Mathematical biomedicine and modeling avascular tumor growth

In this chapter we review existing continuum models of avascular tumor growth, explaining howthey are inter related and the biophysical insight that they provide. The models range in complexity and include one-dimensional studies of radiallysymmetric growth, and two-dimensional models of tumor invasion in which the tumor is assumed to comprise a single population of cells. We also present more detailed, multiphase models that allow for tumor heterogeneity. The chapter concludes with a summary of the different continuum approaches and a discussion of the theoretical challenges that lie ahead

Multiscale mechano-morphology of soft tissues : a computational study with applications to cancer diagnosis and treatment

Cooperation of engineering and biomedical sciences has produced significant advances in healthcare technology. In particular, computational modelling has led to a faster development and improvement of diagnostic and treatment techniques since it allows exploring multiple scenarios without additional complexity and cost associated to the traditional trial-and-error methodologies. The goal of this thesis is to propose computational methodologies to analyse how the changes in the microstructure of soft tissues, caused by different pathological conditions, influence the mechanical properties at higher length scales and, more importantly, to detect such changes for the purpose of quantitative diagnosis and treatment of such pathologies in the scenario of drug delivery. To achieve this objective different techniques based on quasi-static and dynamic probing have been established to perform quantitative tissue diagnosis at the microscopic (tissue) and macroscopic (organ) scales. The effects of pathologies not only affect the mechanical properties of tissue (e.g. elasticity and viscoelasticity), but also the transport properties (e.g. diffusivity) in the case of drug delivery. Such transport properties are further considered for a novel multi-scale, patient-specific framework to predict the efficacy of chemotherapy in soft tissues. It is hoped that this work will pave the road towards non-invasive palpation techniques for early diagnosis and optimised drug delivery strategies to improve the life quality of patientsJames-Watt Scholarship, Heriot-Watt Annual Fund and the Institute of Mechanical, Process and Energy Engineering (IMPEE) Grant

Elastic free energy drives the shape of prevascular solid tumors

It is well established that the mechanical environment influences cell functions in health and disease. Here, we address how the mechanical environment influences tumor growth, in particular, the shape of solid tumors. In an in vitro tumor model, which isolates mechanical interactions between tumor cells and a hydrogel, we find that tumors grow as ellipsoids, resembling the same, oft-reported observation of in vivo tumors. Specifically, an oblate ellipsoidal tumor shape robustly occurs when the tumors grow in hydrogels that are stiffer than the tumors, but when they grow in more compliant hydrogels they remain closer to spherical in shape. Using large scale, nonlinear elasticity computations we show that the oblate ellipsoidal shape minimizes the elastic free energy of the tumor-hydrogel system. Having eliminated a number of other candidate explanations, we hypothesize that minimization of the elastic free energy is the reason for predominance of the experimentally observed ellipsoidal shape. This result may hold significance for explaining the shape progression of early solid tumors in vivo and is an important step in understanding the processes underlying solid tumor growth.Comment: Six figures in main text. Supporting Information with 6 additional figure

On an inverse problem from magnetic resonance elastic imaging

The imaging problem of elastography is an inverse problem. The nature of an inverse problem is that it is ill-conditioned. We consider properties of the mathematical map which describes how the elastic properties of the tissue being reconstructed vary with the field measured by magnetic resonance imaging (MRI). This map is a nonlinear mapping, and our interest is in proving certain conditioning and regularity results for this operator which occurs implicitly in this problem of imaging in elastography. In this treatment we consider the tissue to be linearly elastic, isotropic, and spatially heterogeneous. We determine the conditioning of this problem of function reconstruction, in particular for the stiffness function. We further examine the conditioning when determining both stiffness and density. We examine the Frechet derivative of the nonlinear mapping, which enables us to describe the properties of how the field affects the individual maps to the stiffness and density functions. We illustrate how use of the implicit function theorem can considerably simplify the analysis of Frechet differentiability and regularity properties of this underlying operator. We present new results which show that the stiffness map is mildly ill-posed, whereas the density map suffers from medium ill-conditioning. Computational work has been done previously to study the sensitivity of these maps, but our work here is analytical. The validity of the Newton-Kantorovich and optimization methods for the computational solution of this inverse problem is directly linked to the Frechet differentiability of the appropriate nonlinear operator, which we justify

