Modelling the growth of solid tumours and …majc/chaplain_sleeman_jmb...Modelling the growth of...

J. Math. Biol. (1993) 31:431-473 Journal of Mathematical

Biology © Springer-Verlag 1993

Modelling the growth of solid tumours and incorporating a method for their classification using nonfinear elasticity theory

M. A. J. Chaplain*, B. D. Sleeman Department of Mathematics and Computer Science, The University of Dundee, Dundee DD1 4HN, Scotland, UK

Received September 3, 1991; received in revised form March 3, 1992

Abstract. Medically, tumours are classified into two important classes- benign and malignant. Generally speaking, the two classes display different behaviour with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumour growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid turnout using membrane and thick-shell theory. A central feature of the model is the characterisation of the material composition of the model through the use of a strain-energy function, thus permitting a mathematical description of the degree of differentiation of the tumour explicitly in the model. Conditions are given in terms of the strain-energy function for the processes of invasion and metastasis occurring in a tumour, being interpreted as the bifurcation modes of the spherical shell which the tumour is essentially modelled as. Our results are compared with actual experimental results and with the general behaviour shown by benign and malignant tumours. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumours (in particular, the Gaussian and mean curvatures of the surface of a solid tumour) in an attempt to produce a mathematical formulation and description of the important medical processes of staging and grading cancers. We hope that this approach may form the basis of a practical application.

Key words: Solid tumour g rowth- Nonlinear elasticity theory - Strain-energy function - Invasion - Metastasis

1 Introduction

Tumours can arise from the cells of virtually all types of tissue, and the diversity of origin is largely responsible for the great variety of structural appearances of tumours. This diversity, along with the complexity and multi-faceted nature of in vivo tumour growth, is reflected in the many mathematical and theoretical

* Current address: School of Mathematical Sciences, The University of Bath, Claverton Down, Bath BA2 7AY, UK

432 M . A . J . Chaplain, B. D. Sleeman

models which have been used in an attempt to describe tumour growth and spread. A wide variety of techniques encompassing several mathematical disci- plines, from applied analysis to computer generated models, has been used to tackle the problem.

One of the simplest approaches is to view the turnout as a population of cells characterised by some variable, N, say, where N is a measure of the size of the tumour. An ordinary differential equation in N is then formulated to describe the basic tumour growth curve. In particular the growth of solid tumours appears to closely follow a Gompertzian growth law and good agreement has been obtained between the predictions of the model and the available experimental data (cf. Laird 1964; Aroesty et al. 1973; Norton et al. 1976). A more sophisticated approach to this type of growth in turnouts has recently been proposed by Gyllenberg and Webb (1989). In their model two different types of cells exist within the turnout, namely proliferating ceils and quiescent cells. Thus, quiescence is proposed to explain the characteristic Gompertz-type growth curves exhibited by many solid tumours.

Other mathematical models, working in ideal geometries, include the diffusion of a growth inhibitor which controls the growth of the tumour, and the diffusion of a vital nutrient, such as oxygen (cf. Burton 1966; Greenspan 1974, 1976; Shymko and Glass 1976; Adam 1986).

Finally, models which employ a stochastic approach at the cell level have also been used in describing the long-term rate of spread of cancer cells (Williams and Bjerknes 1972; Richardson 1973).

In this paper we propose a new approach to the problem of modelling the growth of a solid tumour using non-linear elasticity theory. With the aid of membrane and thick-shell theory, we hope to incorporate the salient features of solid tumour growth into our model. By associating a particular strain-energy function with the tumour (in effect charaeterising the material that the turnout is composed of), not only is the growth of the tumour simulated, but also, explicit in the model, a theoretical basis is provided for describing, from a mathematical point of view, the difference between the two important classes of tumours from a medical standpoint - benign and malignant. We hope to show that the type of material which goes to make up the tumour as well as the total number of live, proliferating cells within the tumour determine whether or not the processes of invasion and metastasis take place. By linking these results with certain aspects of the surface geometry/surface morphogenesis of solid turnouts, we also hope to describe in a mathematical way the medical processes of staging and grading of cancers, and hope that our theoretical suggestions may be of future practical assistance to those in the medical profession, furnishing them with more objective criteria upon which to make their decisions.

As well as being applicable to the complete tumour (i.e. on a "global" level) we believe that this approach could also be helpful at the cellular level (i.e. "local" level), by helping to describe mathematically the marked differences which exist between normal cells and cancer cells. In the following section, we describe the differences between normal and neoplastic tissue and cell growth. In Sect. 3, after a description of the underlying theory and techniques of nonlinear elasticity which will be used in the subsequent sections, we introduce our model for the growth of a solid turnout. Section 4 introduces the medical terms of staging and grading cancers and attempts to describe these mathematically using techniques from differential geometry, while in Sect. 5 various concluding remarks are made and shortcomings of the model discussed. Finally, several

Modelling the growth of solid tumours 433

appendices are included which contain certain mathematical details omitted from the main body of the paper.

2 Neoplastic disease and the growth of tumours

The size of an organ and the number of cells it contains are normally maintained at approximately constant and optimal values by the activity of control mechanisms that regulate the mitotic activity of cells. The control mechanisms are geared to permit repair of many tissues when they are injured, to allow continuous proliferation and replacement of the cells that undergo continuous wear and tear (e.g. the cells of the intestinal mucosa or of the skin), and hyperplasia of tissues in response to increased functional requirements (hyperplasia is an excess of proliferation of cells occurring in, for example, the liver, the salivary glands, pancreas, thyroid, adrenal cortex and ovaries).

In contrast, neoplasia is a state in which the control mechanisms become deficient and an excessive proliferation of cells continues indefinitely without relation to normal growth and tissue repair. The neoplasia, (literally, new growth), gives rise to a neoplasm, that is, an abnormal tissue mass or tumour, (from the Latin tumor = swelling), consisting of the tumour cells and their products and also some blood vessels and supporting stroma. The abnormal rate of cell proliferation may range from slightly above normal to wild, uncontrolled growth in which the neoplasm extends into adjacent tissues causing serious damage. Although cell multiplication in neoplasms is abnormally excessive, it rarely exceeds the fastest rate of normal multiplication of which some tissues are capable, for example, during embryonic development, and even in the adult in the regeneration of cells of the intestinal mucosa, blood and skin. The nature of the change which makes tumour cells behave thus is not fully understood, but it is irreversible and persists after the removal of the causal factors which have induced the changes leading to the development of a tumour from what was originally a normal cell. Such neoplastic change is also heritable in the sense that when the tumour divides, it produces more tumour cells of the same type.

Since they exhibit varying degrees of uncontrolled proliferation, tumours are sometimes called autonomous, but they are, of course, dependent on the host for their blood supply, nutrition and supporting stroma. Also, their escape from the mechanisms which control proliferation of normal cells is not always complete, for the growth of some turnouts can be retarded or arrested by changes in their hormonal environment. Thus, in summary, we may define a tumour as follows:

A tumour is a mass of tissue formed as a result of abnormal, excessive and inappropriate (i.e. purposeless) proliferation of cells, the growth of which continues indefinitely and regardless of the mechanisms which control normal cellular proliferation.

There are certain other types of abnormally rapid growth which, however, are generally excluded from the above definition of a neoplastic tumour. Thus, reactive growth responses to clearly defined chronic irritant stimuli are not considered to be true turnouts. These are described as metaplasias if there is a change from one cell type to another (usually the change is from more specialised to less specialised cells), and dysplasias where there is a disorganisation of the pattern of the squamous epithelium in tissues such as the skin, oesophagous and uterus in response to chronic irritation or inflammation (the

434 M.A.J. Chaplain, B. D. Sleeman

cells increase in number with a thickening of the epithelium). Both metaplasias and dysplasias are usually reversible if the source of chronic irritation is removed. However, if they persist, they may ultimately progress to neoplasias i.e. they are often precancerous states (cf. Melicow 1982).

2.1 The classification of tumours

Tumours can be conveniently classified into two types: benign (or simple) and malignant, although these really represent the extremes of a spectrum of characteristics of tumour growth and tumours occur which combine the properties of both types. Some tumours may slowly change from one type to the other. However, as a general rule, benign tumours are highly differentiated, their cells are usually uniform, grow slowly and, by definition, remain localised, neither invading the adjacent tissues nor giving rise to secondary tumours (metastases) elsewhere in the body. Malignant tumours, on the other hand, are usually less well differentiated, their cells tend to grow rapidly, to show differences in size and shape and, by definition, invade the tissues locally and spread to other parts of the body (metastasise) where secondary tumours may develop. Thus it can be seen that benign and malignant tumours differ in three main areas: their degree of differentiation, their rate of growth and the manner in which they grow. These three areas are now elaborated upon in the following sections.

(I) Differentiation A tumour is said to be highly differentiated when its structure bears a close resemblance to the tissue of origin. This requires that the tumour cells should resemble the adult cells of the tissue of origin in their morphology, in their arrangement in relation to one another and to the stroma and blood vessels, and in their functional activities. For example, a well-differentiated tumour of thyroid epithelium forms follicles, produces and stores thyroglobulin, and may secrete thyroid hormone. A well-differentiated tumour of fat cells (a lipoma) is remark- ably similar, both grossly and microscopically, to normal adipose tissue. In both these examples, as well as retaining the functions of the original, non-neoplastic cells, the tumours are readily distinguishable from the non-neoplastic thyroid or adipose tissue only because they form a discrete mass. However, even benign tumours are often imperfectly differentiated.

At the other extreme, some malignant turnouts show no recognisable attempt at differentiation, and are termed anaplastic - the individual tumour cells have a primitive, undifferentiated appearance, producing a highly cellular mass in which little attempt at forming special structures or cell products can be discerned, so that it is often not possible to determine their histogenesis. Anaplastic tumours are nearly always highly malignant, but most malignant turnouts show some evidence of differentiation, even though it is usually much less perfect than in benign tumours. For example, the cells of malignant tumours derived from lipoblasts (adipose tissue stem cells) often produce and store droplets of fat.

Not only do the cells of malignant tumours fail to differentiate fully, but they also vary abnormally in shape and size (in the same tumour), in their arrangement in relation to one another, in having relatively large, sometimes distorted, nuclei and a high nuclear/cytoplasmic ratio. These changes, and also their mitotic activity, are sometimes collectively termed cellular atypia. They are, in general, most pronounced in highly malignant, poorly differentiated cancers in


which cellular variability (pleomorphism) is sometimes extreme. This can be seen in certain tumours in which some of the cells are enormous and contain a single, very large nucleus or multiple nuclei. Such cells have arisen by replication of the cellular DNA without the completion of mitosis. Abnormal mitoses e.g. with unequal division of DNA between the daughter cells or with the formation of three mitotic spindles instead of two, are also observed in highly malignant tumours.

(II) Rate of growth The rate of increase in the number of tumour cells depends on the rate of cell production and the rate of cell loss. The rate of cell production depends on the number of cells undergoing mitosis and on the time they take to complete the cell cycle. The proportion of cells seen to be in mitosis in histological sections of tumours (the mitotic index) is a useful guide to the rate of cell production, but it can be misleading for it does not take account of the cell cycle time which varies considerably for different tumours. For example, there are some tumours e.g. basal cell carcinomata of the epidermis, in which mitoses are numerous and yet the tumour grows very slowly, partly because the cell cycle time is long. Using techniques such as microdensitometry or microfluorimetry, it is possible to measure the nuclear DNA of individual cells and hence an indication of the number of cells in the S and M phases of the cell cycle. However, in pleomorphic tumours, a high DNA content does not necessarily mean that a cell is in the mitotic cycle and this gives rise to an error.

More accurate estimates of cell production can be obtained experimentally upon administration of colchicine, vincristine or radio-active (tritiated) thymidine. Using these methods, it has been shown that malignant tumours in experimental animals show logarithmic growth at first, but as the tumour enlarges, cell loss increases and the rate of growth gradually slows down. This increase is partly due to the cells of malignant tumours being abnormal and having a rather short average lifespan, which varies for individual tumours. Another cause of cell loss is ischaemia- as a tumour enlarges, it tends to outgrow its blood supply, particularly if growth is rapid, as in many malignant tumours. Ischaemia inhibits mitoses and causes the death of individual cells and groups of cells. In some rapidly growing tumours, extensive necrosis is seen macroscopically in the central part of the tumour, and microscopically in those cells which lie furthest from small blood vessels. By the time a human cancer is discovered, it has usually passed the initial period of logarithmic growth and the rate has slowed down considerably because of increasing cell loss. It follows that, even if the present rate of growth can be accurately assessed, it cannot be used to calculate the age of the tumour. In some cancers, for example, the basal cell carcinoma of the epidermis, many cells undergo shrinkage necrosis (apoptosis) and the high rate of loss is another reason why this tumour grows very slowly in spite of a high mitotic index.

