A systematic approach to the scale separation problem in the development of multiscale models

Abstract

Throughout engineering there are problems where it is required to predict a quantity based on the measurement of another, but where the two quantities possess characteristic variations over vastly different ranges of time and space. Among the many challenges posed by such ‘multiscale’ problems, that of defining a ‘scale’ remains poorly addressed. This fundamental problem has led to much confusion in the field of biomedical engineering in particular. The present study proposes a definition of scale based on measurement limitations of existing instruments, available computational power, and on the ranges of time and space over which quantities of interest vary characteristically. The definition is used to construct a multiscale modelling methodology from start to finish, beginning with a description of the system (portion of reality of interest) and ending with an algorithmic orchestration of mathematical models at different scales within the system. The methodology is illustrated for a specific but well-researched problem. The concept of scale and the multiscale modelling approach introduced are shown to be easily adaptable to other closely related problems. Although out of the scope of this paper, we believe that the proposed methodology can be applied widely throughout engineering.

A theoretical analysis of the scale separation in a model to predict solid tumour growth

“…The aim of this study was to find the scale separation of a new multiscale tumour growth model that minimises the modelling complexity, while respecting the experimental resolution and computational constraints that limit the scale ranges. To this end, we used an approach first proposed in [27], which tackles the problem by considering a multiscale model as an engineering construct, optimised on the basis of the experimental and computational limitations imposed by the available methods, rather than on the basis of abstract mathematical considerations.…”

“…However, space-time is a continuum, and the decision to partition it into separate scales must be made based on the model's purpose, the resolution of the instrumentation used to inform and validate the model, and the computational power available. The approach we propose to define scale separation, first described in [27], requires as the first step, the definition of a mathematical model that provides the infinite resolution idealisation of the multiscale model being developed.…”

“…More recently, a theoretical framework was proposed, but for a much narrower problem [25]. One of the authors introduced the problem in [26], and proposed a general approach in [27]. This paper uses a similar theoretical approach to analyse the scale separation of a tumour growth model [28].…”

A stochastic hierarchical model for low grade glioma evolution

“…As the variables involved in these interactions are fast-acting compared to the macroscopic scale, we make use of a scale separation variable and the Hilbert expansion method to derive the corresponding macroscopic scale equation for the time and space variables (for a more general discussion of multiscale modelling and moment closure techniques, we refer the reader to Bhattacharya et al. ( 2021 ), Kuehn ( 2016 ), Hunt ( 2018 )).…”

Effect of particularisation size on the accuracy and efficiency of a multiscale tumours' growth model

“…For example, with reference to the modelling of solid tumours growth, the space-time scales to span are 10 À8 to 10 À1 m, and 10 À2 to 10 7 s. 1 Hereinafter, we define range of a model, the largest portion of space or time the model accounts for, and grain, the smallest portion of it. 2 The grain is frequently referred to also as the representative volume element (RVE). A neuroblastoma can reach volumes as large as 4,188,790 mm 3 (a sphere of 20 cm of diameter).…”