ISSN 1994-0416, E-ISSN 1994-0424, Vol. 13, p. 1843-1859Article in journal (Refereed) Published
National Category
Physical Geography
Identifiers
urn:nbn:se:uu:diva-334156 (URN)10.5194/tc-13-1843-2019 (DOI)000474653300002 ()
Available from: 2019-07-09 Created: 2017-11-21 Last updated: 2019-09-01Bibliographically approved
van Dongen, E. C. H., Kirchner, N., van Gijzen, M. B., van de Wal, R. S. W., Zwinger, T., Cheng, G., . . . von Sydow, L. (2018). Dynamically coupling full Stokes and shallow shelf approximation for marine ice sheet flow using Elmer/Ice (v8.3). Geoscientific Model Development, 11, 4563-4576
Open this publication in new window or tab >>Dynamically coupling full Stokes and shallow shelf approximation for marine ice sheet flow using Elmer/Ice (v8.3)
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2018 (English)In: Geoscientific Model Development, ISSN 1991-959X, E-ISSN 1991-9603, Vol. 11, p. 4563-4576Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-363123 (URN)10.5194/gmd-11-4563-2018 (DOI)000450295700001 ()
Projects
eSSENCE
Available from: 2018-11-16 Created: 2018-10-12 Last updated: 2019-01-24Bibliographically approved
Projects
Algorithms for simulation of stochastic models in molecular biology [2008-05056_VR]; Uppsala UniversityAlgorithms for simulation of stochastic models in molecular biology [2011-03148_VR]; Uppsala University
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-2143-3078

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