(III) The spread of tumours Benign tumours proliferate locally and grow by expansion: they compress the surrounding tissue, causing atrophy and disappearance of its cells. The stroma of the surrounding tissue is more resistant and becomes condensed to form a fibrous capsule around the tumour and this may increase in thickness as a result of desmoplastic reaction stimulated by the tumour. The edge of the tumour is well defined. Some benign tumours have little or no capsule and yet the margin


between the tumour and the surrounding tissue remains sharp, without evidence of local invasion. Clinical disorders produced by benign tumours arise mainly from mechanical effects, such as obstruction of viscera or pressure on nerves and organs (e.g. the brain). A benign tumour may remain in situ for years without causing ill effects and if detected it is easily removed from its capsule by surgery.

Malignant turnouts (cancers) grow both by expansion and by infiltrating the surrounding tissues. They are not incapsulated and their edges are ill-defined. Projections of tumour cells extend from the central mass into surrounding tissues like the legs of a crab (the word cancer in fact comes from the Latin for crab). Their cells also invade the walls of the lymphatics and blood vessels in and around the tumour and are carried away to other parts of the body where they may proliferate, giving rise to secondary tumours or metastases. Malignant tumours can be eliminated if they are recognised and successfully removed by surgery before the stage of metastasis. However, the aggressive behaviour of malignant tumours presents the major obstacle to their complete removal, and once metastatic spread has occurred, surgical removal of the primary turnout is usually to no avail. A typical example of a malignant tumour is lung cancer which spreads from the lung tissue to the lymph nodes and then metastasises in the brain, bone, liver and other organs.

Although there are exceptions, it is a general rule that malignant tumours which are poorly differentiated tend to grow rapidly, invade extensively and metastasise early. More highly differentiated malignant tumours tend to behave less aggressively, but individual types of tumour show differences in their behaviour, and a detailed knowledge is necessary to draw conclusions from the histological features on the likely behaviour of each particular type of tumour.

The contrasting features of benign and malignant turnouts are summarised in Table 1.

Table 1. Contrasting features of benign and malignant tumours

Benign Malignant

(i) Evidence of rapid growth Mitoses Few and normal Nuclei Little altered

Nucleoli Little altered Cytoplasmic basophilia Slight Haemorrhage and necrosis Rare

(ii) Differentiation Naked-eye resemblance to tissue Often close

of origin Microscopic resemblance to Usually close

tissue of origin Function e.g. secretion Usually well

maintained

Numerous, often abnormal Enlarged, often vary in size

(i.e. pleomorphic) Usually large Marked Often extensive

Variable, from close to none

Usually poor

May be retained, lost or abnormal

(iii) Evidence of transgression of normal boundaries Capsule intact Frequent Rare (usually none) Local invasion Absent Very frequent Metastases Absent Frequent


As previously stated, tumours can arise from the cells of virtually all types of tissue, and the diversity of origin is largely responsible for the great variety of structural appearances of tumours. However, the cells of most turnouts show some degree of differentiation towards the adult cell type of the tissue of origin. The tendency for the cells of a tumour to differentiate along a particular pathway usually allows recognition of the type of tissue cell from which it is derived and this forms the basis of the histogenetic classification of tumours.

A malignant turnout of epithelial tissue is called a carcinoma if it is derived from surface epithelium, and an adenocarcinoma if it is derived from glandular epithelium. A benign tumour of surface epithelium is called a papilloma, while that of glandular epithelium an adenoma. A malignant tumour of connective tissue cells or of muscle is called a sarcoma (e.g. chondrosarcoma of cartilage, osteosarcoma of bone etc.), while benign tumours of connective tissue or muscle are simply named from their tissue of origin (e.g. fibroma of fibrous tissue, chondroma of cartilage, osteoma of bone etc.). Tumours arising from other types of cells are classified in a similar manner.

2.2 The cell surface and neoplasia

The features by which tumours are recognised - the morphological abnormalities of their cells, uncontrolled growth and in the case of malignant turnouts, their invasiveness and ability to form secondary tumours - have all been described in the previous section. We now describe, in greater detail, the differences at the cellular level between normal and tumour cells, focussing in particular on the plasma membrane. Indeed, a vast catalogue of differences has been built up between normal and tumour cell surfaces (i.e. the plasma membrane and attached intra- and extracellular membrane-associated components) and it is now a well recognised fact that the cell surface is involved in a number of important aspects of neoplastic growth including mitosis, cell motility, cell recognition, cell contact, escape from host immune surveillance and many other properties (cf. Nicolson 1976; Gallez 1984). In malignant disease (cancer), the surface properties of the tumour cells have an effect on the cell mechanical properties and are not only important in tissue invasion, but also in determining the subsequent patterns of cell distribution and establishment of distant metastases. For example, in the case of carcinoma cells, the diminished formation of spot desmosomes is thought to lead to a reduced adhesiveness to one another. This lack of adhesiveness is an important factor in the infiltration of surrounding tissues by the cells at the periphery of a cancer and in their penetration of the walls of lymphatics and blood vessels.

The cell surface is thus involved in a variety of physiological properties which directly relate to neoplastic transformation with the result that it is one of the potentially most useful routes to combat cancer (cf. Willmott et al. 1991). It would also be convenient if these properties were definable by unique cell surface characteristics which were readily identifiable. Unfortunately, such a simplified approach has not yielded many results which are clinically beneficial. However, the potential to gain a deeper understanding of many of the basic cellular characteristics remains. By focussing attention at the cellular level on the plasma membrane and on the cells at the periphery of the tumour itself and characterising their behaviour using a strain energy function, we hope to be able to describe some of the important differences mentioned above in a way which may have potential clinical application.


2.3 The vascularisation of tumours

There is one other important aspect of solid tumour growth which we shall now briefly describe before elaborating upon the mathematical techniques which we shall use throughout the paper - angiogenesis, or the ability of solid tumours to stimulate the growth of neighbouring blood vessels, which provide the tumour with an additional supply of nutrient. Solid tumour growth then is believed to proceed through two distinct phases, known as the avascular and the vascular. During the first, or avascular, phase, the turnout obtains its nutrients and disposes of waste products via diffusion processes alone. Lacking its own blood supply and network of blood vessels, it cannot grow beyond a few miUimetres in diameter. Examples of this type of solid tumour include in situ carcinomas and spheroids grown in vitro. These avascular nodules typically develop a central necrotic core which is surrounded by a thin layer of live cells at the tumour surface. From experiments performed on the cornea or anterior chamber of the eye of test animals, where small fragments of various human and animal malignant tumours are implanted, it has now been established that in order to make the transition from avascular to vascular growth, solid tumours secrete a diffusable chemical compound known as tumour angiogenesis factor into the surrounding host tissue. This has the effect of stimulating the growth of capillary vessels from the adjacent vascular tissues and the new vessels penetrate the tumour, thus providing it with essential nutrient and enabling the vascular phase to take place. Rapid growth now follows and it is during this vascular phase of growth that invasion of the surrounding tissues may take place and secondary tumours (metastases) may form at distant sites of the body. A comprehensive survey of all the events which take place during angiogenesis can be found in the review article of Paweletz and Knierim (1990), while mathematical models can be found in the papers of Balding and McElwain (1985) and Chaplain and Sleeman (1990).

2.4 Nonlinear elasticity theory

As we have seen in the previous section, the processes of invasion and metastases involve the deformation of the tumour. Many crucial events in a cell cycle involve the deformation of the cell membrane (cf. Gallez 1984). Deformations also occur in many other areas of biology and in biomechanics. The motion and deformation of a body (assumed to be a continuous distribution of matter in space), and how forces are transmitted in the body are best described using the methods of continuum mechanics, and for our purposes here, elasticity theory. Elasticity theory seeks to describe the basic relationship between forces in a continuous body and its shape. When forces are imposed on the body stresses (which can be thought of as the generalisation of pressure) arise as it deforms and changes shape. An equilibrium configuration is attained when these stresses balance the applied forces and this relationship is expressed mathematically as a set of equilibrium equations. These describe a fundamental relationship between the force and the stress, irrespective of the particular material properties of the body. The deformation of the body can be described in terms of the strain, which is directly related to the stress through a constitutive relation. It is this relation which "characterises the material" and it depends upon the specific properties of the material.


We shall assume that the underlying deformations involved in the growth of solid tumours are large, and thus we shall use nonlinear or finite deformation elasticity theory to model the growth of solid tumours (cf. Greenspan 1972, for a discussion of scale). Unlike the perhaps more familiar linear theory of infinitesimal elasticity, where the deformations are assumed to be small, the relationship between the stress and the strain in finite deformation theory is nonlinear and depends upon the characteristics of the material under consideration. Another point to note is that as the body changes shape under large deformations, changes in the geometry (negligible in linear theory) become important and these must be taken into account in the modelling. These assumptions mean that it is more convenient to characterise the material using a strain-energy function, W, and to relate the stress to the strain via W. A strain-energy function approach and membrane theory have been used to successfully describe individual cell characteristics (cf. Hettiaratchi and O'Callaghan 1974, 1978; Skalak et al. 1973; Wu et al. 1988), liposome morphogenesis (Sekimura and Hotani 1991) and also in the modelling of plant cell growth and morphogenesis in the unicellular alga Acetabularia (Chaplain and Sleeman 1990; McCoy 1989), while the application of large-deformation theory has been used in modelling soft-tissue growth and swelling (Bogen 1987; Demiray 1976, 1981; Gou 1970; Vito 1973). A mathematical description of the concepts introduced above, such as stress, strain, constitutive equation etc. can be found in Appendix 1, while a complete account of the theory may be found in, for example, Fung (1981).

3 The mathematical model

3.1 Radially symmetric growth

Models of tumour growth by diffusion processes alone lead to a steady-state, in which the tumour reaches an ultimate size and where there is a balance between nutrient supply and growth. The model for tumour growth which we shall base our formulation upon is the model of Greenspan (1976), which considers the tumour to be composed of a large, central necrotic core, surrounded by a layer of live, proliferating cells on the tumour surface several cells thick (cf. epithelial tissue). Initially, the turnout is assumed to be spherical. All the living turnout cells are assumed to be identical and each is considered to be an incompressible structure of constant volume. The gross internal forces in the necrotic core are characterised by a pressure distribution P. Cell adhesion produces a surface tension force at the boundary of the tumour i.e. in the thin layer that maintains the compactness of the tumour and counteracts internal expansive pressure. Unstable development of the tumour arises when the internal pressure forces overcome the surface tension and inter-cellular adhesion. The instability is initially manifested as a pinch or a corrugation of the boundary surface at the equator of the tumour and more elaborate instability configurations may lead to further subdivision or disintegration, with subsequent invasion of the surrounding host tissue.

Having described the make-up and characteristics of benign and malignant tumours, and pointed out the salient features and assumptions of the Greenspan model, we now present our formulation of tumour growth from the point of view of elasticity theory. More precisely, we initially treat the growing turnout as

440 M. A, J. Chaplain, B. D. Sleeman

an inflating balloon. In this way, the essential features of the Greenspan model are preserved. In particular, the thin layer of live, proliferating cells at the tumour surface is modelled by the membrane of the balloon and as such is characterised by a strain-energy function W which describes the properties of the layer. The internal pressure produced by the necrotic core is modelled via the inflationary pressure inside the balloon, and the pressure/surface-tension balance on the boundary of the tumour is reproduced as a consequence of the equilibrium equations from membrane theory.

Subsequently, the same type of analysis is used in determining whether or not the turnout may metastasise and invade the surrounding tissue i.e. the introduction of small perturbations superposed upon the underlying basic state of motion (in elasticity this is the theory of small deformations superposed on large ones cf. Greenspan 1976; Schwegler et al. 1985). The onset of invasion of the host tissue by the tumour will be characterised by the bifurcation from the radially symmetric configuration of the membrane into an aspherical equilibrium configuration. Similarly, at the cellular level, the onset of the process giving rise to two daughter cells from the parent cell is characterised by a pinching of the cell around the equatorial belt.

