INTRODUCTION

1.1 Fractal

analysis

Fractal is term, coined by Mandelbrot, from the Latin

adjective fractus (fragmented, irregular) that derives from Latin verb frangere

meaning to break, to create irregular fragments.

Fractal analysis can be described as a repeated

form of an equation of a geometric transform of structure, shape and space at

infinite level by a property known as auto-similarity. An auto-similar structure

has the same geometrical properties when observed at different levels. The living world consists of an immense diversity of forms and

structures, most of which are generated by irregular morphogenetic processes.

The functional and metabolic phenomena which take place in living organisms

follow non-linear dynamics, rather than the linear law of cause and effect

which are irreversible and occur far from thermodynamic equilibrium 1. A

fractal geometry shows how each individual structure resemble other individual

and form a whole structure that in turn resembles individual structures over a

complex surface.

To understand a fractal set we need specify three things:

1) the

shape of a starting object; (the initiator), 2) the

algorithm enabling its iterative application on the initiator and then

repeating the same on all obtained geometrical objects (the generators), and 3)

the conditions which these generators should satisfy, before all the properties

of geometrical similarity.

To understand fractals better let us consider an

example of Koch Fractal set.

Figure 1. Koch fractal set. (a)

Initiator-equilateral triangle, r0 is

side length; (b) The first stage of

construction (z = 1), detail

below the drawing is the

corresponding generating element;

(c) The second stage of

construction (z = 2), detail below the

drawing

is the corresponding generating element

In the above figure we start our description with a simple equilateral

triangle (Initator) with edges length r0. The iterate algorithm to

generate the set of Koch curves (prefractals) consists of recursive reduction

of the straight-line segments (or the scales) by 1/3 exchanging repeatedly the

middle third of each side of the initiator, or a preceding generator, with two

sides of a smaller, equilateral triangle whose side

is one-third the length of the previous side. The result after the first

iteration (the stage of construction z = 1) is shown in Figure 1(b), and

that after the second iteration (the stage of construction z = 2), in Figure 1(c). For the

Koch prefractals, the length of a segment at the zth stage of

construction (rz) and the number of segments at the same stage (Nz)

are respectively,

In other terms fig 1a can be subdivided into 3

generating elements to form fig 1b and can be further divided into 12

generating elements to form fig 1c.

1.2 Fractals

in Biology

The new concept of fractal

geometry created by Mandelbrot acquires high development in biology and

medicine. It can describe and quantitatively characterize complex natural

structures which present highly irregular shapes, impossible to be defined by Euclidean

geometry Losa, 2012. The extension of the

concepts of fractal geometry towards biology unleashed to significant progress

in understanding the complex functional properties and architectural/morphological/structural features that

characterize cells and tissues during in

both normal and pathological cycle of development. From the direct observations

of Nature, it emerges that most cells, tissues, organs – in either the animal

or vegetal worlds – are systems in which the component parts and unit fragments

assemble with different levels of complexity and organization. This means that

a single fragment or element may, on various scales, reproduce the whole object

from which it is derived; in other words, it is self-similar, albeit in a

statistical sense.

Statistical fractals are present also in cardiac muscles Bassingthwaighte

JB et al, 1994 and in those organs provided with multiple folds, such as

small intestine and cerebral cortex, or in the vascular branches of placenta Bergam

DL et al, 1998.In lungs, fractal geometry provides an exceptional solution

to the problem of maximizing the available surface thus simplifying the

respiratory exchanges Weibel ER, 1991; fractal branches provide to

create a thick net for the distribution of nutrients and oxygen and for the

removal of waste substances Goldberger et al, 1990.

2.0 Adenomatous

polyps

Adenomatous polyps (Adenomas) are abnormal growth

of colon and rectum and may be precursor lesions for colorectal cancer (CRC).

Most benign polyps are classified as one of two types: Adenomatous (Adenoma)

and Hyperplastic. Polyps greater than 1cm in diameter are associated with

greater risk of cancer.

Colorectal cancer is rather common with 50%

people of age 50 or older have one or more adenomatous polyps; however, with

only 6% people develop colorectal cancer. The chance of polyps increases with

patients having family history of colorectal cancer, including inherited

disorders such as Gardner’s syndrome and familial adenomatous polyps.

