1
|
Nikas AN, Curcio EJ, Nascone-Yoder N, Lubkin SR. Morphoelastic models discriminate between different mechanisms of left-right asymmetric stomach morphogenesis. Cells Dev 2024; 177:203902. [PMID: 38281683 DOI: 10.1016/j.cdev.2024.203902] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2023] [Revised: 01/19/2024] [Accepted: 01/19/2024] [Indexed: 01/30/2024]
Abstract
The mechanisms by which the vertebrate stomach undergoes its evolutionarily conserved leftward bending remain incompletely understood. Although the left and right sides of the organ are known to possess different gene expression patterns and undergo distinct morphogenetic events, the physical mechanisms by which these differences generate morphological asymmetry remain unclear. Here, we develop a continuum model of asymmetric stomach morphogenesis. Using a morphoelastic framework, we investigate the morphogenetic implications of a variety of hypothetical, tissue-level growth differences between the left and right sides of a simplified tubular organ. Simulations reveal that, of the various differential growth mechanisms tested, only one category is consistent with the leftward stomach curvature observed in wild-type embryos: equal left and right volumetric growth rates, coupled with transversely isotropic tissue thinning on the left side. Simulating this mechanism in a defined region of the model over a longer period of growth leads to mature stomach-like curvatures.
Collapse
|
2
|
Peng Q, Vermolen FJ, Weihs D. Physical confinement and cell proximity increase cell migration rates and invasiveness: A mathematical model of cancer cell invasion through flexible channels. J Mech Behav Biomed Mater 2023; 142:105843. [PMID: 37104897 DOI: 10.1016/j.jmbbm.2023.105843] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/23/2022] [Revised: 03/28/2023] [Accepted: 04/07/2023] [Indexed: 04/29/2023]
Abstract
Cancer cell migration between different body parts is the driving force behind cancer metastasis, which is the main cause of mortality of patients. Migration of cancer cells often proceeds by penetration through narrow cavities in locally stiff, yet flexible tissues. In our previous work, we developed a model for cell geometry evolution during invasion, which we extend here to investigate whether leader and follower (cancer) cells that only interact mechanically can benefit from sequential transmigration through narrow micro-channels and cavities. We consider two cases of cells sequentially migrating through a flexible channel: leader and follower cells being closely adjacent or distant. Using Wilcoxon's signed-rank test on the data collected from Monte Carlo simulations, we conclude that the modelled transmigration speed for the follower cell is significantly larger than for the leader cell when cells are distant, i.e. follower cells transmigrate after the leader has completed the crossing. Furthermore, it appears that there exists an optimum with respect to the width of the channel such that cell moves fastest. On the other hand, in the case of closely adjacent cells, effectively performing collective migration, the leader cell moves 12% faster since the follower cell pushes it. This work shows that mechanical interactions between cells can increase the net transmigration speed of cancer cells, resulting in increased invasiveness. In other words, interaction between cancer cells can accelerate metastatic invasion.
Collapse
Affiliation(s)
- Qiyao Peng
- Mathematical Institute, Faculty of Science, Leiden University, Neils Bohrweg 1, 2333 CA, Leiden, The Netherlands.