In elasticity theory, in order to describe the deformation of a body, two coordinate systems are required for a complete description - one is known as the reference configuration (which we shall take to be the initial parametric represen- tation of the tumour at t = 0 i.e. R(0) = a) and the other the current configuration (see Appendix 1 for a more complete description). Following the analysis of Haughton and Ogden (1978a), the two basic variables employed will be the deformation gradient tensor a, which is a measure of how much the tumour de fo rms- this can be thought of as the "rate of change" of the current configuration with respect to the reference configurat ion- and the nominal stress tensor s, which describes the state of stress in the thin layer of live ceils. The stress within a membrane may be thought of as a generalisation of the surface tension.

In thin shell (i.e. membrane) theory, it is convenient to work in terms of averaged variables by averaging over, in this case, the reference thickness of the shell. This lends itself to a description of points in terms of the middle-surface of the shell (see Appendix 2 for details). Averaged variables will be denoted throughout the paper by a bar.

Let R and r, respectively, be the initial and current radii of the tumour, and H and h be the corresponding thickness of the layer of live, proliferating cells i.e. the membrane thickness (hence, we take our reference configuration to be the initial state of the tumour at t = 0 i.e. R(0) = a and the initial reference thickness H to be the thickness of the layer of proliferating cells at t = 0).

With spherical coordinates 0 and q0 in the current configuration, we take al and a 2 to be the corresponding unit basis vectors and a3 to be the unit outward normal to the spherical membrane.

The average ~t of the deformation gradient a (our measure of the deformation of the membrane i.e. the layer of viable cells, and hence of the tumour itself) may be decomposed on the above basis as

= ~lal @ al q- ~2a2 @ a2 q- ~3a3 ~) a3. (3.1)

The averages ~.~ of the principal stretches )~ (i = 1, 2, 3) which are measures of how much the layer of live cells stretches or compresses along each direction


i = l, 2, 3 (corresponding to the (0, q~, r)-direction respectively), are such that

~17.223 = 1, (3.2)

since we are treating the membrane (i.e. the layer of live cells at the outer boundary of the tumour) as an incompressible material because it is composed of incompressible cells (cf. Greenspan 1972, 1976). We emphasise that this assumption of incompressibility applies to the membrane only and not the tumour as whole, which does of course increase in volume due to the internal expansive pressure.

In view of the symmetry of the deformation (i.e. radial expansion only), we write

= ~1 = ~2, ~3 = 2--2 (3.3)

Then, we have r = 2R, h = 2-2H. (3.4)

For a growing tumour, we have 2 > 1, and so as the tumour increases in size, the layer of proliferating cells (the membrane) decreases in thickness (cf. Tubiana 1971; Greenspan 1976).

Let the average # of the Cauchy stress tensor a have principal components 511 , 522 , 533. Then we may write

6i, = 6, -/3, (i = I, 2, 3), (3.5)

where/~ is the average of the arbitrary hydrostatic pressure p, and 6~ is the average of ae. For an incompressible, isotropic (i.e. the stress is assumed to act in no particular preferred direction) elastic solid, a~ is given by

= 2i 8W ( i = 1, 2 , 3) , ( 3 . 6 ) O'i 02 i '

where the strain-energy function W = W(2~, 22, 23) is a symmetrical function of 25,22, 23. This can be thought of as the three-dimensional counterpart of the elastic energy stored in a spring. Thus, for the tumour, it is a measure of the elastic potential energy stored in the layer of live, proliferating ceils at the tumour surface.

From a result of Haughton and Ogden (1978a), specialised to the case of isotropy, it follows from (3.6) that

_ S W 6e = 2i O)//' (3.7)

where 1~ = W(Z1,22, 23). In the present circumstances, the membrane approximation is that

533 ~ ~3 - - f f = 0, (3.8)

so that the stress can effectively be thought of as a generalisation of surface tension. It follows from (3.3), (3.5) (3.7), (3.8) that

0 ~ _ o f f "


where

r~(~) = w(~, ~, ~-2),

and

d,~"

From the equilibrium equations (see Appendix 1 for details), we then have

p = h(~n + #a2) 2h#1~, (3.10) r r

where P is the internal, expansive pressure producing the inflation. Thus, we can see that in the spherically symmetric equilibrium configuration, the internal pressure P is exactly counterbalanced by the product of the surface tension force in the membrane and the curvature (cf. Thomson 1961; Isenberg 1978). This is precisely the boundary condition used by Greenspan (1976). With the help of (3.4) and (3.9) this can be written as

P = 22 , (3.11)

where ~ = H/R. The essential features of the initial phase of solid turnout growth are thus captured by the model - the gross internal forces are characterised by an internal expansive pressure distribution which is counteracted by a surface tension at the outer boundary of the turnout. Of course the cause of the growth is the intake of nutrient by the cells and cells in the interior die and necrosis takes place here due to the lack of a vital nutrient. Although not explicit in the model, we assume that the expansive pressure is caused by these factors, and hence this is an implicit assumption of the model. We now examine the effect of small perturbations (which are always present in a real environment) on the stability of the tumour. If the expansive pressure overcomes the surface tension then the turnout can develop an aspherical shape and further growth would then be asymmetric. The function of 2 on the right-hand side of (3.11) plays a critical role in the perturbation analysis and so for tumour growth, the activity of the live, proliferating cells in the thin surface layer, which is characterised by the strain-energy function W, determines whether or not any subsequent invasion or metastasis takes place.

3.2 Perturbation analysis - The incremental equations

We assume that a point a in the finitely-deformed, radially-symmetric configuration is now given an incremental displacement a, say (cf. Greenspan 1976, where exactly the same procedure was followed i.e. the tumour was considered to be subject to small deviations from its radially-symmetric state and small perturbations were superposed upon the basic state of motion variables, resulting in a new set of equations to be solved). Thus, we have

a ~ a + a. (3.12)


Correspondingly, the average deformation gradient 6t and the average of the nominal stress $ will also undergo incremental displacements i.e.

o~-~o~+o~, g ~ g + ~ . (3.13)

The incremental quantities are then evaluated in the current configuration giving ~t0, which we denote by 1], and 30, respectively. These quantities are now used in the equilibrium equations which are given by

S~oui,,u "Jr- S o u i a v • au,v + ~ o u v a i " a~.~ + S o ~ 3 a i • a3, ~ - - h - 1 p 0 3 i -~- h -- 1/~t~i3 : 0

i = 1,2,3 (3.14)

now expressed relative to the given orthonormal basis (see Appendix 3 for details).

With the unit basis vectors al, a2, a3 taken to correspond with the (0, q~, r)- directions respectively, in spherical polar coordinates, the incremental displacement ~i is written as

(l = v a I + w a 2 + u a 3 . (3.15) Assuming that the deformation of the membrane is axisymmetric i.e. inde-

pendent of ~0 (cf. Greenspan 1976), then the equilibrium equation corresponding to i = 2 leads to the solution

w = c sin 0. (3.16)

Since this deformation does not alter the spherical shape of the membrane, and hence the tumour, it will not be discussed further. The equilibrium equations corresponding to i = 1, 3 then reduce to

(2272 - 273)Uo + S2(Voo + Vo cot 0) - (272 - S3 + $2 cot 20)v = 0, (3.17)

(2272 - S3)(Vo + v cot 0 + 2u) - Sl(Uoo + uo cot 0 + 2u) - r2h-1/~ = 0, (3.18)

where S1, S2,273 are defined as

271 = G l l , (3.19)

272 ~ /~111 1 -~ /~3333 - - 261133 + 217, (3.20)

273 = 2/~1212 =/~1111 - - / ~ 1 1 2 2 + 0"1, (3.21) with B~_xl the components of the average of the instantaneous elastic moduli tensor B (see Appendix 3 for details). We can relate $1, $2, $3 to our strain- energy function W as follows

1 A 271 = 2 2 W 2 , (3.22)

1 2 ^ 2272 + $1 - 273 = ~2 W~. (3.23) The solution of the above incremental boundary value problem is unique at

any stage along the considered stable path up to a critical configuration at which uniqueness fails and the solution path bifurcates. The critical configuration at which this occurs is called a bifurcation point and the solution here changes its behavioural character. The stability of the equilibrium solution will now be conditional on a parameter of the model (in this case the stretch ratio 2) and changes in this parameter will produce abrupt changes in the behaviour of the model. In the following section, we examine the effect of varying the stretch ratio 2 on the existence and stability of the steady-state solutions.


3.3 Bifurcation criteria

In order to solve Eqs. (3.17) and (3.18), we now write

u = ~ A,P,(cos 0), (3.24) n = 0

v = B, ~ [P,(cos 0)1, (3.25) n = l

where A, and Bn are arbitrary constants, and / ' , ( co s 0) is the Legendre Polyno- mial of degree n (el. Greenspan 1976),

Substituting the above two equations into (3.17) and (3.18) then gives us the following

dP, ( ( 2 ~ ' 2 - - ff'3)An -'}- (~'3 - - m~'z)Bn} ~ - - - O, ( 3 . 2 6 )

r t = l

{[42; 2 - 2Z 3 - (2 -m)Z~IA, - m(2Z2 - Z3)B, }Pn = r2h -1p , (3,27) n = 0

where m = n(n + 1). The above equations can now be solved and non-trivial solutions to the above system for a particular mode n, correspond to the bifurcation points.

Now considering the bifurcation mode n = 0 first, we see from (3.27), using (3.22) and (3.23) that

A0(22ffza~ _ 22 lg-a) = r2h- 1p. (3.28)

This, of course, corresponds to the radial mode of deformation and (3.28) can be deduced from (3.11) directly using (3.4). Then Ao = {, as it should be.

Pressure turning points can be found by differentiating (3.1 l) and are given by

u_123 dP )3~.z - 2W~ = 0. (3.29) - ~ = .

The case of the n = 1 mode bifurcation necessitates the solution of the equation

(A1 - B1)(2Z2 - 2;3) = 0. (3.30a)

Three possibilities arise in this mode depending upon the choice of (arbitrary) constants A1, B~. The solution A~ = B1 corresponds to a rigid body translation and shall not be discussed further (cf. Greenspan 1976). If A~ is taken to be zero (A~ = 0), then the mode v = - B ~ sin 0 is possible. This mode corresponds to a local thinning at one pole and a thickening at the other pole of the membrane (i.e. layer of live cells), with the radial shape of the membrane being preserved (cf. Feodos'ev 1968). If instead, B~ is taken to be zero, (B~ = 0), then the mode u = A~ cos 0 is a possibility. In this deformation, the equatorial radius is maintained but the membrane becomes asymmetric about its equatorial plane. If A1 > 0, then the membrane thins (thickens) in the upper (lower) hemisphere and vice-versa for A~ < 0.

It is clear that the condition for bifurcation into one of these modes is

(2S2 - 273) = O, (3.30b)


which can be written in terms of the strain-energy function as

2 I ~ - W~ = 0. (3.31)

The aspherical configurations associated with this mode are not applicable to the onset of invasion and metastasis as characterised by the pinching of the equatorial belt (cf. Greenspan 1976), and so wilt not be discussed here in this context. However, they are instructive with regards to indicating when the onset of these processes within the tumour occur, and so we shall examine the criteria necessary for these modes to take place. They may however be applicable to the phenomenon of capping observed in cells which is discussed in Sect. 3.4.

We now consider the shape changes involved for the higher order modes, n ~> 2. Here, as in the models of Greenspan (1976) and Schwegler et al. (1987), the first mode of instability to become possible is the mode n = 2. As in these models, this mode is characterised by its "peanut-like" shape i.e. a furrowing at the equator coupled with a bulging at the poles. This shape is found for the mode n = 2 in the above model. Higher mode bifurcations (n t> 3) produce correspondingly more complicated aspherical shapes (cf. Feodos'ev 1968).

For higher order modes, i.e. n ~> 2, we see from (3.24) and (3.25) that bifurcation into a particular mode n will occur (i.e. a non-trivial solution (An, B,) of (3.24) and (3.25) exists) provided that

n(n + 1)SIX2 + X3(2X 2 - S 1 - - X 3 ) = 0. (3.32) Haughton and Ogden (1978b), have shown that the values of 2n

(n = 0, 1, 2 . . . . ) of 2 for which the mode n bifurcations are initiated (from a radially-symmetric configuration) form a monotonic sequence, increasing with n. This compares favourably with the results of Greenspan (1976) and Schwegler et al. (1987) where a similar situation occurred i.e. the mode n = 2 was the first mode of bifurcation to exist. Also, the results of Schwegler et al. (1987) showed a monotonic increasing sequence of critical radii occurred. Of course, here the increasing 2 values also correspond to increasing current radius r since

r = 2R. (3.33) Also from Haughton and Ogden (1978b), the bifurcation into a particular

mode n must occur between a pressure maximum and a pressure minimum. Finally, there is an interesting physical interpretation of Eq. (3.31). The surface tension in the membrane (i.e. the layer of live cells) is given by h8n, which by (3.4) and (3.9) is seen to be proportional to W~/2. Hence, we can see that (3.31) corresponds to the turning points of surface tension. In particular, bifurcation into mode n >~ 2 can only occur after the surface tension has reached its maximum value.

