Abstract
Mesoscale modeling is often considered merely as a practical strategy used when information on lower-scale details is lacking, or when there is a need to make models cognitively or computationally tractable. Without dismissing the importance of practical constraints for modeling choices, we argue that mesoscale models should not just be considered as abbreviations or placeholders for more “complete” models. Because many systems exhibit different behaviors at various spatial and temporal scales, bottom-up approaches are almost always doomed to fail. Mesoscale models capture aspects of multi-scale systems that cannot be parameterized by simple averaging of lower-scale details. To understand the behavior of multi-scale systems, it is essential to identify mesoscale parameters that “code for” lower-scale details in a way that relate phenomena intermediate between microscopic and macroscopic features. We illustrate this point using examples of modeling of multi-scale systems in materials science (steel) and biology (bone), where identification of material parameters such as stiffness or strain is a central step. The examples illustrate important aspects of a so-called “middle-out” modeling strategy. Rather than attempting to model the system bottom-up, one starts at intermediate (mesoscopic) scales where systems exhibit behaviors distinct from those at the atomic and continuum scales. One then seeks to upscale and downscale to gain a more complete understanding of the multi-scale system. The cases highlight how parameterization of lower-scale details not only enables tractable modeling but is also central to understanding functional and organizational features of multi-scale systems.
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Source: Batterman (2012)
After Selinger (2015)
After: Nemat-Nasser and Hori (1999)
Figure republished with permission of The Royal Society (U.K.), from (Sabet et al. 2016), permission conveyed through the Copyright Clearance Center
Figure republished with permission of The Royal Society (U.K.), from (Cristofolini et al. 2008), permission conveyed through the Copyright Clearance Center
Figure courtesy of James C. Weaver
Figure republished with permission of The Royal Society (U.K.), from (Cristofolini et al. 2008), permission conveyed through the Copyright Clearance Center
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Notes
Epstein and Forber (2013) critically discuss the merits of common assumptions about microfoundational models, but primarily consider the role of higher-level models and parameters as means to tweak parameters to improve model fit. For a discussion of microfoundationalist vs. macroscale models, see also (Neal in preparation).
MacLeod and Nersessian (2018) seem to indicate this when highlighting that the representations in mesoscale models “do not capture accurately the control structures of biological networks” as they are highly abstract and simplified representations of systems that likely have a loose relationship with underlying system mechanisms” (p. 17). Whereas mechanistic details indeed are important for some predictive purposes (as in their example), predictability and explanatory power often do not rest on getting the lower-scale details right (Batterman 2002). We therefore aim to clarify the importance of identifying mesoscale features and parameters that—within certain boundaries—are relatively insensitive to lower-scale changes.
A central assumption in hierarchy theory is that many complex systems are hierarchically organized in a way that makes higher-scale levels independent of some lower-scale details. A hierarchy of dynamically uncoupled models can hence be constructed through considerations of which lower-level details are necessary to include when upscaling from lower to higher scales.
We have more to say about this below.
Stoneham and Harding (2003, p. 77) highlight that the attempt to span length scales often results in the “worst of both regimes”, and that a simpler strategy will often suffice. Although their article offers resistance to the aim of “bridging” between scales, the article supports our argument that the mesoscopic features are critical for many phenomena of practical and scientific interests (and that starting from a lower-scale models is often not feasible or strategic). We agree with this and hope that our examples will shine further light on why mesoscales are often the most relevant starting point of analysis.
The number of publications on multi-scale modeling in medicine is growing rapidly. A recent search for the term ‘multiscale’ in the PubMed database (03-05-2020) gave 10,861 hits (among these 3682 with ‘multiscale’ in the title). In March 2016, the numbers were 5457 and 2180, respectively (Bhattacharya and Viceconti 2017).
“An engineer also selects the level and detail of simulation he needs according to the problem he is tackling. It is not necessary to understand all the molecules in order to model and construct a bridge, for example” (Noble 2006, p. 81).
A detailed discussion of representative volume elements will be provided in Sect. 3. For now, simply think of them as mesoscale regions that statistically represent the features of composite systems taken to be important at that scale.
The equations treat systems as continuous blobs with no structure all the way down to the infinitesimal. Of course, materials are composed of atoms, and have mesoscale structures that those equations completely ignore.
For another discussion of this autonomy and the idea of relatively autonomous levels of reality see (Chibbaro et al. 2014).
