Transcript Prediction of Regional Tumor Spread Using Markov Models

Prediction of Regional Tumor
Spread Using Markov Models
Megan S. Blackburn
Monday, April 14, 2008
Background
• 2006 Conference Paper by Benson et al.
• Describes use of Markov Chains to model
cancer spread in patients
• Specifically studied head and neck cancers
• Comparisons made to surgical data
• Goal: Try to reproduce model proposed by
Benson et al.
Background
• Cancer impacts our society as a whole
• Everyone is affected in some way by cancer
in their lifetime
Cancer
• Generalized name describing more than 100
different disorders
• Cancer cells can be considered immortal
• Divide many more times than normal cells
thus growing out of control
• Do not interact normally with other cells
resulting in invasion of regions of the body
other cells could not
Cancer Treatment
• Because of the far-reaching effects, much
effort has been put into the treatment of
cancer
• Goal of treatments: Kill the maximum
cancer cells while killing the minimum
normal cells
• Typical Treatments: Radiation Therapy
and/or Chemotherapy
Radiation Therapy
• Beam of photons is
incident on the
patients
• Photons deposit their
energy within the body
• Kills both healthy and
diseased cells
http://www.srhc.com/services/oncology/image/Clinac.jpg
Radiation Therapy
• Before treatment can
begin, CT scan is
taken of the patient
• CT scan is used to
plan the patient’s
treatment
http://asiaonc.com/files/images/H&N%20Lat2.img_assist_custom.jpg
Radiation Therapy
• ICRU 50 describes
several definitions for
treatment planning
• GTV
• CTV
• PTV
• Treated Volume
• Irradiated Volume
Room for Improvement
• In Radiation Therapy, many different margins
must be used in order to assure the tumor gets a
therapeutic dose
• Margins result in more healthy tissue being dosed
• Microscopic disease must also be accounted for
• Markov Model could help to determine where the
cancer has spread
Head and Neck Cancers
•
•
•
•
Unique compared to other cancers
Typically very irregular in size
Early treatment was surgical removal
Radiation Therapy and Brachytherapy are
now useful tools
• Diseased Lymph nodes must be treated as
well
Lymph Node Regions
Region
I
II
III
IV
V
VI
http://www.iscb.org/rocky06/presentations_pdf/29Kalet-rev2.pdf
Nodal Group
Submental &
Submandibular
Upper Jugular
Middle Jugular
Lower Jugular
Posterior Triangle
Anterior
Compartment
Cancer Staging
• Tumor-Node-Metastasis (TNM) Staging
• T – anatomical description of the primary
tumor site
• N – involvement of the lymph nodes
• M – metastasis to other regions of the body
• Staging is very diverse for cancers of the
head and neck due to varied anatomical
regions
Markov Model
• Named after Russian Mathematician
Andrey Markov
• Mathematical Model describing a random
progression of a system
0.1
0.5
S1
0.5
S2
0.3
0.7
S3
0.9
S4
0.8
0.2
S5
Markov Model
• Probabilities of transitioning from one state
to another
• Discrete time steps
• Cannot be in more than one state at a time
0.1
0.5
S1
0.5
S2
0.3
0.7
S3
0.9
S4
0.8
0.2
S5
Markov Model
• Clock – describing number of steps to take
t  1,2,..., T 1, T 
• N States – number of locations in which one could be
Q  1,2,..., N  1, N 
• N Events – One event associated with each state
E  e1 , e2 ,...,e N 1 , eN 
0.1
0.5
S1
0.5
S2
0.3
0.7
S3
0.9
S4
0.8
0.2
S5
Markov Model
• Initial Probabilities – probability of starting state
 j  Pq1  j
• Transition Probabilities – probability of moving from one state
to another aij  Pqt  j | qt 1  i  1  i, j  N
aij  0 i, j

N
a
j 1
ij
1
i
0.1
0.5
S1
0.5
S2
0.3
0.7
S3
0.9
S4
0.8
0.2
S5
Applications to Cancer
• Apply Markov Models to the movement of
cancer – specifically lymph nodal
involvement
• Goal is to determine the lymphatic spread
with only the starting location of the cancer
and the staging of the cancer
• Used a series of Markov Chains rather than
a single Markov Chain
Applications to Cancer
• Single Markov Model represents each nodal
region
• Within each Model, there are 5 states –
0 (no cancer), 1, 2, 3, 4 (extensive disease)
• Unique aspect is the linking of the Markov
Models
• Links represents future nodal involvement
Applications to Cancer
• s defined states (0,1,2,3,4) in each node
region
• Probability qs of each state metastasizing to
next nodal region (state 1)
• Probability distribution, pi, of being in a
specific state within each lymphatic region
• Probability, p’, of the next nodal region
becoming affected by microscopic disease
4
p'  psqs
s 0
Applications to Cancer
• Initially, probability distribution for each
lymphatic region is set
• For initial tumor region, probability is set to
1 for state 1, 0 in all other states
• For all other lymphatic region, probability is
set to 1 for state 0, 0 in all other states
Applications to Cancer
• Initial Tumor Region
– Transition Matrix
1 0
0
0
0 


0 
0 0.9 0.1 0
P' 0 0 0.9 0.1 0 


0 0.9 0.1
0 0

0
0
1 
0 0


– Metastasis Vector
0
0.2
 
m'  0.4
 
0.6
0.8
• Lymphatic Regions
– Transition Matrix
0
0
0
1 0
0 0.5 0.5 0
0 

P  0 0 0.5 0.5 0 


0 0.5 0.5
0 0
0 0
0
0
1 
– Metastasis Vector
0 
 0 .6 
 
m   0.7 
 
0.8
0.9
Application to Cancer
• Model is iterated on 4*Cancer Staging
• Probability of microscopic cancer occuring in each
nodal region uses: p'  p q
• Value must be adjusted against the probability that
metastasis has already occurred

• Added to the probability that the downstream
nodal region is already in State 1
4
s s
s 0
Implementation
• Implemented in MATLAB
• Benson et al. used FMA (Foundational
Model of Anatomy) for all anatomical
regions
• Unable to match the anatomical data that
Benson et al. obtained
• Could not directly compare with his results
Results
• Assumed 6 nodal regions downstream from primary tumor
• Obtained probabilities of cancer spreading to this regions
given an initial cancer stage
Lymph
Region
A
Lymph
Region
B
Lymph
Region
C
Lymph
Region
D
Lymph
Region
E
Lymph
Region
F
1
28%
22%
17%
12%
9%
7%
2
36%
30%
25%
21%
17%
13%
3
43%
37%
32%
27%
23%
19%
4
50%
44%
38%
33%
28%
24%
Cancer
Stage
Conclusions
• Much tweaking is needed to the concept
• Too many arbitrarily chosen values
• Interesting idea BUT very unlikely to be
accepted by the medical community
anytime soon