It is a known fact that in malignant tumours, there is a decreased number of spot desmosomes, the lack of which means a lack of adhesiveness of the cancer cells. The Greenspan model (1976) modelled this intra-cellular adhesiveness via a surface-tension/curvature force, and the contribution that surface tension played in determining whether or not the turnout split up was of crucial importance. If the surface tension force was too low, then the internal pressure force could overcome it, leading to the break-up of the carcinoma. Thus, in our above model a decrease in surface tension, corresponding to a lack of spot desmosomes, will lead to the possibility of bifurcation into a mode n ~> 2 and could indicate the onset of metastasis and invasion of the surrounding host tissue by the malignancy.


3.4 Strain-energy functions

We now consider the above analysis for the class of strain-energy functions introduced by Ogden (1972) i.e.

W = #,q~(~,), (3.34)

where

q~(~) = (2]' + 2~ + 2~ - 3)/ct, (3.35)

with #,, ct, real constants and summation over a finite number of terms applied by the repetition of the index number r in (3.34). According to Ogden (1972), these constants should be such that

#,ct, > 0, for each r (no summation). (3.36)

In the present context, we thus have

if/'(2) = #,(22 ~, + 2--2~tr - - 3)/ct,. (3.37)

It can then be shown that

Z 1 = #,(2 ~r - 2 -2~,), (3.38)

2; 2 = #, {(e, - 1)2 ~ + (e, + 1)2 --2er}, (3.39)

2;3 = #r~r 2 ~r' (3.40)

It then follows that we can now write

22ff '~ - 221~ = 2#, {(~, - 3)2 ~ + (2~, + 3)2--2~r}, (3.41)

221~.~ -- )~I~Z = 2#, {(~, -- 2),~ ~' + 2(~, + 1),~ --2~,}. (3.42)

Now a necessary condition for a pressure maximum to exist is - 3 - - < ct, < 3, for some r, (3.43)

2 while a necessary condition for the surface tension to become a maximum is

- 1 < ~r < 2, for some r, (3.44)

and finally, a necessary condition for modes of all orders to exist is

- 1 < ~r < 1, for some r. (3.45)

For realistic forms of (3.34) to describe a material, three terms are required. What we propose is that benign and malignant tumours can be characterised and classified by different strain-energy functions. Through the strain-energy function, the inherent differences in the structural make-up of these tumours can be taken into account i.e. we can take into account the high differentiation of benign tumour cells and the low differentiation of malignant, cancer cells using different strain-energy functions. Also, the scope that exists in the variety of strain-energy functions of the type of (3.34) can be used to describe the spectrum of malignancy that exists in tumours i.e. those tumours that lie in between the well defined extremes of benign and malignant tumours. For benign tumours, we suggest that the strain-energy function is such that bifurcation, if it should even occur at all, into the mode n = 0 mode only is possible i.e. the radial mode of bifurcation. Thus, the tumour will remain a compact, spherical mass, with no


possibility of invasion of the surrounding tissue arising. We note that both the neo-Hookean material (cq =2) and the Mooney-Rivlin material (~l-=2, ~2 = - 2 ) satisfy only the first of these three conditions, and hence if the strain-energy function of a benign tumour were of one of these forms, then only n = 0 mode bifurcation is possible. Hence, for benign tumours, we forward the hypothesis that the strain-energy functions associated with these tumours satisfy the condition of Eq. (3.43) but not the conditions of (3.44) and (3.45). We propose then, that the parameters c~r of the strain-energy function W should satisfy (at most) the condition

G ~ ( ~ , - l l w [ 2 , 3).

Benign tumours could thus be thought of as being characterised by neo- Hookean-like or Mooney-Rivlin-like substances.

However, for malignant tttmours, we propose that the strain-energy functions in these cases should allow for the (real) possibility of invasion and metastasis, and so we suggest that the conditions for the surface tension reach- ing its maximum and the possibility of high order modes of bifurcation should be met i.e. for malignant tumours, the strain-energy functions should satisfy the conditions of (3.44) and (3.45), and in particular (3.45), since the conditions of (3.44) are then automatically satisfied. Then, for malignant tumours we have

- I < G < I , for somer.

For tumours which display an intermediate degree of malignancy, then we put forward the hypothesis that the strain-energy functions of these tumours, like the degree of malignancy, can be described as lying somewhere between those of the benign tumours (neo-Hookean-like, Mooney-Rivlin-like) and those of the malignant tumours.

3.5 Thick shell theory: Tumours with a greater number of live cells

In concluding his study of tumour growth, Greenspan (1976) noted that the model constructed could easily and readily be adapted to more complex models of cell aggregation including a "thick" layer of proliferating cells. It is now our purpose to show that the same degree of flexibility can also be achieved using non-linear elasticity theory and extending our attention from membranes to thick spherical shells. In this way, we model turnouts which have a greater number of tumour cells in them. Once again, we assume that these cells are confined to an external shell. To this end then, we now consider the tumour to be described by the thick, spherical shell

A<~R<~B, O<~O<~rc, 0~< • ~<27z,

in its initial (i.e. undeformed and unstressed) configuration, where R, O, q~ are spherical polar coordinates.

We assume that initially the tumour grows in a radial direction only i.e. the shell is inflated so that its spherical shape is maintained. In its current configuration, the tumour is now described by

a<<.r<~b, 0~<0 ~<rc, 0~<~o ~<2~,


where r, 0, ~0 are also spherical polar coordinates. The layer of viable, proliferating cells is now assumed to occupy the region a ~< r ~< b with the interior of the tumour 0 ~< r < a being composed of dead and dying cells, stroma, blood vessels etc.

Now since we are assuming that the tumour is composed of incompressible cells (cf. Greenspan 1976) then we can confine our attention to incompressible materials, and hence the deformation described above may be given, from the equilibrium equations, as

r = (R 3 + a 3 - A 3 ) 1/3, (3.46)

with 0 = O, q~ = q~. The principal stretches 21 and 42, corresponding to how much the tumour

stretches in the 0- and q~-directions respectively, are equal and denoted by 2, where

r 2 = R > 1. (3.47)

We note that 4, defined thus, is variable. Also, we have that dr

43 - dR = 2--2 (3.48)

by virtue of the incompressibility of the tumour (i.e. r 2 dr = R 2 dR), where 43 is the principal stretch corresponding to the radial direction.

We assume that the surface R = B is free of traction and that there is a pressure P ( > 0 ) per unit current area on R = A, corresponding to the internal pressure in the tumour interior.

For an isotropic elastic solid (i.e. the layer of viable cells) possessing a strain-energy function W per unit volume, the principal Cauchy stresses aii (i = 1, 2, 3) are given by

0"11 ~" 0"22 = 0"1 - - P , 0"33 = 0"3 --P, (3.49) in the present circumstances. Also, al = o'2 and 0"3 are given by (3.6) evaluated for 21 = ~,2 = • and 43 = 2-2, and p is the arbitrary hydrostatic pressure.

Hence we can show that 1

0"11 - - 0"33 ~--- 2 2 W 2 , (3.50) where 1~ is as defined in the previous sections. Equation (3.50) is similar to (3.9), but here 0"33 :fi 0 in general.

The only equilibrium condition which is not satisfied identically is do'33 2

dr + (0"~3 - 0"11) = 0, (3.51)

in the absence of body forces. Now since 0"33(a) --- - P and o-33(b ) = 0, integration leads to

P = ,~ . (3.52) r

Using (3.46), (3.7) we can rewrite this (see Appendix 4) as

P = ~ ( Z - - ~ - ) ' (3.53)


where 2~ and )~b are given by

2~ = a/A, with

Now we note that

~b = b / B = (~3 _[_//3 __ 1)1/3///, (3.54)

//= B/A > 1. (3.55)

//3(2~ 1 ) - 3 - - ) ~ - 1, (3.56)

so that, in view of (3.55), we have

2a ~> 2b t> 1, (3.57)

with equality if and only if 2a = 1. P can now be regarded as a function of 2a, the tangential stretch at the inner

surface (cf. Greenspan 1976). Differentiation of (3.53) with respect to 2a and then use of (3.56) (see Appendix 4) leads to

dP lYv'~(2,) l?V~().b) (/~a - - ,~ ~-2) dt~a ~2a I~ 2 (3 .58)

This should be compared with the corresponding expression (3.29) for a membrane to which (3.58) reduces in the limit as b ~ a .

Once again, it follows, from the requirement of the function ~')./2 to be non-monotonic for all 2 > 1, that a necessary condition for a pressure maximum to exist is that 2W~ - 2W~ = 0, for some 2 > 0. However, for a thick shell this condition is not sufficient to ensure that dP/d2a vanishes at some 2a > 1 since 2b depends on 2a. As a result, the thicker the shell i.e. the greater the number of tumour cells within the tumour, the less likely a pressure maximum is to exist.

3.6 The incremental equations

We now suppose that the tumour is subject to the influence of external forces and is such that the layer of cells undergoes an incremental deformation from the current, spherically symmetric configuration. Let x be the incremental displacement (cf. a) of the material currently at position x, and let ~t and i be the associated increments in the deformation gradient and the nominal stress, respectively. Referred to the current configuration, i is denoted by So and & by ot o which we denote by I!, where

q = ~xx" (3.59)

The general, incompressible elastic constitutive law (see Haughton and Ogden 1978a) in its incremental form, referred to the current configuration is given by

s0 = Bq + p q - / ) & (3.60)

where B is the tensor of instantaneous elastic moduli and 6 is the identity tensor. In view of the incompressibility condition, we have

tr(q) = 0. (3.61)


Now the equilibrium condition gives

div So = 0. (3.62)

The exterior boundary of the shell (i.e. the layer of live cells) is free of traction, while, in the current configuration, the interior pressure boundary condition can be expressed in terms of the nominal stress as

s r N = - P[IN, (3.63)

where N is the unit normal to the boundary in the reference configuration, and I~ is the inverse of a T (see Appendix 1 for details).

The incremental form of (3.63), referred to the current configuration, is

"sin = - P n + e q T n , (3.64)

when n is the unit normal to the boundary in the current configuration. Let el, e2, e3 be the orthonormal basis corresponding to the spherical polar

coordinates 0, ¢p, r respectively. On this basis, (3.62) can be given as

Soji,,-+ s0j;ek " ej.k + Sokje,- • ej, k = 0. (3.65)

The expressions ei • ej, k are exactly the same as the quantities a i " ay, k which have been given in the previous section. Hence, we write the incremental displacement as

~c = re1 + we2 + ue3. (3.66)

In general, u, v, w depend upon 0, ~p, r. However, as was seen in the previous section, the inclusion of ~p-dependence does not affect the bifurcation criteria. Thus, we restrict attention to axisymmetric modes of deformation. The equilibrium equations (3.65) then reduce to one equation for w i.e. i = 2 and two coupled equations for u and v i.e. i = 1, 3. As in the previous section with the membrane, we shall concentrate attention on the latter case, since the w equation does not affect the spherical configuration. In fact, for thick shells it is convenient to take w = 0 as a solution (cf. Haughton and Ogden 1978b). Then the equilibrium conditions become

rut {r(B;333 - B'n33 + P ' ) + F - 2G} + r2urr(B3333 - B1133 - - B3113 )

+ (Uoo + Uo cot 0 + 2u)Bl~13 = r2/~r, (3.67)

(U 0 - - v) {r(B'3113 --}-p') -}- F} + rUro(B31,3 + Bl133 - - B l l l l )

+ rvr(rB'3131 + 2B3131) + riVrrB3131 :-- ri%, (3.68) where ' denotes d/dr. Also

F(r) = B3113 -t- B1313 -Jr- 81122 -- B n l l , (3.69)

G(r) = 2B3113 -{- 2B1133 - - B3333 - - B l l l l , (3.70) while p ' is obtained from (3.51).

The boundary conditions (3.64) are now

r v r + u o - v = O o n r = a , b (3.71) ~ ' - P on r = a t

(B3333 - - 81133 -1- O'3)Ur --1~ : ~0 on r = b. ~' (3.72)

where use has been made of the boundary conditions in the current configuration o'33(a ) ~ o3(a ) - - p ( a ) = - e and o-33(b ) :-~ a3(b) - p ( b ) = O.


