Mathematical modelling of avascular ellipsoidal tumour growth
Breast cancer is the most frequently diagnosed cancer in women. From mammography, Magnetic Resonance Imaging (MRI), and ultrasonography, it is well documented that breast tumours are often ellipsoidal in shape. The World Health Organisation (WHO) has established a criteria based on tumour volume change for classifying response to therapy. Typically the volume of the tumour is measured on the hypothesis that growth is ellipsoidal. This is the Calliper method, and it is widely used throughout the world. This paper initiates an analytical study of ellipsoidal tumour growth based on the pioneering mathematical model of Greenspan. Comparisons are made with the more commonly studied spherical mathematical models. 1. Tumour biology. Cell proliferation is normally a highly regulated process, such that only the required numbers of cells populate a given tissue. If control of proliferation is altered or lost, cells may continue to divide leading to an abnormal mass of tissue – a tumour. The most common cause of tumours is genetic mutation resulting in uncontrolled cell division. Although a single mutation can account for this loss of control, it is far more common for a series of mutations in a number of genes to accumulate, eventually resulting in loss of proliferative control. This increased cell mass can be due to increased cell division or a decrease in programmed cell death, which normally occurs as part of limiting cell numbers, or a combination of both. Two classes of genes that are commonly mutated in tumours are oncogenes and tumour suppressors. Oncogenes are mutated forms of proto-oncogenes,which normally encode Received March 18, 2010. 2010 Mathematics Subject Classification. Primary 92C05, 92C50. c ©2011 Brown University 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf 2 G. DASSIOS, F. KARIOTOU, M. N. TSAMPAS, AND B. D. SLEEMAN proteins involved in growth promoting signal transduction and mitogenesis. Oncogenes are more active, hence increasing the rate of proliferation. Tumour suppressor genes, as their name suggests, are normally involved in slowing cell growth and division; mutations release this control, again increasing proliferation. Mutations in either class of gene, or in both, result in reduced control of cell growth and division, giving these cells a growth advantage over neighbouring healthy cells and leading to the development of a tumour. Tumours can be benign or malignant, dependent on their aggressiveness. A benign tumour stays as a noninvasive cluster, without spreading into its surroundings. The margin of the tumour is very distinct and the whole tumour can usually be removed by surgery. By aquiring more mutations, a benign tumour can become malignant. Malignant tumours grow aggressively and invade into surrounding tissue. Tumour cells that break away from the parent tumour, and move via lymphatic or blood vessels to a distant site to form secondary tumours or metastases, are characteristic of a malignant tumour. Most tumours start as a small mass of rapidly proliferating cells, where nutrients and oxygen are acquired by passive diffusion from the surrounding tissues, the size of the tumour is limited to about 2mm in diameter and tumours can stay in this diffusion limited state, where cell proliferation is balanced by cell death, for months or even years. It was over thirty years ago that Judah Folkman [10] first proposed that in order to develop beyond this dormant state, the tumour must induce the growth of new blood vessels in order to supply the increasing metabolic demands. At that time, the mechanism of new vessel growth, called angiogenesis, was not known, but Folkman suggested that the switch in a tumour towards a pro-angiogenic state is a specific stage in tumour development. Once the tumour becomes vascularised, diffusion no longer limits size, and the tumour can grow and develop. It is now known that control of angiogenesis is orchestrated by a large number of pro-angiogenic and anti-angiogenic factors, and it is the shift in balance from antito pro-angiogenic that elicits the so-called angiogenic switch and induces growth of blood vessels towards the tumour. Many factors contribute to the switch towards angiogenesis, one of which is oxygen deficiency within the tumour. Tissues deprived of oxygen become hypoxic, and express a range of factors to help them survive, some of which are proangiogenic factors, driving the growth of new blood vessels towards that tissue. This whole orchestration of complex events leads to a micro-vascular structure that eventually reaches and penetrates the tumor, vastly improving its blood supply and allowing for rapid and unconstrained growth. For an up-to-date account of the biochemistry of tumour angiogenesis, we refer to Plank and Sleeman [14] as well as the references [3], [13] and [16]. In this paper we consider the growth