| | - Fred J Vermolen
- Computational Mathematics Group, Department of Mathematics and Statistics, Faculty of Science, University of Hasselt, 3590 Diepenbeek, Belgium
| | - Daphne Weihs
- Faculty of Biomedical Engineering, Technion-Israel Institute of Technology, 3200003 Haifa, Israel
| |
Collapse
|
3
|
Walker BJ, Celora GL, Goriely A, Moulton DE, Byrne HM. Minimal Morphoelastic Models of Solid Tumour Spheroids: A Tutorial. Bull Math Biol 2023; 85:38. [PMID: 36991173 PMCID: PMC10060352 DOI: 10.1007/s11538-023-01141-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2022] [Accepted: 03/03/2023] [Indexed: 03/31/2023]
Abstract
Tumour spheroids have been the focus of a variety of mathematical models, ranging from Greenspan's classical study of the 1970 s through to contemporary agent-based models. Of the many factors that regulate spheroid growth, mechanical effects are perhaps some of the least studied, both theoretically and experimentally, though experimental enquiry has established their significance to tumour growth dynamics. In this tutorial, we formulate a hierarchy of mathematical models of increasing complexity to explore the role of mechanics in spheroid growth, all the while seeking to retain desirable simplicity and analytical tractability. Beginning with the theory of morphoelasticity, which combines solid mechanics and growth, we successively refine our assumptions to develop a somewhat minimal model of mechanically regulated spheroid growth that is free from many unphysical and undesirable behaviours. In doing so, we will see how iterating upon simple models can provide rigorous guarantees of emergent behaviour, which are often precluded by existing, more complex modelling approaches. Perhaps surprisingly, we also demonstrate that the final model considered in this tutorial agrees favourably with classical experimental results, highlighting the potential for simple models to provide mechanistic insight whilst also serving as mathematical examples.
Collapse
Affiliation(s)
- Benjamin J Walker
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK.
- Department of Mathematics, University College London, Gordon Street, London, WC1H 0AY, UK.
| | - Giulia L Celora
- Department of Mathematics, University College London, Gordon Street, London, WC1H 0AY, UK
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK
| | - Alain Goriely
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK
| | - Derek E Moulton
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK
| | - Helen M Byrne
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK
| |
Collapse
|
4
|
Egberts G, Vermolen F, van Zuijlen P. Stability of a two-dimensional biomorphoelastic model for post-burn contraction. J Math Biol 2023; 86:59. [PMID: 36964257 PMCID: PMC10038978 DOI: 10.1007/s00285-023-01893-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2022] [Revised: 02/24/2023] [Accepted: 02/27/2023] [Indexed: 03/26/2023]
Abstract
We consider the stability analysis of a two-dimensional model for post-burn contraction. The model is based on morphoelasticity for permanent deformations and combined with a chemical-biological model that incorporates cellular densities, collagen density, and the concentration of chemoattractants. We formulate stability conditions depending on the decay rate of signaling molecules for both the continuous partial differential equations-based problem and the (semi-)discrete representation. We analyze the difference and convergence between the resulting spatial eigenvalues from the continuous and semi-discrete problems.
Collapse
Affiliation(s)
- Ginger Egberts
- Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands.
- Research Group Computational Mathematics (CMAT), Department of Mathematics and Statistics, University of Hasselt, Hasselt, Belgium.
| | - Fred Vermolen
- Research Group Computational Mathematics (CMAT), University of Hasselt, Hasselt, Belgium
- Data Science Institute (DSI), University of Hasselt, Hasselt, Belgium
| | - Paul van Zuijlen
- Burn Centre and Department of Plastic, Reconstructive and Hand Surgery, Red Cross Hospital, Beverwijk, The Netherlands
- Department of Plastic, Reconstructive and Hand Surgery, Amsterdam UMC, location VUmc, Amsterdam Movement Sciences, Amsterdam, The Netherlands
- Pediatric Surgical Centre, Emma Children's Hospital, Amsterdam UMC, location AMC and VUmc, Amsterdam, The Netherlands
| |
Collapse
|
5
|
Egberts G, Desmoulière A, Vermolen F, van Zuijlen P. Sensitivity of a two-dimensional biomorphoelastic model for post-burn contraction. Biomech Model Mechanobiol 2023; 22:105-121. [PMID: 36229698 PMCID: PMC9957927 DOI: 10.1007/s10237-022-01634-w] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2022] [Accepted: 09/05/2022] [Indexed: 11/02/2022]
Abstract
We consider a two-dimensional biomorphoelastic model describing post-burn scar contraction. This model describes skin displacement and the development of the effective Eulerian strain in the tissue. Besides these mechanical components, signaling molecules, fibroblasts, myofibroblasts, and collagen also play a significant role in the model. We perform a sensitivity analysis for the independent parameters of the model and focus on the effects on features of the relative surface area and the total strain energy density. We conclude that the most sensitive parameters are the Poisson's ratio, the equilibrium collagen concentration, the contraction inhibitor constant, and the myofibroblast apoptosis rate. Next to these insights, we perform a sensitivity analysis where the proliferation rates of fibroblasts and myofibroblasts are not the same. The impact of this model adaptation is significant.