In order to solve (3.67), (3.68) we write oo

u = ~ f,(r)P,(cos O) n=0

v = gn(r) -~ Pn(cos O) (3.73) n=l

/)= ~ h,(r)P,(cos 0). n=0

The incompressibility constraint leads to

mg,=rf ' ,+2f , , n =0 , 1,2 . . . . . (3.74)

where m = n(n + 1). Substitution of (3.73) into (3.67) and (3.68) followed by the elimination of g,

and h, leads to two sets of differential equations for n = 1, 2, 3 , . . . the details of which are given in Appendix 4. Expressions for the boundary conditions can also be obtained in a similar manner.

For n = 0 , f0 is found from (3.74) while ho is obtained directly upon integration of (3.67). This then gives

r-2h'o = [{a~ + 2(~3 - al)/r)r-2]'fo, (3.75)

fo = aZa/r2 --= a3~a/r22a, (3.76)

where 2a is given by (3.54). For n = 0, the boundary condition (3.71) is satisfied exactly, while (3.72)

yields onr--a} (B3333 -- Bl133 + °3 ) f ° - - h° [ 0 on r = b " (3.77)

This equation can be obtained directly by taking the increment of (3.53), giving

b - ~r2(/~a)~a ~rJ'(/~b)/~b (3.78) ( 2 3 - 1 ) ( 2 3 - 1 ) '

which is equivalent to (3.58). This n = 0 mode maintains the spherical symmetry of the shell and so

will not be discussed further (cf. mode n = 0 for a membrane). We note, however, that since Eqs. (A4.11)-(A4.14) of Appendix 4 are homogeneous in fn (n = 1, 2 , . . . ) , then the solution f0 is always possible. Hence, bifurcation into an aspherical mode and its subsequent development depends upon the existence of a non-trivial solut ionf , for some n >~ 1. This wil! be discussed in the next section.

3. 7 Bifurcation into aspherical configurations for higher order modes

For mode numbers n ~> 2, analytical progress with Eq. (A4.1) is very limited. However, some progress can be made for the case n = 1, and the main results obtained here can be used to give a qualitative analysis for higher order modes.

From Eq. (A4.1) and with n = 1 we then have

r2B3131~l " -~- (r2B~131 ~- 4rB31~1)~' + r(2B~131 d- cr~3)f ] d- (2G - F)f'l = 0, (3.79)


while the boundary conditions give

r f l ' + 2 f ' l = 0 on r =a,b. (3.80)

Integrating (3.79) twice, followed by the use of (3.80), then leads to

( ;3 _ 1 ) -2 /3 ( 2ff _ ff,) d,: = o, ( 3 . 8 1 )

with ha, 2b as defined previously. This then, is the criterion for bifurcation into the mode n = 1. If a value for

2a > 1 can be found, such that (3.81) holds then bifurcation into this mode can occur when the stretch at the inner boundary has reached this value. It can be shown (see Haughton and Ogden 1978b), that, as in the case for a membrane, the bifurcation points (i.e. those values of 2a) occur in pairs. Also, the thicker the shell, the less likelihood there is of bifurcation into the n = 1 mode occurring. Correspondingly, the likelihood of bifurcation into higher-order modes is also reduced for thick shells. Thus, for tumours which have a larger proportion of turnout cells, the likelihood of metastasis and invasion is reduced. This result is supported by the findings of Parham et al. (1988), where they found that neoplasms with a high proportion of tumour to stroma had a better prognosis than those neoplasms with a low proportion of tumour to stroma. This may be explained by the fact that a high proportion of malignant cells within a tumour shows a greater degree of cell to cell adherence, and hence a reduced ability to metastasise and invade.

3.8 Individual tumour cells

On the cellular level, we also feel that this type of analysis from membrane theory (particularly the characterisation of the material under consideration via a strain-energy function cf. Skalak et al. 1973; Hettiaratchi and O'Callaghan 1974, 1978) is both useful and instructive. The features by which tumours are recognised on a cellular level are mainly in the abnormalities of their cells. Benign tumour cells display a uniformity in size, shape and nuclear configuration. They function normally and have relatively infrequent mitotic figures, producing twin, identical daughter cells. Malignant turnout cells i.e. cancer cells, exhibit a haphazard arrangement, bear little resemblance to their cells of origin, and vary widely in size, shape and nuclear configuration. They also exhibit frequent mitosis, which is often abnormal e.g. three or more daughter cells may be produced. However, tumour cells differ from normal cells not only in their metabolism but also and particularly in the features of their plasma membrane, the plasmalemma. These membrane-differences are most notable in the cells of malignant tumours i.e. in cancer cells, but also manifest themselves to a lesser degree in the cells of benign tumours (it has been suggested (Muir 1985) that the oncogenes present in cells may be in part responsible for some of the changes which take place in the cancer cell membranes).

Thus, as we can see, all tumour cells (i.e. neoplastic cells) are different from normal cells, and within this division, cancer cells are markedly different from benign tumour cells. Hence, we propose that at the cellular level also, different types of tumour cells (ranging from the benign tumour cells to the cancer cells of malignancies) have different types of strain-energy functions, and that all


types of tumour cell strain-energy functions are different from the strain-energy functions of normal cells.

We therefore apply the same analysis of the preceding sections to the process of individual cell growth and division. Treating each individual cell as an incompressible, isotropic elastic solid under inflation, then the same analysis goes through. However, all cells, whether normal or tumour cells must split into two daughter cells. Thus, for all types of cells the strain-energy function must be such as to allow bifurcation into the mode n = 2, and we take bifurcation into this mode as the initiation of the M-phase of the cell cycle i.e. splitting into two daughter cells. Thus, from (3.32), the strain-energy functions describing cells should satisfy the condition

6S,1Z 2 + Z3 (2z~ 2 - - z~ 1 - - z~3) = 0,

which permits n = 2 mode bifurcation. However, it is known that malignant tumour cells exhibit abnormal mitosis, often splitting up into three or more daughter cells. Hence, in the case of cancer cells we feel that the strain-energy function describing the plasma membrane of the cell should be such as to permit higher order modes of bifurcation i.e. satisfying the condition

1 2 Z 1 Z 2 q- Z 3 ( 2 Z 2 - Z 1 - - Z3) = 0,

which allows bifurcation into the mode n = 3, and also perhaps the condition of (3.45)

- l < c ~ r < l for somer ,

which allows for all modes of bifurcation. Thus, we see that at the cellular level, the make-up and structure of the cell

membranes plays a crucial role in determining the subsequent behaviour of the tumour. The way in which the tumour behaves will depend upon, in some way, the behaviour of the cells of which it is composed. A malignant tumour which is made up of cancer cells will thus have a strain-energy function which reflects the behaviour of these cells and so determines its nature. A similar situation will also arise for benign tumours, and those tumours which are of an intermediate nature i.e. they lie in the spectrum of malignancy, the two extremes of which are benign and malignant. Thus, comparing the strain-energy functions of individual cells may provide an early detection system for the onset of cancer.

The inherent make-up of a benign tumour is thus completely different from that of a malignant tumour. The above distinctions, on a cellular- and on a tumour-level strengthen our proposition of characterising both cells and tumours via strain-energy functions. In the following tables, we thus present the classifica- tions, both of cells and of tumours, in terms of strain-energy functions.

Table 2. Cells and their corresponding strain-energy functions

Cell Type Strain-energy Function

Normal W Neoplastic W*

Benign W~' Malignant W* (Cancer cell)

454

Table 3. Neoplasms and their corresponding strain- energy functions

Tumour Strain-energy Function

Benign W~ Malignant WM

M. A. J. Chaplain, B. D. Sleeman

Finally we give appropriate forms of the strain-energy function for W*, W*, W8 and WM in accordance with the restrictions on the parameters ar deduced from (3.32), (3.43), (3.44), (3.45).

w~' = ~ (2~ ~ + ~ - ~ - 3), (3.82)

where e satisfies

(~2 + 3e - 6)22= + (2~ 2 + 3~ + 12)2-" - 6(a + 1)2-4= = 0.

W*m = ~ (22 ~ + 2-2~ _ 3), (3.83) 0~

where a satisfies

(a 2 + 9~ - 12)22= + (2a 2 + 3e + 24)2 -= - 12(e + 1)2-4~ = 0.

wB = -~ (222 + 2 - ' - 3) (3.84)

W~t = 2#(220.5 + 2 - ' - 3). (3.85)

Unfortunately, the conditions permitting n = 2 and n = 3 mode bifurcation for benign and malignant cells respectively, deduced from (3.32), do not yield simple conditions on ar and so in these cases (Eqs. (3.82) and (3.83)) we have chosen to illustrate for single-term strain-energy functions only. Conditions for strain- energy functions involving more terms become considerably more complicated and will not be given here. The strain-energy function chosen to characterise a benign tumour (Eq. (3.84)) is a single-term function with e 1 = 2 (neo-Hookean material) while that chosen to characterise a malignant tumour is a single-term function with ~ = 0.5. In the latter case, a single-term function was chosen because the necessary condition (3.45) (for modes of all orders of bifurcation to exist) is also sufficient here (Needleman 1976). For strain-energy functions involving a higher number of terms it is not possible to deduce simple necessary and sufficient conditions on ~r.

Although we have given examples of single-term strain-energy functions only, in general, the number of terms n in the strain-energy function can be as little or as many as is necessary to characterise the particular tumour under consideration. The precise values of the parameters ~r would be determined by experiment. The parameters #~ would then be chosen to satisfy

~'~#rO~r = 2# (3.86) r = l

where # is the shear modulus of the particular material under consideration (cf. Ogden 1972), which would vary from tumour to tumour.


4 Surface morphogenesis of tumours

4.1 Introduction

As we have seen in the previous sections, the main attention of our model has been focussed upon the outer surface of the tumour and the activity of the thin layer of live, proliferating cells there. The vast majority of cancers are carcinomas i.e. cancer of the epithelial tissue, which is formed by thin sheets of tissue, usually one cell thick. When individual cells or groups of cells break off from the main body of a turnout, or when invasion of the surrounding tissue takes place, these events occur at the tumour surface. Thus, the essential surface character of our models would also seem to lend itself to attempt to study tumour growth and development as a surface phenomenon, using the tools of surface geometry. The changes in the shape of a thin sheet due to growth processes (or forces) generated within the sheet itself, is known as surface morphogenesis. In turnouts, we can imagine that the growth processes are caused by the mitotic surface tension force present in the thin layer of live cells (cf. the Greenspan model, 1976, referred to in the previous sections; Hart and Trainor 1989). The geometry of a biological surface is closely related to the growth regime which produces the surface, and we here distinguish between two types of growth: uniform growth, which increases the size of a surface but does not affect the intrinsic geometry (i.e. shape) and differential growth, which may affect both size and shape. In the present context of tumour growth, we may think of uniform growth being associated mainly with benign tumours (since these tend to remain localised), and differential growth as being associated with malignant tumours (since these both grow and also change shape). With advances in technology, the equipment available to pathologists (3D-scanners, etc.) makes it easier to look at a turnout from the surface geometry point of view. This approach, then, allows for general shapes to be studied and is thus a logical progression from the previous sections where only specific shapes i.e. spheres, were considered. Linking the results of the previous sections on classification via strain-energy functions with the results which we will show here will provide a complete theoretical and mathematical method which can be used to grade and stage tumours. In general, it is necessary to describe a surface at any point with two particular quantities- the Gaussian curvature and the Mean curvature. These are related to what are known as the first and second fundamental forms of a surface and will be described in detail in the following two sections, where we shall describe the turnout surface in terms of differential geometry.

4.2 The Gaussian curvature of a tumour surface

A surface can be described by a coordinate system which involves two independent parameters, here u and v. The surface is uniquely defined if for every coordinate point there is associated a 3-dimensional vector determining the position of the point with respect to a fixed origin. We thus let (u, v) be the cartesian coordinates of a point on a plane and let these be mapped isotropically onto a surface x(u, v). An isotropic map has the same local growth in all directions (cf. treating the tumour as an incompressible, isotropic elastic solid in Sect. 3, where the stress was assumed to be isotropic, and the model of Greenspan 1976). The first fundamental form relates distance on the surface to


parametric distance and is given by I = d s 2 = E du 2 + 2F du dv + G dv 2, (4.1)

where E = x , "xu, F = x u .x~, x~ "xv. (4.2)

The first fundamental form gives us information about the spatial extent of a surface near a given point in terms of the coordinate system. The quantity ds can be taken as the length dx of the vector differential dx on the surface of the turnout, and is known as the element of arc.

