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
To understand a fractal set we need specify three things:
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
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)
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.
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 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.
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.
Classification of Colon Adenoma
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)
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
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
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.
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.
Figure. 7: Serrated adenoma of
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
of colorectal cancer
There are three
types of CRC which can be distinguished by their forms, origin and expression:
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
colorectal cancer (mCRC)
colorectal cancer is spread to other parts of the body it is described as
metastatic colorectal carcinoma. There are two types of metastatic colorectal
a) Lymphatic metastasis: This occurs via the pericolonic and periaortal
lymph nodes to the thoracic duct and from there to the supraclavicular lymph
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
stages for development of colorectal cancer
Figure 7: Various stages showing development of
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:
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:
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
Stage 2 to stage 3:
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:
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.
colorectal carcinoma is developed it can turn into metastatic colorectal
carcinoma by various means as mentioned above.
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
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
Richardson–Mandelbrot equation provides the mathematical basis for
understanding geometrical and spatial fractal structures, and for measuring and
interpreting them, namely:
L(?) = N(?).(?) (1)
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:
= (1 ? D)log?/lo
equation represents a dimensionless scaling power law indicating that the
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.