Collapse
Affiliation(s)
- Ginger Egberts
- Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands. .,Research Group Computational Mathematics (CMAT), Department of Mathematics and Statistics, University of Hasselt, Hasselt, Belgium.
| | - Alexis Desmoulière
- grid.9966.00000 0001 2165 4861Department of Physiology, and EA 6309, Faculty of Pharmacy, University of Limoges, Limoges, France
| | - Fred Vermolen
- grid.12155.320000 0001 0604 5662Research Group Computational Mathematics (CMAT), Department of Mathematics and Statistics, University of Hasselt, Hasselt, Belgium
| | - Paul van Zuijlen
- grid.415746.50000 0004 0465 7034Burn Centre and Department of Plastic, Reconstructive and Hand Surgery, Red Cross Hospital, Beverwijk, The Netherlands ,grid.509540.d0000 0004 6880 3010Department of Plastic, Reconstructive and Hand Surgery, Amsterdam UMC, location VUmc, Amsterdam Movement Sciences, Amsterdam, The Netherlands ,grid.5650.60000000404654431Pediatric Surgical Centre, Emma Children’s Hospital, Amsterdam UMC, location AMC and VUmc, Amsterdam, The Netherlands
| |
Collapse
|
6
|
Peng Q, Gorter WS, Vermolen FJ. Comparison between a phenomenological approach and a morphoelasticity approach regarding the displacement of extracellular matrix. Biomech Model Mechanobiol 2022; 21:919-935. [PMID: 35403944 PMCID: PMC9132877 DOI: 10.1007/s10237-022-01568-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2021] [Accepted: 02/25/2022] [Indexed: 11/30/2022]
Abstract
Plastic (permanent) deformations were earlier, modeled by a phenomenological model in Peng and Vermolen (Biomech Model Mechanobiol 19(6):2525–2551, 2020). In this manusctipt, we consider a more physics-based formulation that is based on morphoelasticity. We firstly introduce the morphoelasticity approach and investigate the impact of various input variables on the output parameters by sensitivity analysis. A comparison of both model formulations shows that both models give similar computational results. Furthermore, we carry out Monte Carlo simulations of the skin contraction model containing the morphoelasticity approach. Most statistical correlations from the two models are similar, however, the impact of the collagen density on the severeness of contraction is larger for the morphoelasticity model than for the phenomenological model.
Collapse
Affiliation(s)
- Q Peng
- Mathematical Institute, Leiden University, 2333 CA, Niels Bohrweg, The Netherlands. .,Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands. .,Computational Mathematics Group, Discipline group Mathematics and statistics, Faculty of Science, Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, BE 3590, Diepenbeek, Belgium.
| | - W S Gorter
- Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands
| | - F J Vermolen
- Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands.,Computational Mathematics Group, Discipline group Mathematics and statistics, Faculty of Science, Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, BE 3590, Diepenbeek, Belgium
| |
Collapse
|
7
|
Egberts G, Vermolen F, van Zuijlen P. Stability of a one-dimensional morphoelastic model for post-burn contraction. J Math Biol 2021; 83:24. [PMID: 34355270 PMCID: PMC8342404 DOI: 10.1007/s00285-021-01648-5] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2020] [Revised: 06/29/2021] [Accepted: 07/19/2021] [Indexed: 01/02/2023]
Abstract
To deal with permanent deformations and residual stresses, we consider a morphoelastic model for the scar formation as the result of wound healing after a skin trauma. Next to the mechanical components such as strain and displacements, the model accounts for biological constituents such as the concentration of signaling molecules, the cellular densities of fibroblasts and myofibroblasts, and the density of collagen. Here we present stability constraints for the one-dimensional counterpart of this morphoelastic model, for both the continuous and (semi-) discrete problem. We show that the truncation error between these eigenvalues associated with the continuous and semi-discrete problem is of order \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${{\mathcal {O}}}(h^2)$$\end{document}O(h2). Next we perform numerical validation to these constraints and provide a biological interpretation of the (in)stability. For the mechanical part of the model, the results show the components reach equilibria in a (non) monotonic way, depending on the value of the viscosity. The results show that the parameters of the chemical part of the model need to meet the stability constraint, depending on the decay rate of the signaling molecules, to avoid unrealistic results.