2.1

Classification of Colon Adenoma

Adenomas

can have several growth patterns that can be seen under the microscope.

There are two major growth patterns: Tubular and Villous. Many adenomas have mixture of both growth patterns called Tubulovillous adenomas (www.americancancersociety.com)

2.1.1 Tubular

Adenoma

Tubular adenoma is generally found in

rectosigmoid and occur singly. When observed under the microscope it reveals

gland or cyst like structures in submucosa with smooth surface and is discrete.

These small polyps are virtually always benign.

Figure 2: Tubular adenoma of colon

(http://www.dovemed.com/tubular-adenoma-of-the-colon)

2.1.2

Villous adenomas

They are often associated with larger adenomas

and severe degree of dysplasia. They are generally found in rectum and

rectosigmoid but may occur anywhere in colon. When observed under a microscope

a “Cauliflower” like mass is seen due to villi stretching. These adenomas have

higher chances to become malignant (Cancerous). They can also lead to secretory

diarrhoea with large volume of liquid stools and few formed elements. They are

commonly described as secreting large amount of mucus, resulting in hypokalaemia.

Figure. 3: Villous adenoma of

colon

(http://www.slideshare.net/csbrprasad/git-5csbrp)

2.1.3

Tubulovillous adenoma

Tubulovillous adenomas are intermediate between

the tubular and villous lesions. The risk of malignancy or invasive carcinoma

generally correlates with the proportion of the lesion that is a villous one.

Figure. 4: Tubovillous adenoma of colon.

(http://

www2.palomar.edu/users/warmstrong)

2.1.4

Serrated adenomas

The WHO has

classified serrated polyps into three types of lesions namely a) hyperplastic

polyps (HP), b) sessile serrated adenomas/polyps (SSA/P) and c) traditional

serrated adenomas (TSA). Sessile serrated adenomas/polyps and TSA are the ones

strongly associated with the development of CRC. In HP, the expanded

proliferation zone is located at the base of the crypts and cells mature towards the surface.

In SSA/P, the proliferation zone is to the side of the crypts instead of the

base, resulting in maturation of epithelial cells, laterally, towards the

surface and the base, leading to crypt base dilatation. Singh R et al.

2016.

Figure. 7: Serrated adenoma of

colon (http://www.pathologyoutlines.com/topic/colontumorserrated.html)

2.1.5

Hamartomas polyps

They are most common type of polyps found in

children due to faulty development and are rarely found when compared to other

types of polyps. It has abnormal mixture of normal tissues. They contain mucus

filled glands, abundant connective tissue and chronic infiltration of

eosinophils. Complications associated with such polyps can be risk of cancer,

recurrence of polyps and extraintesinal complications.

Figure 6: Hamartomas polyps

(http://www.Basimedicalkey.com)

3.0 Classification

of colorectal cancer

There are three

types of CRC which can be distinguished by their forms, origin and expression:

a) Sporadic

form which does not show any type of family link. The vast

majority of CRC, between 60 and 80%, are of sporadic type

b) The familial

type, constitutes 20–40% of the cases. Population studies shows

that there is a greater chance of developing a tumour when family members of

primary consanguinity have suffered from sporadic colon cancer, and the risk is

two to three times higher than in the normal population Wong H.H. et al

2012. Environmental factors play important role in the development of this

type of cancer.

c) The hereditary

type, with two tumour variants depending on the presence of adenomatous

polyps. We can distinguish hereditary polyposis colon cancers (HPCC) and

hereditary non-polyposis colon cancers (HNPCC). Categories of HPCC include

Familial adenomatous polyposis (FAP), MUTYH-associated polyposis, hyperplastic

polyposis syndrome (HPS), Peutz-Jeghers syndrome (PJS); and Juvenile polyposis

syndrome (JPS).

4.0 Metastasis

colorectal cancer (mCRC)

When

colorectal cancer is spread to other parts of the body it is described as

metastatic colorectal carcinoma. There are two types of metastatic colorectal

carcinoma,

a) Lymphatic metastasis: This occurs via the pericolonic and periaortal

lymph nodes to the thoracic duct and from there to the supraclavicular lymph

nodes.

b) Hematogenous metastasis: This follows the vena

cava pattern in rectal carcinoma and the portal vein pattern in colon

carcinoma. Later, hematogenous metastases may follow the liver pattern of

dissemination.