of avascular tumours, the first step being in understanding the growth of complex processes involved in angiogenesis and vascular structures. There exist in the literature several mathematical models of avascular tumour growth. These include (i) models that describe continuum cell populations and their growth by considering the interactions between cell density and the chemical species that provide nutrients as well as inhibitors, (ii) models that describe mechanical interactions between tumour cells and their surroundings and (iii) individual cell based models that allows one to track cells in both space and time. From in vitro experiments and License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf MATHEMATICAL MODELLING OF AVASCULAR ELLIPSOIDAL TUMOUR GROWTH 3 some observed in vivo studies, it is known that avascular tumours may grow as symmetric spheroids wherein growth is essentially radial. In this situation many of the above mathematical models admit to analytical treatment and enable one to determine cell movement, track the spheroidal boundary and to assess the roles of growth inhibitors and growth promoters. The stability of tumour growth can also be analysed; see [12] for a review and cited references. A solid mass growing in healthy tissue produces stress. Models [11] have been developed to study this form of mechanical stress in which the tumour deforms the surrounding tissue due to the stress it imposes on the environment, and the environment in turn alters tumour growth dynamics. In these models, tumour growth inhibition depends on the stiffness of the surrounding environment. In an in vivo setting this corresponds to the stiffness of the extracellular matrix environment. In an in vitro setting, this corresponds to the stiffness of the agarose gel. The effects of physical confinement on tumour growth have been studied experimentally. In [8], human colon adenocarcinoma cells were grown in cylindrical glass tubes with a radius that was much smaller than the length of the tube. It was found that cell aggregates in 0.7% gel placed in a capillary tube grew to take on an ellipsoidal shape driven by the geometry of the capillary tube. The same cells grown outside a capillary tube developed into a spherical shape. This experiment highlights that geometric confinement alters the shape and growth dynamics of a developing tumour. Mathematical models which treat avascular tumours as visco-elastic materials and discuss the effects of mechanical stress are considered in [4, 15]. Breast cancer is the most frequently diagnosed cancer in women. From mammography, magnetic resonance imaging (MRI) and ultrasonography it is well documented that breast tumours are often ellipsoidal in shape. Indeed the World Health Organisation (WHO) established in 1979 criteria based on tumour volume change for classifying response to therapy as progressive disease, partial recovery or no change. Typically the volume of the tumour is measured on the hypothesis that growth is ellipsoidal. This is the socalled calliper method and is widely used throughout the world in assessing and grading gliomas. See [2, 5, 17]. In this paper we initiate an analytical study of ellipsoidal tumour growth based on the pioneering mathematical model of avascular tumour growth due to Greenspan [6, 7, 15]. In section 2 we formulate the mathematical model in terms of ellipsoidal geometry and explicitly solve for the pressure field and nutrient concentration within a growing avascular ellipsoidal tumour made up of a live cell layer, a quiescent layer and a necrotic core. Because the analysis depends extensively on the use of Lamé functions, their relevant properties together with a description of ellipsoidal coordinates are outlined in Appendix A. Section 3 contains a resumé of the well-known spherical tumour problem and emphasises the modelling differences with the Greenspan model. In section 4 we carry out a number of numerical simulations. The paper concludes in section 5 with a discussion of the results. 2. The mathematical model. The tumour is assumed to have a three-layer structure consisting of a thin outer layer of live proliferating cells that envelops an inner layer of quiescent live but not proliferating cells which in turn envelops a large necrotic core of License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf 4 G. DASSIOS, F. KARIOTOU, M. N. TSAMPAS, AND B. D. SLEEMAN dead cells and debris. Cells proliferate as long as the available concentration of nutrient supply, denoted by σ(r, t), remains above a critical level σ1. A cell dies due to apoptosis or otherwise when σ falls below a critical level σ2. In the quiescent region, nutrient supply varies over the interval σ2 ≤ σ ≤ σ1. The characteristic thickness s of the layer of live proliferating cells depends on σ1 and the value of σ at the outer surface of the tumour. It is assumed that the tumour boundaries evolve as members 