As x(u, v) is mapped isotropically from (u, v), then E = G = 22(u, v) and F = 0, where 2 is the linear scale factor of the surface growth. Now when F = 0, the Gaussian curvature K of the surface can be written as follows (cf. Todd 1985; Cummings 1989)

K = ~ ~uu au ] +Ovv ~vv / / ' (4.3)

and so we can write

¢ K = V 2 log ,L (4.4)

We can see that V 2 log 2 is negative, zero, or positive as K is positive, zero, or negative. The Gaussian curvature of the tumour surface determines whether the growth rate of the surface is subharmonic or superharmonic. These properties correspond to the notions of convexity and concavity and thus to local excess growth and local deficiency of growth.

As defined above, 2(u, v) is the scale factor associated with the surface growth at the point (u, v). A growth process occurring at a given point in a body can be characterised by a local growth rate #, say (cf. Hart and Trainor 1989; Todd 1985), representing the rate at which new material is being added to the body e.g. via cell division. Using this result a thin shell (cf. the present model for tumour growth) undergoing growth may be likewise characterised in terms of a surface growth rate averaged over the shell thickness. It can then be shown (cf. Hart and Trainor 1989; Todd 1985) that the relationship between 2 and # is given by

2 = exp(#T) when it is assumed that the growth is accomplished over a time T. This relationship describes how the surface deforms locally. From the above expression, it is easily seen that log )~ ( = # T ) is thus proportional to the average local growth rate over a period of time T.

Now from Eq. (4.4), we see that any maximum point of growth rate must occur in a region of positive Gaussian curvature, since here V 2 log 2 < 0. Similarly, any minimum will occur at a point with negative Gaussian curvature.

By analogy to the one dimensional case, a function f (u , v) is known as a subharmonic function if V2f > 0 and as a superharmonic function if V2f < 0. It can be shown (Todd 1985) that for a subharmonic functionf(u, v), the following two inequalities hold

1 fo '~ f(u, v) <~ ~ f (u + r cos 0, v + r sin 0) dO = f(u, v), (4.5)

f(u, v) <~ re--- 5 f (u + 0 cos O, v + 0 sin 0)0 dO do = F(u, v). (4.6)


Thus, for any subharmonic function the function value at a point (f(u, v)) is less than or equal to the average value round a circle centred at that point (f(u, v)), and also is less than or equal to the average value over a circular disc centred at (u, v) (F(u, v)). Correspondingly for a superharmonic function, we have that f (u , v) >~ f (u , v) and f (u , v) >1 F(u, v).

In conclusion, we have from Eq. (4.4) the following results: (i) Regions of positive Gaussian curvature correspond to superharmonic growth rates.

(ii) Regions of negative Gaussian curvature correspond to subharmonic growth rates.

We can thus say that regions of the tumour surface which have positive Gaussian curvature denote regions where the growth rate at a point is greater than the average growth rate about that point. Thus this may give an early indication as to those regions of the tumour surface where invasion is most likely to take place. Similarly, regions of the tumour surface with a negative Gaussian curvature denote regions where the growth rate at a point is greater than the average growth rate about that point. Hence, relatively speaking (i.e. in relation to the average growth of the whole tumour), areas of the tumour surface which have positive and negative Gaussian curvatures denote those areas of differential excess growth and differential deficient growth respectively. A complete 3D analysis of the tumour surface from this point of view thus also gives a more complete picture than perhaps the mitotic index does. It is a known fact that this index can vary over the surface of an individual tumour and that this does not always provide the best possible description of how a tumour is developing. The surface geometry approach offers a more complete picture at the global level, while using local information to build up this picture e.g. for carcinoma arising in organs, such as the lung and the breasts, growth is seen as a nodular mass, whose margin, although in places sharp and well-defined, is elsewhere irregular due to the local invasion of the surrounding tissue.

4.3 The significance o f the mean curvature

Having defined the local growth rate on the surface of a tumour in terms of the Gaussian curvature K, we now turn attention to the mean curvature H. As we saw in the previous section, the first fundamental form of a surface V = x(u, v) does not give us any information about how the surface bends away from the local tangent plane. This is given by the second fundamental form, and allows a quantitative study of the shape of V in the neighbourhood of a point. Whenever surfaces are considered for practical applications, the second fundamental form is of paramount importance: in optics, it determines the caustics and the image one sees when one looks into the surface; in hydrostatics it provides the notion of a metacentre which is vital in the design of ship hulls; and in mechanics, it is used in the design of large smokestacks.

Denoting by n a unit vector on the surface of the tumour, the second fundamental form is defined as

H = L du 2 + 2 M d u dv + N d v 2, (4.7) where

L = xu,, " n, M = x,,~ • n, N = x,~ - n. (4.8)


With this definition, the mean curvature is then given as ~1+ ~c2 I E N - 2 F M + GL

H 2 2 EG - F 2 ' (4.9)

where tq, ~2 are the principal curvatures. As we have seen in previous sections the surface tension force on the tumour

surface has been taken to be proportional to the mean curvature there. Thus a measure of the mean curvature on the tumour surface is an indirect way to measure the local surface tension force since

T = ~ (xl + x2) = ell , (4.10)

(cf. Eq. (3.10)). Thus from a surface map of a tumour we can measure the local variation in the mean curvature and hence the local variation in the surface tension at individual points on the tumour surface, thus giving an indication of those regions where the surface tension has become weak and whence individual cells or group of cells may invade the surrounding tissue or break off to form metastases. We suggest that for carcinomas, there exists a minimum local surface tension, T * = e l l* , say, such that if the surface tension at any point on the tumour surface falls below this critical value, then we can say that there is a definite likelihood that invasion and/or metastasis have occurred i.e.

{T>~ ~H*, no invasion/metastasis (4.11) < a H * , invasion/metastasis

where for carcinomas, the parameter ~ could depend upon the number of spot desmosomes.

The significance of the mean curvature from a geometrical point of view can be found by considering each point (u, v) on an oriented surface V = x(u, v), and measuring along the normal v to V at (u, v) a distance g(u, v). The points thus determine a new surface, which we shall denote by Vg. For instance, if g is a constant, Vg is parallel to V. Then

d [area(Vtg)](0) = 2 f g(u, v)H(u, v) du dr. (4.12) dt Jv

Thus, for a growing turnour, the mean curvature at each point of its surface also gives an indication of the rate of change of the local surface area around each point. Once again this information could indicate where points of invasion and metastasis are likely to occur or have already occurred. From a surface map of a tumour, we can find out those regions of the tumour surface where the mean curvature is either locally positive or negative in the neighbourhood of a point. Regions of positive mean curvature denote where the local area growth is increasing relative to the underlying global tumour growth rate, while those regions of negative mean curvature denote where the local area growth rate is decreasing relative to the overall tumour growth rate. Thus, theoretically, the surgeon/pathologist is given a better knowledge as to whether cells have sepa- rated from the tumour or not and so can gauge better whether or not invasion and metastases have taken place. For benign tumours, we would expect there to be little local variation in the mean curvature o f the tumour since, as a general rule, benign tumours remain compact masses with well-defined boundaries, whereas for malignant tumours, we would expect a lot of local variation since here invasion and metastasis often occur and the tumour boundary is ill-defined.


The Gaussian and mean curvatures of a tumour surface thus provide information about the local growth rate and the local surface tension force over the tumour surface and hence give an indication as to the likelihood of invasion and metastasis having occurred. These proposals, taken together with the results of the previous section on classification via strain-energy functions, can theoretically go together to form a new mathematical description of turnouts from the point of view of what is known in the medical profession as staging and grading. This is discussed in the following section.

4.4 Staging and grading of cancers

Improvements in the treatment of various forms of cancer are dependent on the accuracy of diagnosis of the presence and type of tumour and the extent of its spread. The information provided by routine histological techniques is by no means always precise. A quantitative measure of the factors which are known to influence the prognosis of a particular type of malignant tumour is often required. Sometimes it is used in deciding on the best method of treatment for a particular patient, but it is also very useful in statistical studies, notably in the comparison of the effects of different forms of treatment. It is essential to ensure that a group of patients treated by one method does not contain a higher proportion with smaller, earlier or better-differentiated tumours than does a group treated by the other method. It is difficult to establish firm criteria, applicable to cancers in general, for this purpose, and elaborate sets of criteria have been set up for most of the common individual forms of cancers. It is, however, generally agreed that there are two main factors involved: the grade and stage of the tumour.

Grading is based on histological examination of the degree of differentiation of the tumour cells and the mitotic index. There is no precise system of grading, for assessment of the degree of differentiation is empirical and subjective. Also, different parts of the same tumour vary in the degree of differentiation and mitotic activity, and a biopsy specimen may not be representative of the whole tumour. In general, tumours are graded by the pathologist into well, moderately and poorly differentiated: the system is most useful if the criteria used place 25% of tumours into the good group, 50% into the middle group, and 25% into the bad group.

Staging provides an estimate of the degree of spread of the tumour. Many systems use four or more stages, the criteria being modified to suit cancers of various sites. For example, squamous-cell carcinoma of the uterine cervix might be staged as follows

Stage Extent of spread 0 Carcinoma in situ (no invasion) I Confined to the cervix II Limited local spread III Greater local spread e.g. to the lower vagina or pelvic wall IV Lymph-node or distant metastases

Alternatively, the TNM system may be used, in which staging depends on the size of the primary tumour, graded as T1-4, absence of lymph-node involvement (NO) or involvement of few (N1) or many (N2) nodes, and distinct metastases absent (M0), few (M1), or many (M2), e.g. T2, N1, M0.


Assessment of the extent of spread of a tumour will depend on the methods used in its detection. The simplest, but least precise method of staging is based on clinical examination and has the advantage that it can be applied to all patients. More accurate staging is provided by combining clinical examination and imaging techniques, including lymphangiography, or by surgical exploration. Whatever methods are used, it is, of course, essential that they should be applied to all the groups of patients involved in the clinical trials of different forms of therapy.

In some conditions, accurate staging is of great clinical importance to the individual patient. For example, in Hodgkin's disease (a malignant tumour mainly composed of reticulum cells), ratiotherapy is commonly used when the disease is localised, and drug therapy when it has spread to one or more distant sites. Where distant spread is not evident on physical examination or radiogra- phy, it is common practice to perform a laparotomy and, if necessary, excise the spleen and examine it histologically for evidence of the disease.

Not surprisingly, tumours of high grade (i.e. poorly differentiated and with a high mitotic index), are likely to have spread more extensively than low-grade tumours at the time of diagnosis, and so there is some correlation between grade and stage.

From the results obtained in Sect. 3 concerning the classification of tumours via strain-energy functions, and those obtained in this section concerning the measurement of the local growth rate and local surface-tension force via surface geometry, we now propose a theoretical basis for a method of the processes of grading and staging of tumours, involving the measurement of three parameters which we have associated with tumours:

(i) S t r a i n - e n e r g y f u n c t i o n - as we have seen, this gives a measurement of the degree of malignancy of the tumour and the degree of differentiation of the tumour cells.

(ii) G a u s s i a n c u r v a t u r e - this gives a measurement of the local growth rate of points on the surface of the tumour. (iii) M e a n c u r v a t u r e - this gives a measurement of the local variation in the surface-tension force over the tumour surface, and so indicates the likelihood and degree of spread of the tumour v i s - h - v i s invasion and metastasis.

Classification of turnouts using the three parameters defined above will enable the pathologist to identify all of the main factors which are determined by grading and staging i.e. the degree of differentiation of the tumour ceils, the mitotic index, and the degree of spread and likelihood of metastasis.

Finally, we mention that to give a complete description of the degree of malignancy/differentiation of the tumour, we allow for the possibility of the strain-energy function changing with time i.e. as the tumour becomes less well differentiated, the strain-energy function will also change. In order to allow for this, we assume a time variation of the parameters #r, and at. We thus write

W(/~, #r, O~r, t) ~--- ~'~ p~(t) ~r( t ) {2]'r(0 + 2~r(`) + 2~r(,) -- 3}, (4.13)

which becomes, under the assumption of incompressibility

I~()., t) = x7 #,(t) {22~(,) + 2-2~(° _ 3}. (4.14) ";r ~(t)

Thus W(2, #r, G) ~ W*(2, #*, ~*) as the turnout becomes less differentiated.