4.1 Different

stages for development of colorectal cancer

Figure 7: Various stages showing development of

colorectal cancer

(https://www.qiagen.com/us/shop/genes-and-pathways/pathway-details/?pwid=133)

In a

classic “adenoma-carcinoma sequence”, every step from the normal healthy mucosa

towards the carcinoma was subjected to specific and well-defined genetic

alterations, among which (adenomatous polyposis coli) K-RAS oncogene, APC, DCC

(deleted in CRC) and p53 oncosuppressors and bacterial/viral infections,

stress, toxins and polyamines could be the major cause.

Stage 0 to stage 1:

Progression

of Dyplastic adenoma from healthy mucosa could occur due to various reasons

including bacterial/viral infections, toxins, stress, polyamines including Ctnn

and APC mutations. Any alterations or mutations in MSI pathway could also

result in this transformation.

Stage 1 to stage 2:

From

Dyplastic adenoma to early adenoma transformation, causes could be

Microsatellite instability (MSI) or alterations in Prostaglandin-endoperoxide

synthase also known as cyclooxygenase which is a key enzyme for prostaglandin

biosynthesis. Other than these KRas mutations can also be the cause for this

transformation.

Stage 2 to stage 3:

Progression

from early adenoma to late adenoma can be due to various mutations in KRas,

Deletion in Colorectal Cancer gene (DCC) and microsatellite instability along

with mutations in SMAD4 in TGF-? growth inhibitor pathway.

Stage 3 to stage 4:

Progression

of late adenoma to colorectal Carcinoma is due to various mutations in cell

mediated pathway like MSH3/6, BAX, p53, TGFbR2, E2F4 gene expression and

alterations in various genes.

Once

colorectal carcinoma is developed it can turn into metastatic colorectal

carcinoma by various means as mentioned above.

5.0 Fractal

analysis in Cancer

From direct observation from nature it is seen that most cells,

tissues and organs in either animal or vegetal worlds in which component parts and unit fragments assemble

with different levels of complexity and organization-

which means that this unit fragment when analysed in different scales

represents the sum of whole object from which it is derived.

Though there is vast knowledge of the molecular mechanisms of cancer,

most diagnosis is still done by visual characteristic of radiological images,

microscopic observation of biopsy specimens, direct observation of tissues,

etc; which gives us the qualitative analysis of the disease we need to have

more detailed description and status of the progression and current state of

the cancer. Hence to overcome this challenge we need to develop a computational

tool to get quantitative coupled with qualitative result to understand the

progression of disease better. Several reviews of the use of fractal dimensions

in pathology have recently appeared in the literature (3– 6). There is a

growing research that shows fractals to be useful measures of the pathologies

of the vascular architecture, tumour/parenchymal border, and cellular/nuclear

morphology.

5.1 Criteria for fractal analysis

Mandelbrot stated in his book that, ”A fractal set is a

set in metric space for which the Hausdorff–Besicovitch

dimension D is greater than the topological dimension Dt.” Fractal

object is basically defined by its structural properties, mainly by its lack of

smoothness. Additional important properties of a fractal object are roughness or

shape irregularity at every scale, high level of organization, iterative

pattern, a peculiar non-integer fractal dimension (FD), and self-similarity or scale

invariance.

The

Richardson–Mandelbrot equation provides the mathematical basis for

understanding geometrical and spatial fractal structures, and for measuring and

interpreting them, namely:

L(?) = N(?).(?) (1)

where

L(?) represents the contour (perimeter) length of the biological

component under investigation, (?) the unit length of measure, and N(?)

the number of unit lengths (?) required to cover the contour L(?).

By substituting N(?) with loD? ?D in Eq.

(1), where lo is a reference scale without influence on the

determination of D, the above equation can be transformed by logarithmic

procedure and rewritten as:

logL(?)/lo

= (1 ? D)log?/lo

(2)

This

equation represents a dimensionless scaling power law indicating that the

estimated

contour,

perimeter, or curve length L(?) changes as a power function of

the scale unit length (?). This dimensional exponent D defines the

fractal dimension that determines the nature of the curve.