Fractal Geometry of Breast Cancer 1

 Breast cancer is considered to be the most commonly diagnosed cancer amongst women in the world. Sadly, breast cancer also seems to be the leading cause of cancer death in women. Incident rates of breast cancer appear to be higher in developed countries, while mortality rates due to breast cancer appear to be disproportionately higher in developing countries. This may be a reflection of asymmetries between the countries in access to early detection and subsequent treatment.

Breast cancer can be considered as a complex system where a number of cells, both healthy normal and cancer cells, interact with each other to create emergent properties. Geometrically, the growth of breast cancer exhibits complex patterns. The characteristic shape appears to depend on the type of breast cancer. In this article, we analyse the geometric properties of breast cancer growth. We characterise two main types of breast cancer as fractals and examine their fractal structures.

To put our approach to a test, we simulate the growth of these two types of cancer by using the dielectric breakdown model. Our approach demonstrates a sufficient ability to realistically replicate various tumour growths. We also analyse the prediction of cancer tumour growth.

As an introductory part, this article aims to establish a common background for further investigations to ensue in the sequels.

Common types of breast cancer

There are two common types of breast cancer, (1) invasive ductal carcinoma (IDC) and (2) invasive lobular carcinoma (ILC).

1) IDC is the most common type of breast cancer and accounts for 70% to 80% of breast cancer. IDC starts in milk ducts and spreads to other breast tissue.

2) ILC is the second most common type of breast cancer and accounts for 10% to 20% of breast cancer. ILC begins in milk-producing lobules and spreads to other breast tissue.

There are also other types of cancer but they are much rarer than IDC and ILC. Geometrically, these two types of invasive carcinoma account for the vast majority of breast cancer growth patterns. Therefore, we model the growth patterns of these two types and numerically simulate them in this article.

Geometric properties of IDC and ILC tumours

As the bottom of Figure 2 shows, cancer cells and normal healthy cells greatly differ from each other in various aspects: structure, shape, colour, volume, size, cell division rate and metastasis (how they spread to other organs).

The geometries of two types of breast cancer, IDC and ILC, are also very different. As shown in the top of Figure 2, IDC forms a tumour which looks like a lump. Every cancer cell is densely radially arranged around a single focal cell immediately adjacent to the other cancer cells. On the other hand, an ILC tumour looks stringy without creating a prominently concentrated cluster.

The main reason for the cause of the two shapes is due to the protein, E-Cadherin. E-Cadherin is basically a cellular “glue” which helps cancer cells bind together. While E-Cadherin is present in IDC, it is almost absent in ILC. In addition, though not ubiquitous, one could possibly find a combination of both characteristic shapes in present, e.g., invasive ductal carcinoma with lobular features.

Figure 3 shows a mammographic image of IDC. The two dotted circles in green show the primary presence of IDC tumours. On the left-hand side, a distinct circular mass is observed inside the dotted circle. It looks like a dense lump radiated from a single focus in the centre of the tumour. The margin of the tumour seems clearly defined and wobbly. On the right-hand side, a prominent lump is observed. Notice that many spikes project from the edges of the tumour (“spiculated”) akin to the pattern of a-hub-and-spokes.

Figure 4 shows mammographic images of ILC. The tumours observed inside the circles in Figure 4 have many branches which are filamentary. The branches look more like a pronounced vascular structure similar to a network of capillaries rather than a lump that one may be accustomed to seeing in a tumour. Unlike the IDC tumours in Figure 3, the ILC tumour in Figure 4 does not appear to have a core or a significant focus in itself.

In general, it is easier to detect IDC than ILC because of a lumpy tumour which may be physically tangible. Many people often notice a lump in their breast, thereby prompting them to consult a physician. Detecting IDC by remote sensing, such as mammograph and MRI, is also visually easier because of the much higher density of cancer cells concentrated in a specific area.

ILC is trickier to detect because of its stringy characteristic shape. An ILC tumour does not usually appear as a lump. Hence, a simple physical examination often fails to spot the presence of an ILC tumour at a relatively early stage, since the tumour is often not palpable. ILC tumours tend to be more linear and very thin as if cancer cells leapt from one area to another and the margin of the tumour might remain very unclear. Because of this, it is also more challenging to visually detect ILC tumours by the standard detection methods, such as mammogram or MRI. Consequently, the lobular form of breast cancer tends to be diagnosed at later more advanced stages ([BCRF]).

Cancer as a complex system

Researchers have been looking into all aspects of breast cancer. Much of the existing investigative effort has been made to understand breast cancer at the genetic and molecular level. Such research typically tries to understand the cause of breast cancer. There are many reasons, perhaps, too many reasons for breast cancer to occur. It appears infeasible to uncover all possibilities and their combinations. To put it another way, no two cancer cases have exactly the same cause.

In this article, we regard breast cancer as a complex system. We often use the word “complex” to describe something. However, complexity is not necessarily easy to define. We use the following definition. “A complex system is a system which exhibits phenomena emerging from a collection of interacting components.” Breast cancer fits the bill as a complex system for the following reason.