Modelling the growth of solid tumours

Table 4. Variation of strain-energy function and degree of differentiation

461

Well-differentiated Moderately differentiated Poorly differentiated W~ g/~ W*

4.5 Application of the theory

In this section we examine how the theory of the previous sections can be applied in a real biomedical situation. In particular we look at ways of making quantitative measurements of the strain-energy, Gaussian curvature and mean curvature.

Recent technological advances now make it possible to extract objective and quantitatively accurate information from 3-I3 images currently produced by techniques such as CT, MRI, PET, and SPECT. There now exist recently developed software packages such as ANALYZE (cf. Robb and Barillot 1989) which allow the properties of solid turnouts to be viewed from the surface geometry point of view. ANALYZE is a comprehensive package which permits detailed investigation and evaluation of 3-D biomedical images. It can be used to display, manipulate and measure multidimensional image data obtained from any one of the aforementioned imaging modalities. Accurate and reproducible measurements of curvature is also a topic of much interest in the general computer vision and image analysis community and already has a wide range of applications. The field of image analysis is still developing rapidly and its continuing evolution promises even greater capabilities for accurate and objective non-invasive diagnoses, as well as for genuine quantitative biological investi- gations. The relevant data/information required to implement our technique is already available from surface intensity images of biomedical objects. In particular the coefficients of the first and second fundamental forms (E, F, G and L, M, N of the previous sections) can be calculated from an intensity image of the surface using, for example, the Plessey operator. From this information it is possible to construct what is known as a topographical primal sketch of the surface and to classify every feature in the entire image. This, in turn, enables one to identify directly interesting local regions in an image which are associated with two dimensional even t s - in our case, local surface growth. Thus the differential geometry of the surface is used to identify the desired features (Haralick et al. 1983).

The recent work of Duncan et al. (1991) concerning the measurement of cardiac shape deformity uses the bending energy of a surface (similar to our strain energy) as a means of measuring cardiac shape deformability. The ideas forwarded in this paper and tested on clinical specimens and on theoretical shapes such as ellipsoids are very similar to our proposals regarding the strain energy function of materials. Information required from the 3-D images is the curvature of the surface and a contour map of surfaces in order to calculate changes in shape. As well as testing the method on 3-D images of the left ventricle, the method was also shown to work well in detecting the shape change in computer-generated images of ellipsoids. Using our proposal of characterising a tumour surface with a strain energy function, a modification of this technique may therefore be particularly useful in detecting breast tumours which are often spherical or ellipsoidal in shape (Parham et al. 1988). The use of bending energy


in modelling surface morphogenesis has also been successfully applied at the cellular level in modelling the shape changes of red blood cells (Svetina et al. 1982) and at the sub-cellular level in modelling the morphogenesis of liposomes (Sekimura and Hotani 1991). We believe that modification of the above technique of Duncan et al. (1991) is certainly possible as the work of Terzopoulos et al. (1987) in obtaining realistic computer graphic images of elastically deformable shapes shows. Here progress has been made by using precisely the methods suggested in this paper - i.e. characterising the elastic solid using a strain energy function. By modifying the parameters associated with the strain energy function different materials can be characterised (as we have demonstrated in the previous sections) and hence computer representations of a variety of deformable objects can be obtained. Some of the most important parameters in the modelling concern the effect of surface tension in the solids. The techniques of this model can be easily extended to simulate nonlinear elastic phenomena such as the bending of thin shells. Knowledge of the strain energy function thus provides a tool for the calculation of the shape. Different strain energy functions will permit different shape changes, which is the basis of our theory.

An attempt to design an automated system to detect and classify breast tumours (a leading cause of death among all cancers for women of middle-age and older) using data/information obtained from mammograms has recently been proposed by Brzakovic et al. (1990), building upon earlier work by Dhawan et al. (1986) and Lai et al. (1989). At present the interpretation of such information from mammograms is performed by experts who visually examine the results for the presence of deformities that can be interpreted as cancerous changes. However, these manual readings are labour intensive and multiple readings of a single mammogram are necessary in order to increase the reliability of the diagnosis. Using the AMA (Automated Mammogram Analysis) system, Brzakovic et al. (1990) developed a method for classifying breast tumours. This utilises information from the mammogram such as size, shape and intensity changes of extracted regions and involves three steps:

(i) enhancement of pre-selected features and removal of irrelevant details using application-dependent techniques.

(ii) localization of suspicious areas. (iii) classification of these areas into non-tumours, benign or malignant tumour areas.

The method was shown to be quite accurate and predictions were reasonable. In particular, the AMA system was shown to be very successful in detecting regions which warrant subsequent attention i.e. it was able to distinguish successfully between normal tissue area and neoplastic tissue area, but it was less successful in actual recognition i.e. it was not so efficient at distinguishing between benign and malignant tumours. Thus further refinement of the method is still necessary. As we have seen, the information regarding the first and second fundamental forms of a surface can be obtained from intensity images of surfaces i.e. from data that is available with state-of-the-art technology. Thus information regarding the Gaussian and mean curvatures of the turnout surface can be obtained and using this information, as we have proposed in the previous sections, the tumours can be staged and graded. This would improve and refine the present AMA system. Work is currently being carried out to investigate this development.


From the above results on actual clinical specimens we believe that it is certainly possible to implement the theory of our models. Applying our theory to the clinical specimens of the breast tumours in the paper of Brzakovic et al. (1990), the information regarding changes in the local Gaussian and mean curvature of the surfaces of the breast tumours is available and an application of our theory would enhance further the classification process forwarded by Brzakovic et al. (1990) into a grading and staging process, thus providing a more objective and more accurate picture of the breast tumours. We believe that this work could be the starting point for new and interesting developments.

5 Conclusions

Throughout this paper, we have used the same approach to both tumour and cell growth as Greenspan (1976) and Schwegler et al. (1987), respectively i.e. the superposition of small perturbations upon large deformations. In our models, we have attempted to describe the growth of a turnout (and also the expansion and division of cells) via elasticity theory, considering the growth of the tumour (cell) to be similar to the inflation of a balloon. In this model, the thin layer of live, proliferating cells which surrounds the necrotic core of the tumour, is treated as the balloon membrane. The gross internal forces are indeed modelled by a pressure distribution i.e. the inflationary pressure, as was the case in Greenspan (1976). This approach also allows for consideration of a more varied composition of the interior of the tumour i.e. the possibility of including stroma, blood vessels etc. within the tumour centre, rather than simply a completely necrotic core, composed entirely of dead and dying cells. This approach concentrates on the activity of the layer of live cells on the tumour periphery or surface, since it is from here that invasion of the external tissues takes place and from here that individual cells or groups of cells may break off and form metastases. The criterion for bifurcation into aspherical configurations is given, indirectly, in terms of this layer (since the bifurcation criterion can be written in terms of the strain-energy function which is itself a measure of the proliferating cells' activity).

Also, all of the main features of Greenspan's model are reproduced i.e. the pressure/surface-tension/curvature balance of forces on the tumour surface, the inflationary, internal pressure P, and the implicit role played throughout by the surface tension while in the end determining whether or not the turnout will invade the surrounding host tissue, the onset of which is characterised by bifurcation into an aspherical equilibrium configuration associated with a high order mode, n i> 2.

The idea of classifying and characterising benign and malignant tumours by their different types of strain-energy functions is new and the class of strain energy functions considered here (i.e. those introduced by Ogden 1972) offers a wide variety of choices to be made corresponding to the wide variety of tumours that exist in widely different stages of differentiation and malignancy.

As was suggested by Greenspan (1976) that his model could be adapted to model tumour growth with "thick" layers of proliferating cells, this use of elasticity theory can also be adapted to model these situations by using thick shells. This was done in Sect. 3 and as was demonstrated there, the results


predicted correspond well with the findings of Parham et al. (1988) which showed that the number of tumour cells present is a significant factor in determining whether or not the patient will survive. The ratio of the number of tumour cells to the amount of stroma present is critical. Turnouts with a greater proportion of cells to stroma (i.e. thick layers of cells) were found to be less likely to invade and metastasise. These results and findings would seem' to confirm our approach and support our hypothesis and results. In our model, we take this ratio to be 2o/2b, i.e. the number of tumour cells is proportional to the outer shell thickness, while the amount of stroma present is proportional to the size of the interior core. If this ratio is too great, then no bifurcation into any mode number n is possible, which means that the tumour cannot invade the host tissue.

At the cellular level too, the differences characterising normal cells from neoplastic cells, and then those distinguishing cancer cells from benign tumour cells, can be given in terms of different strain-energy functions, which then go on, at the global tumour-level, to influence the nature and course of action of the whole tumour itself. Once again our attention is focussed on the cell membrane (plasmalemma) since at the cellular level this is where there is the greatest difference between normal cells and neoplastic cells (Muir 1985). We do ac- knowledge, however, that the process of cell division involves much more complicated processes which occur in the interior of the cell, and our model of cell division via membrane theory is in no way intended to be a definitive one which includes all of these internal processes. These are given in greater detail, for example, by Greenspan (1977a, 1977b, 1978), Wu et al. (1988), Zinemanas and Nir (1988).

Finally, we note that the main results to emerge from the analysis of the growth and development of tumours and their cells via membrane and shell theory are confirmed from the medical evidence- bifurcation from a radially symmetric shape into an aspherical equilibrium condition (signifying the onset of local invasion of the surrounding tissue and metastasis) depends upon both the type of material of which the membrane/shell is composed and characterised by a particular strain-energy function (corresponding to the degree of malignancy of the tumour) and upon the thickness of the shell (corresponding to the number of cells present).

In modelling such complicated phenomena as turnout growth and cell growth and division, it is impossible to capture all of the inherent features without obviating the subsequent mathematical analysis and the present model is no exception. As with any other model of turnout growth, this one has its own limitations and there are other factors which have a bearing upon the processes of invasion and metastases and cell shape change other than the mechanical properties considered in this paper. However, by focussing upon these mechanical properties, a novel approach is achieved which retains many of the essential features of previous models, while introducing a new qualitative measure of both the tumour and the tumour cells. This work is in no way intended to be a definitive model, but rather the foundation for subsequent work in this direction, with future developments including time-dependent variables, and an investigation into the exact nature of the constitutive equations for different tumours.

Acknowledgment. The work of one of us (MC) was supported by a SERC research studentship.


Appendix 1: Elasticity theory

(i) Deformation

Let an elastic body occupy a certain region £2 of R 3. The position of each particle in the body can be described in terms of a set of coordinates, referred to a fixed frame of reference. When the body is deformed, each particle occupies a new position which is described by a new set of coordinates. Let X = (X~, X2, I"3) be the position vector of a typical material point in the fixed reference configuration, and suppose that this particle (initially at X) moves to a point with position vector x = (x~, x2, x3) in the current configuration. Thus the deformation of the material can be described according to the mapping

x : X ~ x ( X ) , (AI.1)

or, using indicial notation,

x i = xi(Xj), i , j = 1, 2, 3. (A1.2)

In the reference configuration, we choose ~i (i = 1, 2, 3) as a triad of curvilinear coordinates. Basis vectors associated with this are then given by

0X E i = ~ , ( i = 1 , 2 , 3 ) . (A1.3)

In a similar manner let ~i (i = 1, 2, 3) be a triad of curvilinear coordinates in the current, deformed material, with basis vectors defined as

Ox ei = ~ , (i = 1, 2, 3). (A1.4)

There are thus two coordinate systems which can be used to describe the deformation and the problem may be formulated with the material coordinates Xj as independent variables, giving a material or Lagrangian description, or with spatial coordinates x~ as independent variables, giving a spatial or Eulerian description. In a material description of a problem, attention is focussed on what is happening at, or in the neighbourhood of, a particular material particle. By contrast, in a spatial description, events at, or near to, a particular point in space are important.

Associated with the deformation we can define what is known as the deformation gradient tensor ~:

dx = ~-~. (A1.5)

This is what is used throughout the paper as the measure of the deformation of the body. It can be thought of as the "rate of change" of the current configuration with respect to the reference configuration.

We assume that the Jacobian

J = det a, (A1.6)

the local ratio of current to reference volume, exists at each point and that J > 0. The Green (or Lagrangian) strain tensor e can be defined in terms of e as:

e = ½(~r~ _ 6). (A1.7)

where 6 is the identity tensor.


