This document discusses topological data analysis and its applications in oncology. It provides definitions of topological space and rough set theory. It also describes different network topologies like bus, star, ring, mesh and tree. The document suggests that topological data analysis provides a promising approach for extracting meaningful information from cancer data using tools from algebraic topology. It proposes utilizing topological data analysis methods like the Mapper algorithm and persistent homology to analyze cancer time-series data and investigate relationships between vascular structure and treatment efficacy.

1. A STUDY ON TOPOLOGICAL DATA
ANALYSIS IN Oncology
BY
G.NITHYA .,M.SC.,M.PHIL.,(PH.D)
HEAD AND ASSOCIATE PROFESSOR
DEPARTMENT OF MATHEMATICS
SRI ADI CHUNCHANAGIRI WOMEN’S COLLEGE
CUMBUM.
Mail id : nithyarajkumar15@gmail.com

2. ABSTRACT
.
This paper presents an introduction to Network Topology. Definitions
of the Physical and Logical Topologies are provided. Additionally
common Network realizations of Physical Topologies are reviewed. This
is followed by a discussion of its relation topological analysis.
Topological data analysis provides a promising path forward using tools
from the mathematical field of algebraic topology. TDA provides a
framework to extract from we also provide suggestions on avenues for
future research including utilizing TDA to analysis cancer time-series
data such as that geometric and topological connectly implies functional
connectivity in the context of cancer.

3. TOPOLOGICAL SPACE
A topology on a set X is a collection τ of sub collection of X having the
following properties.
 ϕ and X are is in τ
 The union of the elements of any sub collection of τ is in τ .
 The intersection of the elements of any finite sub collection of τ is in τ.
.

4. ROUGH SET
Rough set is based on the establishment of equivalence classes within the
given training data. All the data tuples forming an equivalence class are
indiscernible, that is, the samples are identical with respect to the attributes
describing the data.
BASIC CONCEPT OF ROUGH SET:
• Information system/Tables
• Indisceribility
 Set approximation

5. INFORMATION SYSTEM:
AGE LEMS WALKS
X1 16.30 50 Yes
X2 16.30 0 No
X3 31.45 1.25 No
X4 31.45 1.25 Yes
X5 46.60 26.49 No
X6 16.30 26.49 Yes
X7 46.60 26.49 No

6.  Information system is a pair (U,A) where
U- finite non empty set of objects
A- finite non empty set of attributes
 The elements of A are called conditional attributes
U= (X1,X2,X3,X4,X5)
• One more special attribute called decision attributes
• Information system with conditional attributies and
decision attributes called decision system its a pair of
(U,A U {d}) where d€ A.

7. INDISCERNIBILITY RELATION
 Indiscernibility means Indistinguishable
 Two objects are indiscernibility if they are same or
have same value with respect to one or more
attributes.
 We can compute indiscernibility Relation with respect
to each attributes or with respect to all the attributes.

8. SET APPROXIMATION
 Equivalent classes or IND (Age/Lems)
 {(X1, X2) (X3,X4) (X5,X7) (X6)}
 Find a set of equivalent classes such that each object
belongs to ‘Yes’
{X1,X6}- lower approximation [͟BX]
 At least one object belongs to ‘Yes’
[͞BX] {X1,X6,X4} – Upper approximation.
upper – lower = Bounder region
{X3,X4}

9. SOFT SET, MULTI TOPOLOGY
SOFT SET :
The concept of soft sets is introduced as A general mathematical tool for
dealing with uncertainty. In this work, we define the soft topology on a of
set and present its related properties. We present the foundations of the
theory of soft topological spaces.
MULTI TOPOLOGY:
Multi topology routing enables you to configure class-based forwarding
for different types to traffic, such as voice, video, and data. Each type of
traffic is defined by a topology that is used to create a new routing table for
that topology.