Collapse
Affiliation(s)
- Ginger Egberts
- Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands. .,Research Group Computational Mathematics (CMAT), Department of Mathematics and Statistics, University of Hasselt, Hasselt, Belgium.
| | - Fred Vermolen
- Research Group Computational Mathematics (CMAT), Department of Mathematics and Statistics, University of Hasselt, Hasselt, Belgium
| | - Paul van Zuijlen
- Burn Centre, Department of Plastic, Reconstructive and Hand Surgery, Red Cross Hospital, Beverwijk, The Netherlands.,Department of Plastic, Reconstructive and Hand Surgery, Amsterdam UMC, Location VUmc, Amsterdam Mov ement Sciences, Amsterdam, The Netherlands.,Pediatric Surgical Centre, Emma Children's Hospital, Amsterdam UMC, Location AMC and VUmc, Amsterdam, The Netherlands
| |
Collapse
|
8
|
Egberts G, Vermolen F, van Zuijlen P. Sensitivity and feasibility of a one-dimensional morphoelastic model for post-burn contraction. Biomech Model Mechanobiol 2021; 20:2147-2167. [PMID: 34331622 PMCID: PMC8595192 DOI: 10.1007/s10237-021-01499-5] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2021] [Accepted: 07/18/2021] [Indexed: 01/13/2023]
Abstract
We consider a one-dimensional morphoelastic model describing post-burn scar contraction. Contraction can lead to a limited range of motion (contracture). Reported prevalence of burn scar contractures are 58.6% at 3-6 weeks and 20.9% at 12 months post-reconstructive surgery after burns. This model describes the displacement of the dermal layer of the skin and the development of the effective Eulerian strain in the tissue. Besides these components, the model also contains components that play a major role in the skin repair after trauma. These components are signaling molecules, fibroblasts, myofibroblasts, and collagen. We perform a sensitivity analysis for many parameters of the model and use the results for a feasibility study. In this study, we test whether the model is suitable for predicting the extent of contraction in different age groups. To this end, we conduct an extensive literature review to find parameter values. From the sensitivity analysis, we conclude that the most sensitive parameters are the equilibrium collagen concentration in the dermal layer, the apoptosis rate of fibroblasts and myofibroblasts, and the secretion rate of signaling molecules. Further, although we can use the model to simulate significant distinct contraction densities in different age groups, our results differ from what is seen in the clinic. This particularly concerns children and elderly patients. In children we see more intense contractures if the burn injury occurs near a joint, because the growth induces extra forces on the tissue. Elderly patients seem to suffer less from contractures, possibly because of excess skin.
Collapse
Affiliation(s)
- Ginger Egberts
- Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands. .,Research Group Computational Mathematics (CMAT), Department of Mathematics and Statistics, University of Hasselt, Hasselt, Belgium.