It is useful to note that the stretch ratios "~1, ~'2, /~3 are related to the strain components e. (i = 1, 2, 3) in the following manner:

1 2 e,; = ~(2; - 1). (AI.S)

The stretches 2; are measures of how much line elements are stretched during the deformation of the body. For example, when considering a membrane with spherical geometry, ,~3 is a measure of the ratio of the current thickness of the membrane to the thickness in the initial membrane configuration. 21 and 22 have similar physical interpretations. It can be shown that the deformation of a body can be completely defined in terms of the three principal stretch ratios, provided that the orientations of the principal axes are known.

(ii) Stress and equilibrium equations

We denote by N and n the outward unit normals to the surfaces of the body in the reference and the current configurations respectively. There are two stress tensors which we can define, corresponding to the two configurations. We have

srN, (A1.9)

as the (current) load per unit reference area, where s denotes the nominal stress tensor, while

an, (AI.10)

is the load per unit current area, where ~ is the true (or Cauchy) stress tensor. These two stress tensors are related by the equation

JtT = ats, ( A 1.11)

which for incompressible materials reduces to

= ats, (Al.12)

since incompressibility implies no change in volume and so J = 1 (we note that in our model of solid tumour growth, only the layer of live, proliferating cells is treated as an incompressible membrane and the tumour as a whole can and does undergo an increase in volume as it grows and expands).

Neglecting body forces, the equilibrium equations can be written either as

Div s = 0, (AI.13)

or, equivalently

div a = 0, (Al.14)

where Div and div denote the divergence operator with respect to the reference configuration X and current configuration x, respectively.

(iii) The constitutive equation

As we have stated previously, the constitutive relation is a relationship between the stress and the strain, defined by the properties of the material under consideration. In infinitesimal theory, where the deformations are assumed to be


small, this relationship is linear and takes the form:

a ij = Coktekl. (Al.15)

This is sometimes referred to as the generalised Hooke's Law and the analogy with the familiar spring equation F = kx , where k is a constant depending upon the material of the spring, is clear. If the further assumption that the stress is isotropic (i.e. there is no preferred direction in which the stress acts) is made, then it can be shown that this simplifies further to

crij = 2ekgrij + 2/~e~, (A 1.16)

where 2,/~ are constants depending upon the particular properties of the material. The above relationship has the advantage of being easily invertible and so the strain can also be written in terms of the stress.

However, for finite deformations, the relationship between the stress and the strain becomes nonlinear and a more appropriate formulation is to consider the strain energy function, W, of the material. This is a function of the strain components e;j and may be thought of as three dimensional counterpart of the potential (i.e. stored) elastic energy in a spring. The assumption of isotropy may also be incorporated by considering W as a function of the principal stretches 21,22, 23 only. (In the case of linear theory, it is instructive to note that W has

1 the form W = ~Cogfeijekt, which is seen to be the analogy of the stored elastic energy in a spring, ~kx2.) We thus suppose that our elastic body is composed of a material which is eharacterised by a strain-energy function W and we write the constitutive relation (i.e. the relationship between the stress and the deformation of the body) as:

OW s - O~t " (Al.17)

Appendix 2: Thin shell theory

We write the position vector of a point in the reference configuration of the shell in the form

X = A(~ 1, ~2) .q._ ~A(~ 1, ~2) (A2.1)

where ~ now denotes ~3, A(~l, ~2) denotes a point on the middle surface of the shell, and (~1, ~2) are parameters describing this surface. The unit vector A3 (~ 1, ~2) is the (positive) normal to the middle surface and ~ is the coordinate measured along the normal.

The shell is defined by 1 1 - ~ H ~< ~ ~<~H (A2.2)

and is bounded by the surfaces ~ = _+ 1H where H is the thickness of the shell in the reference configuration.

Similarly, we can write

X = a(~ 1, ~2) + ffa3(~l, ~2) (A2.3)

as the position vector of a point in the deformed shell, with ffi= (~1, if2, ~) the curvilinear coordinates describing the deformed shell.


It is convenient in thin shell theory to give a description of the deformation of the shell in terms of average values - in our case, the average throughout the reference thickness of the shell. Hence, for the deformation gradient 0t, our measure of the deformation of the membrane, its average, denoted by 6t, is defined as

1 I 1~ ~t = ~ J-½H 0t d~. (A2.4)

Appendix 3: The incremental equations

We suppose that a point a in the finitely-deformed, radially-symmetric configuration is now given an incremental displacement a, say i.e.

a ~ a + a. (A3.1)

The average deformation gradient and the average of the nominal stress will also undergo corresponding incremental displacements.

a ~ + & , ~ + } . (A3.2)

In order to evaluate these increments, we first of all write the average deformation gradient ~t (our measure of the deformation of the layer of live cells throughout the paper) in the form

~a Ct = ~ 4- ~3a3 @ ,4 3 , (A3.3)

where a is the position vector of the point of the membrane which had position vector A in the reference configuration. The unit vector .43 is normal to the membrane in the reference configuration at the point A (see Haughton and Ogden 1978a).

Then, taking the increment of (A3.3) and evaluating the result in the current configuration will give us ~to, which we will denote by il. This can be written as

~0 ~ fl = ~ a + ~3~3 la3 @ a3 + a3 @ a 3 . ( A 3 . 4 )

The only non-zero components of a i ' a j , k (appearing in the equilibrium equations (3.14)) are given by

~ - a 3 • ax,1 = - a 3 • a2,2 = a l • a3,1 = a2 • a3,2 (A3.5) ( a l • a2,2 = - a z • a m = - r - 1 cot 0.

The components of /! on the basis ai can now be calculated from (A3.4) and Eqs. (3.15) with the help of (A3.5), giving:

" u + Vo u + v cot 0 + w~/sin 0 ?]11 = - - ' /~22 - - , /733 = --/~11 - - ~22 r r

£h2 = v~o/sin 0 - w cot 0, 621 =--w° (A3.6) r ?"

Uo - v u ~ / s i n 0 - w ~ - - - - ~1~, ~ 3 2 - - r/~3 t" r

with subscripts 0, tp denoting partial differentiation.


The incremental displacement in the average nominal stress, referred to the current configuration, can be found from the incremental constitutive law (see Haughton and Ogden 1978a) giving

= B~] +fi~] - /38, (A3.7) where B is the average of B the tensor of instantaneous elastic moduli. B plays a similar role in relation to 30 and ~t0 as the tensor of elastic moduli plays in relation to the stress and strain.

Now for the growing tip, which we consider to be an incompressible, isotropic elastic solid, the components of B on principal axes have been given by Ogden (1974) in terms of the principal stretches 2~ and the principal components o-e (i = 1, 2, 3), here given by (3.6). For a membrane, the corresponding components of B are given by the same formulae but with 2i and ai replaced by 2~ and if,.. The only non-zero components are as follows (referred to the principal axes of ~)

/~,jj = Bsj,, = ~.,2j O2e 02/' (A3.8)

/~0~ -/~/JJ~ =/~/~ -/~'~o = fit, i Cj. (A3.10) In terms of incremental quantities, the membrane approximation takes the form:

~03~ = 0, (i = 1, 2, 3), (A3.11) (see Haughton and Ogden 1978a, Sect. 4, for details). In fact, S03,u = 0, (~ = 1, 2), follows immediately from (A3.7) on substitution of (A3.10) together with expressions for 03~ (i = 1, 2, 3) derived from (A3.4) and the membrane approximation given in (3.8). The further approximation ~033 = 0 serves to determine/~.

Thus the components of ~ can now be written in full as

S011 = /~II11/'711 + B1122J~22 + Bl133~33 '~-/~/111 - - / ) , (A3.12) ~022 = /~1122t]11 -~- /~2222t]22 "q- /~2233 t]33 "~- fi~22 --/~, ( A 3 . 1 3 )

0 = /~1133/7111 "Ji- B2233/722 +/~3333033 '~-/~633 - - / ) , (A3.14) ~o~2 =/~1212021 + @~221 +/5)f/12, (A3.15) So21 = @2112 +/5)t/2~ + B2121 q12, (A3.16)

~0,u3 = O'#,u/~3p, (~ = 1, 2). (A3.17) For the problem in which the underlying finite deformation is radially

symmetric, Eqs. (3.3) and (3.9) hold, and consequently t~ 1 = t7 2. The expressions (A3.8) and (A3.9) then reduce in number according to:

/~2222 = Bl111, Bll33 =/~2233, B1313 = B2323 , (A3.18) with consequent implications in respect of (A3.10). Also since "~1 = )72, we have

1- B12t2 = ~2~ ~ (~1 - 62) = BEI2~, (A3.19)

in which 22 is set equal to ~1 after differentiation. It is easily shown that /~1212 ~ 1 -- ~(Blm -- B~122 + fix). (A3.20)

470 M . A . J . Chapla in , B. D. Sleeman

Having assumed that the deformation of the membrane is axisymmetric i.e. independent of cp (cf. Greenspan 1976), then the equilibrium equations (3.14), with i = 2 and use of Eqs. (A3.8)-(A3.20) yields the equation

woo + wo cot 0 + (1 - c o t 20)w = 0, (A3.21)

which leads to the solution of Eqs. (3.16). Similarly, use of (A3.8)-(A3.20) in the equilibrium equations (3.14) with i = 1, 2 leads to the two Eqs. (3.24), (3.25).

The three independent elastic moduli, denoted by 2; 1 , Z2, S,3, are defined as

2;1 = ¢~11, (A3.22)

2;2 = B l l l l + / ~ 3 3 3 3 - - 2Bl133 + 2fi, (A3.23)

2;3 = 2B1212 ~ /~1111 - - /~1122 + G1. (A3.24)

From (3.9), we note that we have the following equalities 1 A S_,~ = ~2W;~, (A3.25)

1 2 ^ 22; 2 + 2; 1 - 2; 3 = ~2 W~. (A3.26)

Since the membrane is undergoing inflation, the surface stress 8ll will be positive. Thus

2;1 ~> 0, (A3.27)

with equality holding only for 2 = 1.

Appendix 4: Thick sheH theory

We first derive explicitly formulae (3.53) and (3.58). By definition, we have

0-11 - - 0"33 : ½~W2. (A4.1)

The only equilibrium condition which is not satisfied identically is

d0"33 2 + -r (0"33 - - 0"11) ~- 0, (A4.2)

in the absence of body forces. This can be rearranged to give

d0"33 21~,~ (A4.3) dr r

Now since 0"33 (a) = - - P and 0"33 (b) = 0, integration from r = a to r = b leads to

= -I'°21~ dr (A4.4) P da r "

Differentiation of (3.47) yields dr dR - - = R + 2 - - (A4.5) d2 d2"

Use of (3.47), (3.48) along with the chain rule for differentiation then gives

dr 3 dr (A4.6)

Modelling the growth of solid tumours

which finally gives

Equa t ion (3.53)

then follows s traightforwardly.

471

dr d2 2 r 1 - 2 3 " (A4.7)

(A4.13)

(A4.14)

In deriving (3.58), we first o f all note f rom (3.56) that

d2b - ~ -2~2 (A4.8) d2a

Different iat ion of (3.53) with respect to 2a gives

aP (A4.9)

d & - l ) - 1 )

Use o f (A4.8) and a few lines o f a lgebra then yields the required result o f

(ha -- 2 a 2) d2 a )2 j 2 (A4.10)

Finally, we give the equat ions for n = 1, 2, 3 , . . . upon subst i tut ion o f (3.73) into (3.67) and (3.68), as well as the cor responding expressions for the bounda ry conditions.

[r4B3~3f " + (r4B~131 + 4r3B3~31)f~ + r3(2B~131 + a'33)f', + r2(mG - r ) f ' , ]"

+ (m 2)[r2B~13a 2 . - -- r ~r33 + rF ' -- F + mB1313]f n = 0, (A4.11)

for n = 1 ,2 ,3 . . . . .

mrhn = r3B3131J °" + r2(rn'3131 + 6B3131)J ~ -~- r[2rB'3131 + ra'33 + 6B3131 - - F

+ re(B3113 + Bl133 -- Blll l)]f ; , + (m -- 2)(rB~131 -- ra'33 + F) f~ , (A4.12)

for n --- 1 , 2 , 3 . . . . . While the bounda ry condit ions yields

r3B3131f " n t- r2(rB~3131 -t- 6B313t)30~

-+- r[2rB/3131 Jr- ro-;3 -- F + m (G - 93131 ) + 6B3131]f ~

+ (rn -- 2)[rB~131 - ro'~3 + Flfn = 0 on r = a, b,

r2f" + 2rf', + (m - 2 ) f , = O on r = a, b,

for n = 1 , 2 , 3 . . . . .

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