A two-fluid model for tissue growth within\ud a dynamic flow environment

We study the growth of a tissue construct in a perfusion bioreactor, focussing on its response to the mechanical environment. The bioreactor system is modelled as a two-dimensional channel containing a tissue construct through which a flow of culture medium is driven. We employ a multiphase formulation of the type presented by G. Lemon, J. King, H. Byrne, O. Jensen and K. Shakesheff in their study (Multiphase modelling of tissue growth using the theory of mixtures. J. Math. Biol. 52(2), 2006, 571–594) restricted to two interacting fluid phases, representing a cell population (and attendant extracellular matrix) and a culture medium, and employ the simplifying limit of large interphase viscous drag after S. Franks in her study (Mathematical Modelling of Tumour Growth and Stability. Ph.D. Thesis, University of Nottingham, UK, 2002) and S. Franks and J. King in their study (Interactions between a uniformly proliferating tumour and its surrounding: Uniform material properties. Math. Med. Biol. 20, 2003, 47–89).\ud \ud The novel aspects of this study are: (i) the investigation of the effect of an imposed flow on the growth of the tissue construct, and (ii) the inclusion of a mechanotransduction mechanism regulating the response of the cells to the local mechanical environment. Specifically, we consider the response of the cells to their local density and the culture medium pressure. As such, this study forms the first step towards a general multiphase formulation that incorporates the effect of mechanotransduction on the growth and morphology of a tissue construct. The model is analysed using analytic and numerical techniques, the results of which illustrate the potential use of the model to predict the dominant regulatory stimuli in a cell population

Modeling the growth of multicellular cancer spheroids in a\ud bioengineered 3D microenvironment and their treatment with an\ud anti-cancer drug

A critical step in the dissemination of ovarian cancer cells is the formation of multicellular spheroids from cells shed from the primary tumor. The objectives of this study were to establish and validate bioengineered three-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitro and simultaneously to develop computational models describing the growth of multicellular spheroids in these bioengineered matrices. Cancer cells derived from human epithelial ovarian carcinoma were embedded within biomimetic hydrogels of varying stiffness and cultured for up to 4 weeks. Immunohistochemistry was used to quantify the dependence of cell proliferation and apoptosis on matrix stiffness, long-term culture and treatment with the anti-cancer drug paclitaxel.\ud \ud Two computational models were developed. In the first model, each spheroid was treated as an incompressible porous medium, whereas in the second model the concept of morphoelasticity was used to incorporate details about internal stresses and strains. Each model was formulated as a free boundary problem. Functional forms for cell proliferation and apoptosis motivated by the experimental work were applied and the predictions of both models compared with the output from the experiments. Both models simulated how the growth of cancer spheroids was influenced by mechanical and biochemical stimuli including matrix stiffness, culture time and treatment with paclitaxel. Our mathematical models provide new perspectives on previous experimental results and have informed the design of new 3D studies of multicellular cancer spheroids

An investigation of the influence of extracellular matrix anisotropy and cell-matrix interactions on tissue architecture

Received: 9 September 2014 / Revised: 21 May 2015. Published online: 02 September 2015Mechanical interactions between cells and the fibrous extracellular matrix (ECM) in which they reside play a key role in tissue development. Mechanical cues from the environment (such as stress, strain and fibre orientation) regulate a range of cell behaviours, including proliferation, differentiation and motility. In turn, the ECM structure is affected by cells exerting forces on the matrix which result in deformation and fibre realignment. In this paper we develop a mathematical model to investigate this mechanical feedback between cells and the ECM. We consider a three-phase mixture of collagen, culture medium and cells, and formulate a system of partial differential equations which represents conservation of mass and momentum for each phase. This modelling framework takes into account the anisotropic mechanical properties of the collagen gel arising from its fibrous microstructure. We also propose a cell-collagen interaction force which depends upon fibre orientation and collagen density. We use a combination of numerical and analytical techniques to study the influence of cell-ECM interactions on pattern formation in tissues. Our results illustrate the wide range of structures which may be formed, and how those that emerge depend upon the importance of cell-ECM interactions.R. J. Dyson, J. E. F. Green, J. P. Whiteley, H. M. Byrn