Assume that a single cancer cell appears in the breast all of a sudden for no apparent known reason. If it remains dormant without spreading, it is not life-threatening. However, the cancer cell may start to grow if it gets enough nutrients and finds space to grow into. The nutrients are mainly oxygen and glucose. They need to be supplied to the cancer cell in the form of blood. Therefore, the cancer needs to access blood vessels and capillaries. The cancer cell faces fierce competitions with normal cells in the neighbourhood in pursuit of the essential nutritional supply.

To cheat the body’s regulatory mechanism, the cancer cell releases some chemicals to promote the growth of new capillaries. This phenomenon is called “angiogenesis”. See Figure 5. If successful, the cancer gets the nutrients and grows into available space by uncontrolled cell division. Breast cancer repeats this process to form a colony of cancer cells or a cancer tumour

Tumours as fractals

Just as a single air molecule has no will to be a cloud, a single breast cancer cell has no consciousness to form any particular shape. However, two distinctive shapes emerge from the collective for the two main types of breast cancer tumours. Geometrically, these characteristic shapes define the types of breast cancer.

As it grows, the tumour, with no will, continues to replicate the same characteristic pattern adding more cancer cells to the original tumour cluster. The growth of the tumour is self-organised, self-similar and scale-invariant, i.e., everywhere in the tumour looks akin to itself regardless of zoom scale. Self-similarity and scale-invariance are quintessential characteristics of fractals [MA].

As described in [HD1] and [HD2], a fractal is a self-similar geometry. A cancer tumour repeats the same geometric pattern as it grows and it appears similar on various scales or scale-invariant. Scale-invariance is a specific type of self-similarity. Such a geometry follows a power law

where M is the mass of the geometry, r is the characteristic radius and D is the fractal dimension.

Consider a digital image, such as an X-ray scan image. Think of M as the number of pixels to create a certain shape in a given circle with the radius r. The fractal dimension is given by

In other words, D is the slope when we plot ln(M(r)) on the y-axis against ln(r) on the x-axis. Figure 6 shows an example where the fractal dimension D is identified to be 1.69 for the diffusion-limited aggregate (DLA) model [WI1] based on simulation results with four radii.

Figure 7 shows the relationship between the fractal dimension D and the characteristic geometry. Imagine each shape grows from the centre of the dotted circle. For D=2, the tips of branches uniformly reach the perimeter of the dotted circle and the inner pixels are densely occupied to create a circle inside the circle as if the growth was designed to perfectly blot the circle in pale blue. This is the type of growth demonstrated by the Eden growth model (EGM) [WI2]. Therefore, the geometry is Euclidean or D=2.

For D=1.71, the object inside the dotted circle has a pronounced radial shape. The growth pattern appears as if it assigned equal probabilities to all directions when growing by adding a new pixel to the cluster. This growth pattern can be characterised as a “random walker” which behaves like a Brownian particle randomly charging a new particle in the immediate vicinity to join the cluster. This is the quintessential growth pattern exhibited by the DLA model.

For D=1, this geometry is very filamentary and polar-opposite to the Euclidean example with D=2. The growth is extremely biased towards specific directions on the perimeter of the dotted circle. This shape is no longer two-dimensional but virtually linear.

In terms of surface dimension, a fractal can be defined as a geometry with the dimension D such that 1 < D < 2. Geometrically, the growth patterns of tumours induced by breast cancer are neither Euclidean nor linear. They are fractals.

As seen in Figure 3, a tumour induced by IDC resembles the geometry represented by DLA in Figure 5 or somewhere between Euclidean (solid sheet) and DLA (radial and lumpy with some spikes).

Similarly, as Figure 4 demonstrate, we can characterise the dimension for a tumour induced by ILC to be 1 < D < 1.7, since they are stringy rather than lumpy but are not quite straight lines.

Modelling the growth of breast cancer

[Feel free to entirely skip the below technical description and move on to Summary in the end of this section.]

Figures 3 and 4 are 2D digital mammographic images based on a certain discretisation grid which is very much like a lattice shown in Figure 8 (a). Such an image consists of many pixels on the lattice structure. The refinement of the lattice, i.e., the resolution of the grid, determines the precision of digital images. The size of a breast cancer cell is around 15 micrometres. A typical breast cancer tumour at an early stage is much larger than a millimetre. The resolution of modern mammography systems (a pixel size) ranges from 50 to 100 micrometres [AGFA]. Therefore, in theory the standard scanning device should be able to detect a tumour but not a single cancer cell.

Given the average resolution of the modern devices, each square in Figure 8(a) is equivalent to a pixel which represents available space for a cancer tumour to grow into. As shown in Figure 8 (b), at inception, a breast cancer tumour randomly appears and occupies one of the squares. From Figure 8 (c) to Figure 8 (h), we can see how breast cancer progresses to a larger tumour via step-wise propagation in time series by adding new cancer cells to the existing tumour cluster.