| | - Fred Vermolen
- Research Group Computational Mathematics (CMAT), Department of Mathematics and Statistics, University of Hasselt, Hasselt, Belgium
| | - Paul van Zuijlen
- Burn Centre and Department of Plastic, Reconstructive & Hand Surgery, Red Cross Hospital, Beverwijk, The Netherlands.,Department of Plastic, Reconstructive & Hand Surgery, Amsterdam Movement Sciences, Amsterdam UMC, location VUmc, Amsterdam, The Netherlands.,Pediatric Surgical Centre, Emma Children's Hospital, Amsterdam UMC, location AMC and VUmc, Amsterdam, The Netherlands
| |
Collapse
|
9
|
Almet AA, Byrne HM, Maini PK, Moulton DE. The role of mechanics in the growth and homeostasis of the intestinal crypt. Biomech Model Mechanobiol 2020; 20:585-608. [PMID: 33219879 PMCID: PMC7979635 DOI: 10.1007/s10237-020-01402-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2020] [Accepted: 11/03/2020] [Indexed: 11/29/2022]
Abstract
We present a mechanical model of tissue homeostasis that is specialised to the intestinal crypt. Growth and deformation of the crypt, idealised as a line of cells on a substrate, are modelled using morphoelastic rod theory. Alternating between Lagrangian and Eulerian mechanical descriptions enables us to precisely characterise the dynamic nature of tissue homeostasis, whereby the proliferative structure and morphology are static in the Eulerian frame, but there is active migration of Lagrangian material points out of the crypt. Assuming mechanochemical growth, we identify the necessary conditions for homeostasis, reducing the full, time-dependent system to a static boundary value problem characterising a spatially heterogeneous "treadmilling" state. We extract essential features of crypt homeostasis, such as the morphology, the proliferative structure, the migration velocity, and the sloughing rate. We also derive closed-form solutions for growth and sloughing dynamics in homeostasis, and show that mechanochemical growth is sufficient to generate the observed proliferative structure of the crypt. Key to this is the concept of threshold-dependent mechanical feedback, that regulates an established Wnt signal for biochemical growth. Numerical solutions demonstrate the importance of crypt morphology on homeostatic growth, migration, and sloughing, and highlight the value of this framework as a foundation for studying the role of mechanics in homeostasis.
Collapse
Affiliation(s)
- A A Almet
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.,NSF-Simons Center for Multiscale Cell Fate Research, University of California, Irvine, Irvine, CA, USA.,Department of Mathematics, University of California, Irvine, Irvine, CA, USA
| | - H M Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - P K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - D E Moulton
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
| |
Collapse
|
10
|
Topol H, Demirkoparan H, Pence TJ. On collagen fiber morphoelasticity and homeostatic remodeling tone. J Mech Behav Biomed Mater 2021; 113:104154. [PMID: 33158790 DOI: 10.1016/j.jmbbm.2020.104154] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2019] [Revised: 03/14/2020] [Accepted: 10/21/2020] [Indexed: 11/23/2022]
Abstract
A variety of biochemical and physical processes participate in the creation and maintenance of collagen in biological tissue. Under mechanical stimuli these collagen fibers undergo continuous processes of morphoelastic change. The model presented here is motivated by experimental reports of stretch-stabilization of the collagen fibers to enzymatic degradation. The fiber structure is modeled in terms of a fiber density evolution that is regulated by means of a fixed creation rate and a mechano-sensitive dissolution rate. The theory accounts for the possibly different natural configurations of the fiber unit constituents and the ground substance matrix. It also generalizes previous theoretical descriptions so as to account for finite survival times of the individual fiber units. Special consideration is given to steady state fiber-remodeling processes in which fiber creation and dissolution are in balance. Fiber assembly processes that involve prestretching the fiber constituents yield a homeostatic stress response with a characteristic fiber tone. Fiber density returns to homeostasis after mechanical disruption when sufficient time has passed.
Collapse
|
11
|
Abstract
We consider mechanically-induced pattern formation within the framework of a growing, planar, elastic rod attached to an elastic foundation. Through a combination of weakly nonlinear analysis and numerical methods, we identify how the shape and type of buckling (super- or subcritical) depend on material parameters, and a complex phase-space of transition from super- to subcritical is uncovered. We then examine the effect of heterogeneity on buckling and post-buckling behaviour, in the context of a heterogeneous substrate adhesion, elastic stiffness, or growth. We show how the same functional form of heterogeneity in different properties is manifest in a vastly differing post-buckled shape. Finally, a fourth form of heterogeneity, an imperfect foundation, is incorporated and shown to have a more dramatic impact on the buckling instability, a difference that can be qualitatively understood via the weakly nonlinear analysis.
