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Suman, Rajeev Kumar RT&A No 4 (76)
STOCHASTIC MODEL ON EARLY-STAGE BREAST CANCER WITH 10„ ' „(nJ
TWOTYPES OF TREATME NTS V°lUmS 18' DeCember 2023
STOCHASTIC MODEL ON EARLY-STAGE BREAST CANCER WITH TWO TYPES OF TREATMENTS
SUMAN !, RAJEEV KUMAR 2
1 2 Department of Mathematics, Maharshi Dayanand University, Rohtak 1 suman.uiet@mdurohtak.ac.in, 2 profrajeevmdu@gmail.com
Abstract
The aim of the paper is to study effectiveness of the different treatments of early-stage breast cancer through analysis of a stochastic model. The early-stage breast cancer is a term used to describe breast cancer that is detected at an early stage of development, typically before it has spread to other parts of the body. Early detection of breast cancer is critical as it greatly increases the chances of successful treatment and saves lives. At early-stage breast cancer of the patient, two types of treatment namely, tamoxifen and tamoxifen combined with radiation therapy are commonly used. As it is essential to consider innovative and cost-effective strategies for early detection and treatment. Investigations through analysis of the stochastic model on early-stage breast cancer with these two types of treatments may help the stakeholders. Keeping this in view, in the present paper, a stochastic model is developed for the early-stage breast cancer considering two treatment types, namely tamoxifen and tamoxifen combined with radiation therapy. The model is analyzed by Markov process and regenerative point technique. Mean sojourn time refers to the average amount of time spent by a patient in a particular state before transitioning to another state and mean survival time refers to the average time a patient survives after diagnosis of breast cancer. Mean sojourn time and mean survival time have been calculated. Sensitivity analysis is a technique to understand how changes in input variables or parameters affect the output or outcome of a model and it helps assess the robustness, reliability, and stability of a model by quantifying the impact of variations in input factors. The paper also includes sensitivity and relative sensitivity analyses of the model which explore the impact of different parameters on the survivability of the patient. The MATLAB software has been used for numerical computing and plotting various graphs. The investigation through our analysis of the stochastic model shows that the mean survival time lessens with the rise in the rates of transition and mean survival time from the treatment, tamoxifen plus radiation is higher than the treatment, tamoxifen only. It is concluded that tamoxifen plus radiation is more effective and useful than only tamoxifen for treatment of early-stage breast cancer.
Keywords: Breast cancer, stochastic model, Markov process, regenerative point technique, mean sojourn time, mean survival time and sensitivity analysis
1. Introduction
Cancer is a leading global cause of adult deaths. According to IARC (International Agency for Research on Cancer), India reported approximately 635,000 cancer-related deaths in 2008. Breast cancer ranks as the most frequently diagnosed malignancy among women globally, with 2.3 million cases and 685,000 deaths in 2020. As of the end of 2020, 7.8 million women who had battled were still alive, making it the most common cancer globally.
A lot of work has been done on Markov modelling on breast cancer. Duffy et al. [6] proposed a Markov model indicating that progression to an advanced state is notably faster among individuals aged 40-49. Johnstone et al. [9] analyzed the historical data of untreated breast cancer patients Subsequently, Anthony et al. [1] and Cong et al. [4] found that the combined treatment of tamoxifen and radiation proves more effective in preventing breast cancer recurrence compared to tamoxifen alone. According to Schairer et al. [13], the probability of death, whether from breast cancer or other causes, was frequently higher in black patients compared to white patients. Understanding the parameters of cancer progression is crucial for evaluating screening policies' effectiveness, as highlighted by Harvey et al. [14]. These parameters include transition rates, preclinical sojourn time, sensitivity and the influence of various risk factors on the progression of cancer. Several studies have previously examined breast cancer and its facets. In 2014, Ventura et al. [15] demonstrated the use of a multi-state Markov model to analyze the progression of breast cancer, employing various methods for parametric estimation. Anthony et al. [3] evaluated that radiotherapy reduces the risk of breast and axillary recurrence in early-stage breast cancer when combined with breast-conserving surgery (BCS) and Tamoxifen, but it does not appear to significantly impact distant recurrence or overall survival. Dey et al. [5] provided insights into breast cancer history, risk factors, symptoms, and global and Indian mortality rates, along with available treatments. Grover et al. [7] developed a three-state Markov model based on CA15-3 marker ranges to track disease progression in breast cancer patients. Furthermore, Huang et al. [8] established a breast cancer transition model based on the Chinese population's natural history validating its applicability. Ruiz-Castro et al. [13] developed a discrete-time piecewise Markov model to study the behavior of a multi-state illness. Bayer et al. [2] developed and conducted an analysis of a Markov model designed to simulate and evaluate the treatment strategies for cancer. In 2022, Mubarik et al. [10] made estimations regarding the future trends of breast cancer-related mortality in East and South Asian countries. Moreover Newman et al. [11] discussed the breast cancer burden in low and middle-income countries.
