SofTMech Work Package WP3 Progress Report
N. Hill, R. Ogden, M. Chaplain, P. Stewart, P. Watton, H. Yin, M. Olson, X. Luo,
D. Husmeier, A. Melnik, R. Barry
WP3. Upscaling from Cells to Tissue
3.1 Micro-meso upscaling & reduced-order modelling tools
(RA Dr Andrey Melnik, supervised by Xiaoyu Luo, Ray Ogden)
Work done so far
A theoretical generalized structure tensor (GST) model has been developed for which fibre dispersion has been incorporated into the coupling invariant I8 in order to provide a consistent approach to each term of the basic Holzapfel-Ogden myocardium model. This has been completed and published . The theory is now being implemented numerically and will be applied to the fitting of mechanical data from experiments on myocardial tissue. A new postdoc, Dr. Yangkun Du was appointed in August 2020 to continue this work.
An interesting connection property between stress components for different deformations has been identified which applies, in particular, to orthotropic constitutive models (such as those used for the myocardium) that are of separable form. This can be used as a basis for deciding whether currently used models are consistent with data that have the connection property and therefore whether the models require modification. A paper dealing with this topic was published in 2019 .
In recent work  the novel effect of collagen fibre cross links at the microscale has been accounted for in a macroscopic model of artery tissue. A model for damage accruing in collagen fibres within tissues during deformation has also been developed .
Our plans for the next several months are:
With the grant coming to an end on 31st July 2021 Dr. Yangkun Du will move to an RA position with SofTMechMP to provide continuity as the new Centre-to-Centre collaboration moves forward.
To work on implementation of the model  and prepare a paper in which the theory is fitted to myocardium data, with the help of Dr Melnik, who has now left Glasgow but will continue to contribute to this project.
Additionally, a detailed review of elasticity models that incorporate growth and remodelling for fibrous soft tissue, begun by Dr Melnik, has made good progress and will be prepared for publication.
3.2 Growth and remodelling - meso to macroscale
PhD student Roxanna Barry (supervised by SofTMech CIs Stewart and Hill) has now completed her studies and graduated in Dec 2020. During her PhD she developed new techniques for rationally upscaling one-dimensional arrays of individual rectangular cell models (from WP1) and derived new macroscale continuum models for use in WP5. This model includes nonlinear hyperelastic and visoelastic models for each cell, and allow the cells to deform, grow and divide. In response to forces exerted by their neighbours, cells move along a substrate, to which they bind resulting in a `Stokes-like’ drag proportional to their speed. The discrete models have been solved numerically to evaluate scaling laws for the growth and proliferation of the line of cells. Using discrete-to-continuum asymptotic methods, corresponding upscaled PDEs have been derived from the discrete models. Analytical and numerical solutions for these novel, rationally-derived PDEs show excellent agreement with the discrete models. The PDEs are much faster to solve numerically than the discrete models for large numbers of cells and provide constitutive laws for the tissue-scale continuum models used in other work packages. A paper will be submitted in the next few weeks. Furthermore, Roxanna did some preliminary analysis of 2D tissue layer models, which will be continued by a future PhD student.
 A. Melnik, X.Y. Luo and R.W. Ogden, A generalized structure tensor model for the mixed invariant I8. International Journal of Non-Linear Mechanics 107 (2018), 137-148.
 A. Melnik, X.Y. Luo and R.W. Ogden, A para-universal relation for orthotropic materials. Mechanics Research Communications 97 (2019), 46-51.
 G.A. Holzapfel and R.W. Ogden, An arterial constitutive model accounting for collagen content and cross-linking. Journal of the Mechanics and Physics of Solids 136 (2020), 103682.
 G.A. Holzapfel and R.W. Ogden, A damage model for collagen fibers with an application to collagenous soft tissues. Proceedings of the Royal Society of London A 476 (2020), 20190821.
Upscaling from cells to tissue (update: May 2018)
Structures in soft tissue are rarely periodic, yet many upscaling techniques use homogenisation based on this periodic assumption. We will develop improved upscaling methods independent of the assumption of periodicity and validate them against specifically-designed cellular and tissue experiments.
Project 1: Micro to mesoscale framework and Meso to macroscale framework
Project 2: Growth & Remodelling framework
Project 3: Evaluation of models by development of modules with Simulia (link to the company) and Ansys (link to the company)