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

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Highlights

  • New approach to the scale separation for the orchestration of mathematical models.

  • Each model represents the phenomenon within a specified portion of space–time.

  • This is based on the resolution of the experimental methods and computational cost.

  • Scale separation of a patient specific personalised cancer treatment model as example.

Abstract

Solid tumour growth depends on a host of factors which affect the cell life cycle and extracellular matrix vascularization that leads to a favourable environment. The whole solid tumour can either grow or wither in response to the action of the immune system and therapeutics. A personalised mathematical model of such behaviour must consider both the intra- and inter-cellular dynamics and the mechanics of the solid tumour and its microenvironment. However, such wide range of spatial and temporal scales can hardly be modelled in a single model, and require the so-called multiscale models, defined as orchestrations of single-scale component models, connected by relation models that transform the data for one scale to another. While multiscale models are becoming common, there is a well-established engineering approach to the definition of the scale separation, e.g., how the spatiotemporal continuum is split in the various component models. In most studies scale separation is defined as natural, linked to anatomical concepts such as organ, tissue, or cell; but these do not provide reliable definition of scales: for examples skeletal organs can be as large as 500 mm (femur), or as small as 3 mm (stapes). Here we apply a recently proposed scale-separation approach based on the actual experimental and computational limitations to a patient-specific model of the growth of neuroblastoma. The resulting multiscale model can be properly informed with the available experimental data and solved in a reasonable timeframe with the available computational resources.

Introduction

Cancer's rising prominence as the second leading cause of death partly reflects the declining mortality rates of stroke and coronary heart disease, relative to cancer, in many countries. There were an estimated 19.3 million new cases and 10 million cancer deaths worldwide in 2020 (Sung, 2021). Therefore, cancer has a significant impact in all human societies. It is a complex and heterogeneous disease due to the variety of biological and mechanical factors at different scales: tumour, stroma, cellular, and subcellular/molecular. In solid tumours the stroma includes connective tissue and blood vessels (Connolly et al., 2021). The occurrence and development of cancer are highly regulated by the biomechanical properties and cellular composition of the tissue microenvironment (Liu et al., 2020). Therefore, it is essential to understand the biomechanical cues that favour the development of a primary tumour from isolated or clustered cancer cells.

Solid tumours in vivo exist in three main stages: the avascular, vascular, and metastatic phases. In the initial phase, namely the avascular stage, the primary mass grows quite rapidly due to cellular replication and the production of extracellular matrix. Beyond a certain size, it starts to compress surrounding tissues and organs. This primary tumour mass can achieve a few millimetres in diameter and its growth is strongly dependent on the mechanical properties of the extracellular microenvironment (Gonçalves and Garcia-Aznar, 2021, Plou et al., 2018). As the tumour grows, the cells at its centre undergo cell death due to a lack of nutrients, forming a necrotic core. However, beyond a certain stage, the tumour can develop its own vasculature by a process of angiogenesis. During this vascular phase, new blood vessels supply the tumour with nutrients and thus, enable rapid tumour growth. During the metastatic phase, some cancer cells migrate from the primary tumour, penetrate blood vessels, and ultimately, colonise distant sites (Escribano, 2019).

Cancer cells can give rise to the above phenomena as an emergent outcome of a number of cellular phenotypic changes, or hallmarks: sustained proliferative signalling and evasion of growth suppressors, resistance to cell death, secretion of molecules inducing angiogenesis, replicative immortality, and their metastatic potential (Hanahan and Weinberg, 2011). For example, many cell cycle proteins such as D-type and E-type cyclins are overexpressed or overactive in cancer cells, leading to uncontrolled proliferation (Otto and Sicinski, 2017). The p53 tumour suppressor that triggers apoptosis in transformed cells is frequently mutated and subverted in cancer cells (Mantovani et al., 2019). Various telomere maintenance mechanisms are associated with aggressive cancer types (including high-risk neuroblastoma) (Ackermann, 2018). High expression of angiogenic factors by the cells in the tumour microenvironment is also common (Jiang, 2020).

If the tumour is left untreated, it grows with a rate dependent on the genetic makeup of the tumour cells, the cell-to-tissue volume ratio (cellularity), and the extent of vascularisation. When treated with chemotherapy or radiotherapy, both the replication rates and cell death rates are altered by the treatment, to an extent that again, depends on the above factors and the actual pharmacokinetics (drug delivery in each part of tumour mass) (Pastor and Mousa, 2019).

