Introduction:

Breast cancer is the most commonly diagnosed malignancy among women and has become a big
threat to human beings globally. As per Indian population census data, the rate of mortality due
to cancer in India was high and alarming with about 806000 existing cases by the end of the last
century. Cancer is the second most common disease in India responsible for maximum mortality
with about 0.3 million deaths per year (Ali et al. ,2011). Breast cancer is the most common
hormone dependent cancer in women. The rising graph of breast cancer both in developed and
developing countries is a big problem, challenging all efforts to screening prevention and
treatment aspects to reduce this cancer. It is the first common cancer of urban Indian women and
second of rural women. It has been estimated that by 2030 the number of this cancer cases in
India will reach under 200000 per year. Global breast cancer incidence increased from 641,000
(95% confidence intervals 610,000—750,000) cases in 1980 to 1,643,000
(1,421,000—1,782,000) cases in 2010, an annual rate of increase of 3·1% (Foreman et al.,2011).
About 6% of all deaths in India are due to cancers , which contribute to 8% of global cancer
mortality. The incidence of breast cancer is rising in India and is now the second most common
women cancer after Cervical cancer( Dutta et al., 2012). Over the past few decades, there has
been appreciable progress in therapeutic strategies for early stage (i.e., localized and operable, as
opposed to metastatic) breast cancer, with a well-developed array of treatment options. Due to
increased screening vigilance and disease awareness, currently over 75% of women diagnosed
have early stage tumors. Despite this progress, the clinical course of breast cancer after diagnosis
remains heterogeneous from patient to patient and thus highly unpredictable for individuals. (v.
Dukic and j Dignam 2007). It has been suggested that both earlier diagnosis and treatment
changes contributed to improve breast cancer survival( Webb et al. 2004). Over the past few
decades, there has been appreciable progress in therapeutic strategies for early stage (i.e.,
localized and operable, as opposed to metastatic) breast cancer, with a well-developed array of
treatment options. Due to increased screening vigilance and disease awareness, currently over

75% of women diagnosed have early stage tumors. Despite this progress, the clinical course of
breast cancer after diagnosis remains heterogeneous from patient to patient and thus highly
unpredictable for individuals. (v. Dukic and j Dignam 2007).
Traditional prognostic markers such as axillary lymph node status, tumor size, histological grade
,hormone receptor expression are helpful to stage the disease, predict overall survival of patient and
response to hormonal therapy. Multigene analysis (Park S et al. 2004) and gene expression profiling
(Van’t Veer LJ et al. 2002) have been studied recently to know more about biological behavior of breast
cancer. However, all these factors need tissue sampling ,are costly and results depend on expertise of
histopatholcgist. None of these factors can single handedly predict risk of development of distant
metastasis in individual patient, overall survival of patient and patients needing close survelliance and
follow-up. We also need a marker which can identify group of patients benefitting most from adjuvant
therapy.
Soluble circulating tumor marker if found to be accurate prognostic factors ,would be ideal candidates for
predicting outcome and monitoring treatment response (Colomer R et al 1989). Serum is easily
accessible, result does not depend on individual expertise, test is economical. Serum CA 15-3 has been
the most frequently investigated tumor marker in breast cancer. Due to low sensitivity and specificity ,CA
15-3 has no value for primary diagnosis (Tondini C et al. 1988).It can ,however be useful in predicting
prognosis, measuring treatment response in advanced breast cancer patients
Tumor markers are a potentially powerful means of obtaining information about cancers whilst
causing minimal morbidity, inconvenience and cost. CA15-3 has been suggested as a marker of
distant metastasis in breast cancer. A highly significant correlation existed between elevated
CA15-3 levels (> or = 30 U/ml) and metastasis disease (Tomlinson IP et al. 1995). It is the most
widely used serum marker in breast cancer. Currently, its main uses are in the surveillance of
patients with diagnosed disease and monitoring the treatment of patients with advanced disease.(
Duffy MJ et al.1999). In general, the higher the CA 15_3 level, the more advanced the breast
cancer and the larger the tumor burden. CA 15_3 concentrations tend to increase as the cancer
grows. In metastatic breast cancer, the highest levels of CA 15_3 often are seen when the cancer
has spread to the bones and/or the liver. Elevated pre-operative CA15-3 level is directly related
to tumor burden and independent prognostic factors for breast cancer. It could be considered for
clinical use such as predicting patient outcome and determining adjuvant treatment for better
outcome (Beruttia A. 1994). CA15-3 is an important diagnostic, prognostic indicator and good