This is a rather extreme assumption. For example, it rules out the possibility of the buckling/failure of the beam. A full-on approach would start with the Navier-Cauchy equations. Nevertheless, our purpose here is to focus on the role of the material parameters as the right variables that reflect lower scale details in the relevant representative volume element. (See below for details.) In addition, in the next section on bone, accounting for fracture is of extreme importance.
Shear refers to situations where applied stress results in internal structures sliding past one another. Shear is particularly important in materials with network structures such as bone where shearing of internal structure protects the overall structure from breaking up to certain limits (see below).
A classical definition of the RVE is "a sample that (a) is structurally entirely typical of the whole mixture on average, and (b) contains a sufficient number of inclusions for the effective overall moduli to be effectively independent of the surface values of traction and displacement, so long as these values are macroscopically uniform" (Hill 1963).
Think of randomly dropping line segments of length r throughout the RVE and seeing if the endpoints are in the same material phase.
Of course, the actual railroad track is heterogeneous at lower scales. The aim of homogenization is to show that there could be an equivalent homogeneous system that will exhibit the same behavior as the actual system. Since the continuum equations do not recognize any heterogeneities, this is required to explain why those equations actually work to describe the behavior of the actual system. We are not reifying the “fictitious” homogeneous system.
A few models do start at the lowest scales displayed on Fig. 4. For instance, a five-step homogenization procedure has been used to show that volume fraction of different constituents at the microscale can be used to predict bone stiffness, and that the elastic properties of the basic constituents of bone are universal (Fritsch and Hellmich 2007). As clarified below, it is however necessary to account for higher-scale microstructures to predict fracture risk, which is influenced by many other features than material stiffness. Moreover, the employment of RVEs and advanced homogenization strategies clarify the validity of phenomenological models and connections between scales.
Whereas stiffness is a measure of a material’s resistance to deformation in response to an applied force, toughness is a measure of the capacity of a material to absorb energy without breaking. Toughness is quantified as the area under a stress–strain curve.
Another interesting example is how epithelial branching in the context of mammary organogenesis is not only determined by the stiffness of the extracellular matrix but also by the local and differential organization of collagen fibers (Barnes et al. 2014; Montévil et al. 2016). We would like to thank an anonymous reviewer for bringing this example to our attention.
Elastic moduli have been show to differ for the femur neck, femur midshaft, and femur head (Novitskaya et al. 2011). Generally, experimental values for elastic moduli of bone are dependent on features such as porosity (or bone type), mineralization, age, trabecular architecture (see Sect. 4), but unfortunately also on the testing method and sampling size. The development of predictive models of bone fracture and strength is therefore a hard problem in biology.
The cross-link density is higher in bones of older individuals. This lowers the ability of bone to dissipate elastically stored energy from an applied load before breaking (Ritchie et al. 2009).
Finite element methods refer to computational discretization strategies commonly used to find approximate solutions for complex mathematical problems and (like here) to develop 2D or 3D models of systems. A finite element mesh consists of a number of finite element subunits representing a block of the material (or a subdivision of a mathematical problem). As clarified in Section 4, development of FE models often involves complex homogenization strategies. For other studies using FEM to predict bone behavior, see (Sabet et al. 2016) and references therein.
Thus, the tissue model is considered as a mesoscale model from the perspective of the organ model, whereas the organ model is the intermediate model from the perspective of the whole body model. This underscores Noble’s point that the term “mesoscale” is relative to the modeling task at hand.
Coupling of cellular and continuum scales is a persistent challenge, and most approaches to date rely on hierarchical modeling strategies that selectively use inputs from lower-scale model but do not fully integrate these in a multiscale model where inputs go in both directions (up and down) (Sabet et al. 2016). Section 4 offers insights to some advances in multiscale modeling since the publication of this model.
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Acknowledgements
We would like to thank the organizers of the Active Materials Project Summer School and Workshop (AMP2) in Georgetown, DC, in July, 2018, and the participants for stimulating discussions. In particular, we thank Patrick McGivern for taking the initiative to edit this special issue. We greatly appreciate helpful feedback from Julia Bursten and two anonymous reviewers on an earlier version of this paper.
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Batterman, R.W., Green, S. Steel and bone: mesoscale modeling and middle-out strategies in physics and biology. Synthese 199, 1159–1184 (2021). https://doi.org/10.1007/s11229-020-02769-y
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DOI: https://doi.org/10.1007/s11229-020-02769-y