The aim of the paper is to study effectiveness of the different treatments of early-stage breast cancer through analysis of a stochastic model. The early-stage breast cancer is a term used to describe breast cancer that is detected at an early stage of development, typically before it has spread to other parts of the body. Early detection of breast cancer is critical as it greatly increases the chances of successful treatment and saves lives. At early-stage breast cancer of the patient, two types of treatment namely, tamoxifen and tamoxifen combined with radiation therapy are commonly used. As it is essential to consider innovative and cost-effective strategies for early detection and treatment. Investigations through analysis of the stochastic model on early-stage breast cancer with these two types of treatments may help the stakeholders.
In the present paper, we develop a stochastic model for the early-stage breast cancer considering two treatment types, namely tamoxifen and tamoxifen combined with radiation therapy. Mean sojourn time and mean survival time have been calculated. In section 2, we described the stochastic model and its assumptions. In section 3, we define the states of the model and various notations used in the model. In section 4 and section 5, we find the steady-state probabilities and mean sojourn times. Section 6 and section 7 deal with computation of unconditional mean time and mean survival time (MST). A particular case has been considered in section 8. Numerical calculation and graphical analysis are done in section 9. Section 10 includes sensitivity analysis and relative sensitivity analysis. Finally, conclusion is presented in section 11.
Suman, Rajeev Kumar RT&A No 4 (76) STOCHASTIC MODEL ON EARLY-STAGE BREAST CANCER WITH 10„ ' „(nJ TWO 'I YI'IS OF TREATMENTS_Volume 18, December 2023
2. Model Description and Assumptions
The present paper introduces a comprehensive six-state markov model for breast cancer. Any normal person may be infected with breast cancer symptoms. Whenever an early-stage breast cancer is diagnosed in a person then without taking the possibility of breast surgery, two types of treatments, namely tamoxifen (say, treatment-1) and tamoxifen plus radiation (say, treatment-2) have been considered. Here only these two treatments are assumed to be available to the patient that may perfectly cure the cancer. When the patient recovers using any of the treatments, then he/she will go to normal state, otherwise he/she will be in death state as the facility to carry out breast surgery of the patient is not available. Various assumptions for the model are as under:
• Initially the person is normal.
• Transition rates follow exponential distribution and other rates follow general distribution.
• All random variables are independent.
3. Model Development
Various states of the model and notations of different parameters are described in table 1 and table 2 respectively.
Table 1: States specification
States
Description
Sn
Normal State
The state when breast cancer is diagnosed. The state in which treatment-1 is given to breast cancer patient. The state in which treatment-2 is given to the breast cancer patient The state in which no treatment is given to the breast cancer patient Death state
S
i
S
2
S
3
S
4
S
5
Table 2: Notations
Notations Description
Xi Transition rate from normal to diagnosed state
X 2 Transition rate from normal to death state
X 3 Transition rate from treatment-1 to death state
X 4 Transition rate from treatment-2 to death state
X 5 Transition rate from no treatment to death state
h1(t)/H1(t) p.d.f. /c.d.f. of time of recovery from treatment-1.
h2(t)/H2(t) p.d.f. /c.d.f. of time of recovery from treatment-2.
p/q Irobability that treatment is given/not given to the patient.
Pi/qi Probability that treatment-1/treatment-2 is given to the patient.
: Normal State / \ : Diagnosed State
Figure 1. State -Transition Diagram
: Death State
4. Transition Probabilities
The transition probabilities are
q0i(t) = +Mt ; q05(t) = ^e-(Al +*2)t
q«(t) = qiP v ie -
q25(t) = V 3e-A3tHi(t)
v t
q45(t) = V
qi4(t) = q vie-v
q30(t) = h2(t)e-
qi2(t) = PiP v ie-q2o(t) = hi(t) e-
q35(t) = v 4e-A4tH2(t)
The steady-state probabilities p = limL (q^ (t)] are obtained as
Poi =
Pi4 = q ;
P35 = 1- h2(v4) ;
P05 =
P20 = h*(V 3) P45 = 1 •
Pi2 = PPi P30 = h2(v 4)
Pi3 = Pqi
P25 =i - h*(V3)
Clearly,
P0i + P05 = 1
P30 + P35 =i
Pi 2 + Pi3 + Pi4 = 1 P45 = 1
P20 + P25 = 1
5. Mean Sojourn Time
Expected time taken by the patient in state i before transiting to any other state is called mean sojourn time in that state.
V t
v
v
2
It is denoted by || and is given by
Then, we have 1
lo =
\ + X2
l3 =-
1 - h2(X 4)
X,
l = J P(T >t)dt = J R(t)dt
li =■
l4 =:
I2 =■
1 - h*(X 3)
6. Unconditional Mean Time Unconditional mean time m.. is mathematically stated as m.. =Jtq..(t)dt = - q*(0) .
Then, we have
X,
(Xi +X2)2
pqi
X,
m25 = h1 (X3) + 7--
1 h1(X3) .