Computational models simulating biological processes are widely used to better understand the underlying mechanisms of biological phenomena, including cancer progression (Altrock et al., 2015). There are diverse approaches to model tumour growth, including discrete methods, continuous models, and hybrid models. Discrete models, such as agent-based or Cellular Potts-based approaches, follow the fate of each single cell or each cohort of cells over time. Due to the computational costs associated with implementing these models, they cannot capture aspects of tissue mechanics effectively and they can only model subdomains of the whole tumour (Metzcar et al., 2019). Continuous models describe cancerous tissues as domains composed of multiple phases interacting with each other. Finally, hybrid models incorporate different aspects of discrete and continuous models (Rejniak and Anderson, 2011). Tumour models range from macroscopic models that describe volumetric tumour growth to others that enable simulation of important molecular processes. In this case, a cell’s proliferation and death rates are modulated by its genotype and phenotype, and by the therapeutics that reach the tumour cells. The modulation of cell proliferation and death rates by chemotherapeutic agents is best described in term of signalling pathways within a single cell, e.g., by intracellular models (Kozłowska et al., 2020). The effects of paracrine signalling, cell-to-cell physical interactions, and the local metabolic conditions (most importantly, oxygenation) are best described by multi-cellular models that represent the collective behaviour of a large cellular population (Metzcar et al., 2019). Finally, the biomechanical interactions of the growing tumour with other organs, and the diffusion–reaction of metabolites are best represented at the whole-tumour scale. While in theory, it is possible to describe this entire process with a single mathematical model, in practice, there are limitations due to the resolution of the data used to parameterise the model and the computational power available to solve it numerically. Therefore, such a brute-force approach is impossible, unless the single-scale cancer model is extremely idealised (Bekisz and Geris, 2020). Thus, most models of tumour growth comprise multiple component models, each describing the phenomenon at a specific space–time scale (Vavourakis et al., 2017, Peng et al., 2017, Pourhasanzade and Sabzpoushan, 2021). In this work, we shift our attention from multiscale models to continuum-based models; a detailed review of cancer models can be found here (Deisboeck et al., 2011, Lowengrub, 2010). Continuous models have the potential to absorb patient-specific data, such as those coming from anatomical magnetic resonance imaging, diffusion tensor imaging and perfusion imaging (Angeli et al., 2018). Also, as multiple treatment protocols are made available, it is important to develop so-called Digital Twins, patient-specific computer models capable of predicting how a patient’s tumour will respond to different treatments, thereby enabling the possibility of informing personalised treatment plans.

One major unresolved issue in developing such multiscale models concerns scale separation: how we split a multiscale model of a complex phenomenon into multiple models, each representing the phenomenon at a specific space–time scale. This critical decision is frequently neglected, and a scale separation is frequently adopted without justification, assuming a “natural” scale separation based on vague and qualitative anatomical concepts (cell, tissue, tumour). To the best of the authors’ knowledge, the first paper to raise the issue of scale separation in this context is (Evans, 2008). More recently, a theoretical framework was proposed, but for a much narrower problem (Chakraborty et al., 2014). One of the authors introduced the problem in (Bhattacharya and Viceconti, 2017), and proposed a general approach in (Bhattacharya et al., 2021). This paper uses a similar theoretical approach to analyse the scale separation of a tumour growth model (Norton et al., 2019).

This study aims to explore the scale separation of a new multiscale tumour growth model being developed in the PRIMAGE project (Martí-Bonmatí, 2020) to personalise the treatment of neuroblastoma patients, with the objectives of minimising the model complexity and respecting the experimental resolution and computational constraints that limit scale ranges.

Section snippets

Scale separation

In the following, a scale is defined in terms of grain and extent. The grain is the largest value between the lower limit of spatial/temporal resolution allowed by the instrumentation, and the smallest/fastest feature of interest to be observed. Similarly, the extent is the smallest value between the upper limit of spatial/temporal resolution (i.e., the region of interest in a four-dimensional space) and the size of the largest/slowest feature of interest to be observed. The resolution is the

Discussion

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 (Bhattacharya et al., 2021), 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

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was funded by PRIMAGE (PRedictive In-silico Multiscale Analytics to support cancer personalized diaGnosis and prognosis, empowered by imaging biomarkers), a Horizon 2020|RIA project (Topic SC1-DTH-07-2018), grant agreement no: 826494. JMGA was also supported by the Spanish Ministry of Science, Innovation and Universities (RTI2018-094494-BC21) and the Government of Aragon in the form of grants awarded to SHR (Grant No. 2019-23).