predictor for relapse (Hiba Qassem Ali et al.2013). As far to our knowledge no study has been
conducted for evaluating the disease progression of breast cancer patients using CA15_3 in
multistae markov models as a disease marker. So based on the ranges of CA15_3 a multistate
markov model can be developed which can be used to check the progression of breast cancer
patient into the various stages. Multistate models based on Markov processes are a well-
established method of estimating rates of transition between stages of disease. These models
have been extensively used to evaluate disease progression of breast cancer patients. A lot of
studies had been conducted for Breast cancer patients using Multistate markov Models. Hein
Putter et al. (2006) also developed a 4-state multistate markov model to estimate transition rates
between the states in the model and later the used of these estimates to obtain predictions for
patients with a given history. Jackson et al. (2002) also used a hidden markov model for
estimating transition rates and probabilities of stage misclassification in the study of abdominal
aortic aneurysms. Modelling and Estimation of Different parameters like transitions
probabilities, Mean Sojourn time ,effect of different risk factors are essential steps towards
evaluating the effectiveness of screening policies (Taghipour et al. 2012). From a study it can be
concluded that the multistate model offers us a much more appropriate way to study prognostic
factors for each transition in breast cancer disease( Broet et al. 1999). In Survival Analysis there
are some random effects which act multiplicatively on the baseline hazard functions. These
effects are called frailties. Several authors propose various frailty models, according to the
distribution of frailties and estimation procedure. Aalen and Husebye (1991) used the Laplace
transform to define a marginal likelihood function of a Weibull model with frailty. This theory
was recently extended to multivariate survival data by Hougaard (2000).
This study is carried out to evaluate the disease progression of breast cancer patients based on the
diasease marker CA15_3 . For this a multistate markov model is developed with 3- states based
on the level of CA15_3 . We have tried to estimate the transition intensities between various
states of Breast cancer, defined by the ranges of CA15_3, using multi-state Markov model. Also
we estimated the total length of stay in various states and the effect of covariates. Here we have
also estimated the effect of frailty in cox proportional hazards model.

Materials and Methods:

Patients: All female primary breast cancer patients treated at breast clinic. (Dept of GenSurgery,
IPGMER, SSKM Hospital, Kolkata )from Jan 2009 to Dec 2010 had their pre-op serum CA15-3
measured and it was reported on 7,30 post op day and every 6 months for 2 years. Patients were
excluded if any other malignancy was known from their previous history or if staging
investigations at the time of diagnosis revealed evidence of instant metastasis. A total of 55
patients fulfilled this criteria . 20 female patients with benign breast diseases were taken as age
matched controls and serum CA15-3 measured for comparison.
Treatment : Patients were treated with either modified radical mastectomy (MRM) or
quandrantectomy and auxiliary lymph node dissection with local radiotherapy(RT). After
completion of surgery, RT and appropriate adjuvant chemotherapy or hormone therapy was not
altered according to marker levels but was administered as indicated based on international
guidelines.
Multi State Markov Models:
Multistate Markov models in continuous time are often used to model the course of diseases.
A multi-state model (MSM) is a model for a continuous time stochastic process allowing
individuals to move among a finite number of states. The patient may advance into or recover
from adjacent stages of disease or die at any stage of disease. The stages of disease may be
modelled as a homogeneous continuous time Markov process. These models have markovian
property, i.e. the probability of going to a future state
depends only on the present state and not on the past states. In this study a multistate markov
model is developed with state space S={1,2,3}, where 1 & 2 are transient states, and 3 is an
absorbing state. If X(t)= r be the state of a patient at any time t, then the intensity with which the
patient moves to state s during the interval (t, t+t) is defined as

0

(()/())
()limrstPXttsXtr
t
t







 for r,s=1,2,3

The transition intensity matrix, defined as P=[ rs

] 33 , has the following properties:

(i)

0,rs
sS
forallr




(ii) rrrs
rs





Based on the ranges of CA15_3 , the three states Markov model has been defined as: state1;
CA15_3 <25, state2; CA15_3 ≥ 25 and death as state3.
The Possible transitions of breast cancer patent based on CA15_ 3 marker has been shown in
the following figure 1.