Collapse
|
12
|
Menon SN, Hall CL, McCue SW, McElwain DLS. A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices. Biomech Model Mechanobiol 2017; 16:1743-1763. [PMID: 28523375 DOI: 10.1007/s10237-017-0917-3] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/28/2016] [Accepted: 05/03/2017] [Indexed: 11/26/2022]
Abstract
The mechanical behaviour of solid biological tissues has long been described using models based on classical continuum mechanics. However, the classical continuum theories of elasticity and viscoelasticity cannot easily capture the continual remodelling and associated structural changes in biological tissues. Furthermore, models drawn from plasticity theory are difficult to apply and interpret in this context, where there is no equivalent of a yield stress or flow rule. In this work, we describe a novel one-dimensional mathematical model of tissue remodelling based on the multiplicative decomposition of the deformation gradient. We express the mechanical effects of remodelling as an evolution equation for the effective strain, a measure of the difference between the current state and a hypothetical mechanically relaxed state of the tissue. This morphoelastic model combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. A novel feature of our model is that while most models describe growth as a continuous quantity, here we begin with discrete cells and develop a continuum representation of lattice remodelling based on an appropriate limit of the behaviour of discrete cells. To demonstrate the utility of our approach, we use this framework to capture qualitative aspects of the continual remodelling observed in fibroblast-populated collagen lattices, in particular its contraction and its subsequent sudden re-expansion when remodelling is interrupted.
Collapse
Affiliation(s)
- Shakti N Menon
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai, 600113, India
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia
- Institute of Health and Biomedical Innovation, Queensland University of Technology, Brisbane, QLD, 4001, Australia
| | - Cameron L Hall
- Mathematics Applications Consortium with Science and Industry, University of Limerick, Castletroy, Limerick, V94 T9PX, Ireland
- Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, UK
| | - Scott W McCue
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia.
| | - D L Sean McElwain
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia
- Institute of Health and Biomedical Innovation, Queensland University of Technology, Brisbane, QLD, 4001, Australia
| |
Collapse
|
13
|
Koppenol DC, Vermolen FJ. Biomedical implications from a morphoelastic continuum model for the simulation of contracture formation in skin grafts that cover excised burns. Biomech Model Mechanobiol 2017; 16:1187-1206. [PMID: 28181018 PMCID: PMC5511621 DOI: 10.1007/s10237-017-0881-y] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2016] [Accepted: 01/25/2017] [Indexed: 12/20/2022]
Abstract
A continuum hypothesis-based model is developed for the simulation of the (long term) contraction of skin grafts that cover excised burns in order to obtain suggestions regarding the ideal length of splinting therapy and when to start with this therapy such that the therapy is effective optimally. Tissue is modeled as an isotropic, heterogeneous, morphoelastic solid. With respect to the constituents of the tissue, we selected the following constituents as primary model components: fibroblasts, myofibroblasts, collagen molecules, and a generic signaling molecule. Good agreement is demonstrated with respect to the evolution over time of the surface area of unmeshed skin grafts that cover excised burns between outcomes of computer simulations obtained in this study and scar assessment data gathered previously in a clinical study. Based on the simulation results, we suggest that the optimal point in time to start with splinting therapy is directly after placement of the skin graft on its recipient bed. Furthermore, we suggest that it is desirable to continue with splinting therapy until the concentration of the signaling molecules in the grafted area has become negligible such that the formation of contractures can be prevented. We conclude this study with a presentation of some alternative ideas on how to diminish the degree of contracture formation that are not based on a mechanical intervention, and a discussion about how the presented model can be adjusted.
Collapse
Affiliation(s)
- Daniël C Koppenol
- Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands.
| | - Fred J Vermolen
- Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands
| |
Collapse
|