10. DEFINITION:
Network topology means design of a network. “The layout of how
the network devices are connected to each other in a network is
known as “NETWORK TOPOLOGY”.
TWO CATEGORIES OF NETWORK TOPOLOGIES:
 Physical topology
 Logical topology
PHYSICAL TOPOLOGY:
“The physical topology and its capabilities is determined by network active
and media like cable types, the level of control or fault tolerance desired, and
the Capex or Opex costs related to its passive and active infrastructure.
LOGICAL TOPOLOGY:
Distances between nodes, physical interconnection, transmission rates, or
signal types may differs between two different network, yet their logical
topologies may be identical.

11. THE NETWORKTOPOLOGIES ARE
 Bus Topology
 Star Topology
 Ring Topology
 Mesh Topology
 Tree Topology

12. Bus topology
 Bus topology is one of the different types of topology in which there is
the main cables and all the devices are connected to this main cable
through drop lines.
 One long cable acts as a backbone to link all the devices in a network.
Bus topology Advantages and Disadvantages:
Advantages :
 Cheap and easy to implement.
 Require less cable.
 Does not use any specialized network equipment.
Disadvantages:
 Network disruption when computers are added or removed.
 A break in the cable will prevent all system from accessing the
network.
 Difficult to troubleshoot.

13. STAR TOPOLOGY
 All computers/devices connect to central device called hub or switch.
 Each device requires a single cable.
 Point-to-point connection between the device and hub.
Star topology Advantages and Disadvantages:
Advantages:
 Easily expanded without disruption to the network .
 Cable failure affects only a single user.
 Easy to troubleshoot.
Disadvantages:
 Requires more cable.
 A central connecting device allows for a single point of failure.
 More difficult to implement.

14. RING TOPOLOGY
 Meaning that data travels in circular fashion from one
computer to another on the network
Ring topology Advantages and Disadvantages:
Advantages:
 Cable faults are easily located, making troubleshooting
easier
 Easy to install.
Disadvantages:
 A single break in the cable can disrupt the entire network.
 Each packet of data must pass through all the computers
between source and destination. This makes it slower than
star topology.

15. TREE TOPOLOGY
A tree topology combines characteristics of liner bus and
star topologies.
 It consists of groups of star-configured workstations
connected to a liner bus backbone cable.
Advantages and Disadvantages:
Advantages:
 Combine the benefits of several different types of
topologies.
 Workgroup efficiency and traffic can be customized.
Disadvantages:
 A break in the cable will prevent other systems from
accessing the network.

16. MESH TOPOLOGY
 Each computer connects to every other.
 High level of redundancy.
Mesh topology Advantages and Disadvantages:
Advantages :
 Even if one of the components fails there is always an
alternative present. So data transfer doesn’t get affected.
 High level of redundancy.
Disadvantages:
 Wiring is very complicated.
 Cabling cost is high.

17. TOPOLOGICAL DATA ANALYSIS
Topological Data Analysis (TDA) is an on the rise data
science tool that looks at the shape of data. It is on
approach to the analysis of data sets using techniques
from topology.
 THE MAPPER ALGORITHM:
 The Mapper algorithm translates data into an
interactive graph.
 Mapper is well suited for exploratory data analysis and
visualizing high- dimensional data.

18. PERSISTENT HOMOLOGY
 Topological data analysis is an up and coming
approach to data analysis that studies the shape of
data. TDA approach called persistent homology.
 Persistent homology characterizes the shape of data
via holes.
 Holes are a fundamental topological feature that tend
to be robust to noise

19. BRIEF SUMMARY OF CONSIDERED TOPOLOGIES
Incomplete Hyper cubes:
An extension to hypercube that allows arbitrary node count
with hypercube-like performance. Performance is only slightly
lower than the complete hypercube, but reliability of the
incomplete hypercube can vary substantially.
Tree Topologies:
Tree topologies are poor as a general network topology. A
severe bottleneck and point-of-failure is present at the root of
the tree, and these networks have very large diameter and
average path lengths.