Figure1: Schematic representation of three states Markov model.
where, 11,12,21,22 are the morbidity transition rates and 13, 23are the mortality
transition rates. Based on these 3-states we can define the transition probability matrix as:

11121
21222
33010
PPQ
PPPQ







where P 11 , P 12 , P 21 , P 22 are the transition probabilities and Q 1 & Q 2 are the death
absorbing probabilities which can be calculated as:

P 11 (0,t) = Pr.[that an individual in state 1 at time 0 will be in same state at time t ]

= ]
P 12 (0,t) = Pr.[that an individual in state 1 at time 0 will be in state 2 at time t ]
= t 1 exp[udu] dt 1

=

22
2121
1111211
0

()
exp[/2]exp[]
2
ttt
ttdt



=

22

12121112

0
()
exp[]exp[]
222
tty

dy



, put t
2
1
=y

=

2

21212
1112

1112
()exp()exp[()/2)1]

2
t

t




P21(0,t) = Pr.[that an individual in state 2 at time 0 will be in state 1 at time t ]
= t 1 exp[udu] dt 1

=

22
2211
2212111
0

()
exp[/2]exp[]
2
ttt
ttdt



=

22
21212221

0
()
exp[]exp[]
222
tty

dy



, put t
2
1
=y

=

2

22121
2221

2221
()exp()exp[()/2)1]

2
t

t




P22(0,t) = Pr.[that an individual in state 2 at time 0 will be in same state at time t ]

= ]
Q 1 (0,t) = Pr.[that an individual in state 1 at time 0 will be absorbed at time t ]

=

1
121311

00
exp[]tt
udutdt


=

2
121
1311

0
exp[]
2
tt
tdt




=
2
12

13
0
exp[]
2
ty
dy




put t
2
1
=y

=

2
1312
12
(exp[]1)
2
t


Q 2 (0,t) = Pr.[that an individual in state 2 at time 0 will be absorbed at time t ]

=

1
222311

00
exp[]tt
udutdt


=

2
221
2311

0
exp[]
2
tt
tdt




=
2
22

23
0
exp[]
2
ty
dy




put t
2
1
=y

=

2
2322
22
(exp[]1)
2
t


We model the effect of covariates on the transition ij using cox proportional hazard model
on the transition hazard for a patient with covariate vector z ,then the transition hazard ()ijqt
for

the transition ij can be defined as:
()(0)exp(.z)T
ijijijqtq
where q ij (0) is the baseline intensity, ij

is the regression coefficient, & z ij is vector of
covariate specific transition i to j, defined for a patient based on covariates z, assuming that the
covariates does not change over time .
Incorporation of Frailties:
We define w k as the unobservable random effects on a subject, which are assumed to have
independent distributions, specific to the transition i 

j, with a mean of 1 (in order to get
unique identification of parameters). The frailty term is assumed to describe the dependence of
the transition hazard function on an individual. The Cox Proportional Hazard model after in
corporating the effect of frailty can be defined as:
(/,)(0)exp(.z).wT
ijkijijkqtzwq
where w k is a frailty variable follow say gamma distribution with mean 1 and density as:

1i

v

w
iwe




where v determines the frailty variance. The likelihood can be written as :

1
[q(0)exp[]]()iji
v
wT
ijiji
ij
Lzwe







Results:
Analysis was done on 85 breast cancer patients diagnosed from Jan, 2009 to Dec, 2010. About
82.4% of cases( 70 patients) were alive during follow up time .The mean age of patients at
diagnosis was 50.09 ( SD=12.82) , ranging from 25 to 85 years. The descriptive characteristics is
shown in Table 1. Table 2 shows the observed transitions between follow up visits. It can be
visualized that there are 12 deaths from State1 and 3 deaths from state 2. Also there are 44
censored cases in State 1 and 8 in State2.
Table1: Descriptive Characteristics of Breast Cancer Patients (N=85)
Factors Categories(Code) Frequency Percentage
Age(in years)

<40 (0) 25 29.4
40-50 (1) 21 24.7
≥50 (2) 39 45.9

T Size

<2 (0) 24 27.9
2-5 (1) 48 55.8
≥5 (2) 13 15.1

Nodal M

0-3 (0) 50 58.8
4-9 (1) 19 22.4
≥9 (2) 16 18.8

Tumor Grade

1 23 27.1
2 42 49.4
3 20 23.5
ER Status negative (0) 40 47.1
positive (1) 45 52.9
PR Status negative (0) 48 56.5
positive (1) 37 43.5
HN2 Status negative (0) 53 62.4
positive (1) 32 37.6

Table 2:
Number of
Observed
Transitions
between

States(Rows to Columns)
States State 1 State 2 Death Censored
State 1 239 13 12 44
State 2 74 56 3 8