Barbara de Melo Quintela was born in Resende, RJ, Brazil in 1985 and received the MS degree in computational modelling at the Federal University of Juiz de Fora, MG, Brazil in 2011 and the DSc from the same program in 2015. During the last year of the doctorate, she worked as research assistant at the Los Alamos National Laboratory, NM, USA. She became assistant professor at the Federal University of Juiz de Fora in 2018 and has worked as research fellow at the University of Bologna, Italy,

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  • Cited by (0)

    Barbara de Melo Quintela was born in Resende, RJ, Brazil in 1985 and received the MS degree in computational modelling at the Federal University of Juiz de Fora, MG, Brazil in 2011 and the DSc from the same program in 2015. During the last year of the doctorate, she worked as research assistant at the Los Alamos National Laboratory, NM, USA. She became assistant professor at the Federal University of Juiz de Fora in 2018 and has worked as research fellow at the University of Bologna, Italy, from Nov-2019 to Jan-2021. Her research interests include multiscale modelling for in silico medicine and computational immunology.

    Silvia Hervas-Raluy received the B.S. degree in mechanical engineering from the University of Zaragoza, Spain, in 2017 and the M.S. degree in Biomedical engineering from the University of Zaragoza in 2018. She is currently pursuing the Ph.D. degree in Biomedical engineering. From 2016 to 2018, she was a Research Assistant with the Multiscale in Mechanics and Biomedical Engineering. Her research interests focus on cell mechanics and developing computational models in cancer diseases. Mrs. Hervas’s has participated in three different European projects.

    Jose Manuel Garcia-Aznar was born in Zaragoza, Spain in 1970. He received the B.S. and M.S. degrees in industrial engineering from the University of Zaragoza, Spain, in 1995 and the Ph.D. degree in Computational Mechanics from the University of Zaragoza, Spain in 1999. Since 2008, he is full Professor of Structural Mechanics with the Mechanical Engineering Department, University of Zaragoza, Spain. In these years, he has been visiting researcher at Keele University (2001), KU Leuven (2012), Cambridge University (2015), NUI Galway (2017) and University of Oxford (2019). In 2004, he was elected as Council Member of the European Society of Biomechanics (ESB) (2004-2012), and finally as Vice-President (2008-2012). He has published over 160 peer-reviewed papers and 20 book chapters. His research interests focus on computational modelling of hard tissues mechanics, mechanobiology of skeletal tissue regeneration and tissue engineering, tissue growth and development and cell mechanics. Most recently his research work has also focused on the combination of computational models and microfluidics-based experiments in order to investigate the mechanisms that regulate tumour growth and metastasis.

    Kenneth Y. Wertheim identifies as a global citizen with the United Kingdom as their passport country. They received the MEng degree in chemical engineering from Imperial College London, the United Kingdom, in 2012; the MS degree in chemical engineering from Columbia University, New York, the United State of America, in 2014; and the PhD degree from the University of Southampton, the United Kingdom, in 2017. As an undergraduate, they spent the academic year 2010–11 on exchange at the University of Sydney, Australia. From 2017 to 2019, they were a Research Associate in the Department of Biochemistry, University of Nebraska–Lincoln. Since 2019, they have been a Research Associate with the Insigneo Institute for in Silico Medicine and Department of Computer Science, the University of Sheffield, the United Kingdom. Earlier in their career, they interned at the Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, La Plata, Argentina, in 2011; and the Chinese University of Hong Kong, HKSAR, in 2007. Their major interests include mathematical modelling, scientific computing, complex systems, systems biology, developmental biology, immunology, and oncology. Dr Wertheim was honoured with the 2020 International Intellectual Benefits to Society Award by the Mensa Foundation.

    Dr Dawn Walker completed a BSc in Physics (Hons.) at the University of Durham, UK in 1996 and a Ph.D. in Medical Physics at the University of Sheffield, UK in 2001. She subsequently worked as a postdoctoral researcher where she developed computational models focussing on cellular interactions. Interests in cellular-based and multiscale modelling have been extended through an RCUK fellowship, then lectureship based in the Department of Computer Science in Sheffield with funding from Arthritis Research-UK, Wellcome Trust and EU Ho2020. She is now a Senior Lecturer in the Computer Science and the Insigneo Institute of in silico Medicine, with research projects focussing on the development of cellular- and multi-scale models, particularly in the field of cancer.

    Marco Viceconti is full professor of Computational Bioengineering in the department of Industrial Engineering of the Alma Mater Studiorum – University of Bologna, and Director of the Medical Technology Lab of the Rizzoli Institute. Before he was at the University of Sheffield, UK, where he founded and led for seven years the prestigious Insigneo Institute for in silico Medicine. Prof Viceconti is an expert in In Silico Trials, the use of subject-specific modelling to test new medical products. He is one of the key figures in the in silico medicine international community: he founded the VPH Institute, an international no-profit organisation that coordinates this research community, and drove the creation of the Avicenna Alliance, which represent the biomedical industry interests in this domain. According to SCOPUS he published 355 papers (H-index = 50).

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