20. Star Topologies:
Poor reliability due to point-of-failure at the node located in
the center of the star. A huge bottleneck is present as well. In fact,
a star topology can be generalized down to a type of tree
topology, and thus has similar issues.
Generalized Star:
Good in areas of performance and node complexity. The
generalized star design scales well with increasing performance
and reliability.

21. 2-D MESHES AND 2-D TOROIDS
2-D MESHES AND 2-D TOROIDS:
 2-D meshes are one of the easiest topologies to visualize-nodes are
connected in a “grid” fashion. The simple layout also allow many problems
to map easily to the structure of this network.
 2-D meshes have unequal node degree. The node degree in the corners is 2,
around edges is 3, and in the center is 4. Also, traffic distribution is unequal
among nodes.
 For reliability, 2-D meshes, in the general case, offer many redundant paths
between nodes and can probably withstand a fair number of random
failures. However, in the worst case, two link failures can isolate a corner
node, as could two node failures.

22. 3-D MESHES AND 3-D TOROIDS
 3-D meshes and 3-D toroids are similar to 2-D meshes and toroids, except the 3-D
mesh /toroid is expanded along the Z- axis to provide another dimensional layer of
nodes. In the case of the 3-D toroid, the topmost nodes (along the new Z- axis) are
connected to the bottommost nodes.
 The advantages and disadvantages of 3-D meshes and toroids are similar to those of
the 2-D meshes and toroids, but are amplified proportionally with the height of the
added dimension. However, due to the added dimension, there is added redundancy
and even more paths and loops within the network.
 3-D toroids have a fixed degree of 6 for all nodes.

23. APPLICATION ON TOPOLOGICAL DATA ANALYSIS
ON ONCOLOGY
 The emergence of the information age in the last few decades brought with
it an explosion of biomedical data. But with great power comes great
responsibility: there is now a pressing need for new data analysis algorithms
to be developed to make sense of the data and transform this information
into knowledge which can be directly translated into the clinic.
 We also provide suggestions on avenues for future research including
utilizing TDA to analyze cancer time-series data such as gene expression
changes during pathogenesis, investigation of the relation between antigenic
vessel structure and treatment efficacy from imaging data, and experimental
confirmation that geometric and topological connectivity implies functional
connectivity in the context of cancer.

24. NODE
A point where two or more curves, lines or edges meet. A node
represents a branching point from the ancestral population.
EDGES
In geometry, an edges is a particular types of lines segment joining two vertices in
a polygon. Polyhedron or higher-dimensional polytope. In a Polygon, an edge is
a line segment on the boundary and is often called a Polygon side.

25. TREATMENT RESPONSES AND PROGNOSIS
 What impedes the success of cancer therapies is often the coexistence of
therapy resistant cells along with therapy sensitive tumour cell populations.
 Despite varying resistance mechanism contingent upon therapy-type and
tumour composition, every therapeutic intervention inevitably selects for
resistant cells, which expand and become the dominant cell type of
recurrent tumours, the cease to respond to therapy .
 The increased resolution on the clonal architecture of intermixed tumour
cell populations that has just now become available calls for prognostic and
therapeutic benefits.
 High intra-tumor diversity in pre-malignant lesions has been shown to
predict progression to malignant growths and poor outcome.

26. TUMOR SEGMENTATION AND
COMPUTER-AIDED DIAGNOSIS
 Computerized methods can efficiently identify quantitative
image features otherwise difficult to spot by manual inspection
quantitative morphological features extracted from H&E
stained slides, such as Zernike shape features, have been shown
to predict survival in lung adenoid-and squalors cell carcinoma.
 A quantitative image analysis approach that complements
genomic profiling with geographical information was
developed Furthermore, the authors characterized cellular
heterogeneity by distinguishing between well defined cell-
populations (stromal cells, lymphocytes, cancer cells).

27. THANK YOU