Initially, the Markov model without covariate has been used to study the overall disease
progression. The estimates of transition intensities ( ij ) with 95% confidence intervals (CI) are
presented in Table 3. It reveals that a patient in State 1 is 33.6(0.101/0.003) times more likely to
move to State 2 than of dying in State 1. However, similar transitions is observed in State2, that a
patient in State2 is 47(.47/.001) times more likely to move to State1 than of dying in State 2. The
estimated transitions probabilities are presented in Table4. It can be seen that there are 75%
chances of remaining in State1 as compared to 16% and 8% of moving to state 2 and of dying
respectively. Similarly a patient has 16% chances of remaining in State 2 than 75% and 8% of
moving to state 1 and of dying respectively. Table 5 summarizes the hazard ratios (HR) for each
covariate (i.e Age, TGrade, Tumor Size, ER Status, PR Status, NodalM) on each transition along
with 95% CI. From the zard ratios it can be concluded that Nodal M is only the significant factor
among all the covariates in both the transient states.

Table3: Estimate of transitions Intensities with 95% CI using Multistate Markov Models
State
s 1 2 3
1 -0.101(-0.55,-0.02) 0.101(0.02,0.55) 0.003(0.001,0.006)
2 0.47(0.098,2.24) -0.47(-2.24,-0.098) 0.001(0.00,2.54)
Table 4: Estimation of transition Probabilities with 95% CI
States State1 State2 State3
State1 0.7524065 0.1626337 0.08495984
State2 0.7527775 0.1627139 0.08450855

Table5:

Estimates of Hazard Ratio using
Multistate Cox Model

  Age

Tumor
grade Tumor Size Nodal M ER Status PR Status
  HR(95% CI) HR(95% CI) HR(95% CI) HR(95% CI) HR(95% CI) HR(95% CI)
State1-
State2

0.68(0.001,3.6
45)

3.00(0.15,5
9.51)

0.37(0.01,9
.70)

0.23(0.05,0.
96)

0.18(0.00,22
8.16)

12.59(0.01,113
64.42)

State1-
State3

2.801(0.512,1
5.321)

9.62(0.95,9
6.96)

2.36(0.46,1
2.10)

11.37(0.37,3
4.07)

0.15(0.00,11
.62)

18.83(0.06,211
7.63)
State2- 0.052(0.001,2. 2.48(0.14,4 0.37(0.02,7 0.20(0.04,0. 0.13(0.00,10 21.79(0.03,150

State1 826) 3.46) .03) 89) 1.82) 37.16)
State2-
State3

1.704(0.364,7.
962)

0.17(0.02,1
.27)

1.24(0.24,6
.39)

7.78(1.64,36
.93)

0.40(0.01,11
.85)

0.03(0.00,99.4
6)

Discussion:
Multistate Models are particularly used in medical applications in which stages or levels of a
disease are represented by the states in the model. Such models have been used in a wide range
of medical applications, for instance HIV/AIDS (Aalen et al., 1997), breast cancer (Duffy et al.,
1995), psoriatic arthritis (Cook et al., 2004), dementia (Joly et al., 2002), diabetic retinopathy
(Kosorok et al., 1996), myocardial infarctions (Grover et al., 2010) and smoking prevention
(Cook et al., 2002). In this study we have introduced a multistate markov model for evaluating
the disease progression of breast cancer patients using the preoperative value of CA15_3 as the
disease marker. We used ranges of CA15_3 as the states of markov models. By using the
properties of Markov models, we have illustrated the usefulness of multi stage illness death
model in the analysis of follow-up study of breast cancer patients. We have calculated the
transitions intensities with covariates set at their mean values and hazard ratio. R package is
used for estimation(Jackson,2011). In terms of covariates Nodal Metastasis is found to be the
significant prognostic factor which match with the findings of Duffy et al. 2010. Also it can be
seen that a patient is more likely to have the concentration of CA15_3 less than 25 . As Cancer
antigen 15-3 (CA 15-3) is used to monitor response to breast cancer treatment and disease
recurrence. Its reference range is less than 30U/ml(Thaker N. 2014). Hayes et al. (16), in 1986,
studied CA15-3 levels in 1050 normal, healthy individuals and found only 14 (1.3%) with an
elevated level above a value of > 30 u/ml. Numerous studies have indicated that CA15-3 is a
reliable marker in breast carcinoma and correlates well with the disease stage,(Hayes et al., Kerin
et al.) . Also to estimate the effect of unobserved random factors we introduced the effect of
Frailty in the Cox model.
The limitation of this study is that the data record is less in numbers, if it can be large than we
may get more accurate results.