X 3 X 3
1
m45 =r~
X5
Thus,
m01 + m05 =lo m30 + m35 = l
(X1 +X2)2
m30 = -h2 (X4 )
m12 + m13 + m14 = |
m45 = 14
PP1
m20 =-h* (X3)
u*n ^ 1 h2(X4)
m35 = h2(X4) + 7---
X 4 X 4
m20 + m25 = l
1
X
1
5
0
X
2
m01 =
X
q
m13 =
m14 =
X
1
7. Mean Survival Time
Let ^ (t) denotes the cumulative distribution function of first passage time from S to death state. The following recursive relations are obtained for ^ (t): i(t) = Q01(t)©^(t)+Q05(t)
^(t) = Q12(t)©Mt)+Q13(t)©Mt)+Q14(t)©Mt) Mt) = Q20(t)©i(t)+Q25(t) ^3(t) = Q30(t)(D^(t)+Q35(t)
^4(t) = QJt)
Taking Laplace Stielje's Transform (L.S.T.) on both sides of above equations and solve for **(s), we have
C(s) = N(s) (1)
0 D(s) v '
where ^**(s) is Laplace Stielje's Transform of (t), N(s) = Q05(s) + Q01 (s)(Q*2 (s)Q25 (s) + Q*3(s)Q35 (s) + Q*4(s)Q45(s)) and D(s) = 1 - Q01 (s) Q20 (s) Q*2 (s) - Q01 (s) Q30 (s) Q*3 (s).
Suman, Rajeev Kumar RT&A No 4 (76) STOCHASTIC MODEL ON EARLY-STAGE BREAST CANCER WITH 10„ ' „(nJ TWO 'I YI'IS OF TREATMENTS_Volume 18, December 2023
Now, the mean survival time is given by
T = lim
1 -i (s)
0 s s
Using L'Hospital's rule and putting the value of (s) from equation (1), we get
t = n
t d' where
N = ^0 + ^iPoi + ^2PoiPl2 + ^sPoiPia + ^4PoiPl4 and d = 1 - poipl2p2o - poipiapso •
8. Particular Case
The following Particular case is considered for analysis PurPose:
h(t) = PXe-Pi(t) and h2(t) = P2e-P2(t), where and P2 are recovery rate from treatment-1 and treatment-2, resPectively. The transition Probabilities are given by
Poi =
\ + X2
Po5 =
\ + X2
Pi4 = q
P2o = h*(X 3)
Pi2 = PPi Pso = h2(X4 ) •
Pis = Pqi
Mean sojourn time is given by i
^o =
X +X2
_ i - h2(X4 ) _ i - Pso
"3 = X4 = X 4
where h*(Xs) = P2o =
Xs + Pi
and
h2(X4) = Pso =
X4 +P2
^2 =
i - h*(X - P2
i
X
i
5
P
P
i
2
9. Numerical ComPutation and GraPhical Analysis
For the numerical comPutation and graPhical analysis, the above Particular case is considered. The transition rates (X,X) are taken as given in Harvey et al. (2ois) whereas other Parameters
(X3,X,X5,Pi,qi,P,q) are assumed here. Various graPhs have been Plotted for mean survival time taking varying values to the Parameters involved in its exPression.
In the figures 2, 4, 6, 8 and io, graPhs exhibit the nature of mean survival time (T0) versus transition rates X1, X2, X3, X4 and X5 for varying recovery rate P whereas in the figures s, 5, 7, 9 and ii, graPhs exhibit the nature of mean survival time (T0) versus transition rates Xl,X,X, X4 and X5 for varying recovery rate P2. In figure i2, graPh Presents the nature of mean survival time versus transition rate X for varying values of P and P2.
3800
3750
3700
3650
3600
A =0.00092, A,=0.0025, A =0.0015, A =0.0035,
2. o 4 5
p=0.7, q=0.3, p1=0.6, q^O.4, @2=0.9
0-T 0-4 /3^0.6 /3^0.8
3550 1.76
1.82 1.84 1.86 Transition rate (A1)
1.92
x 10"'
Figure 2. Mean Survival Time (T0) versus transition rate (\) for varying recovery rate (P)
3550 -1-1-1-1-1-1-1-1
1.76 1.78 1.8 1.82 1.84 1.86 1.88 1.9 1.92
Transition rate (Aa) x 10 3
Figure 4. Mean Survival Time (T0) versus transition rate (X2) for varying recovery rate (^)
CD &
■■c >
'E
3
371
3700
3690
3680
3670
3660
3650
3640
3630
A =0.00184, A =0.0025, A, =0.0015, A =0.0035,
1 3 4 5
p=0.7, q=0.3, p^O.6, qj=0.4, /3^0.7
0=OA /3=0.6
P2=0-8
8.4
8.6
8.8
9.2
9.4
9.6
9.8
10
Transition rate (A2)
x 10
3685
3680
3675
u g
M 3670
i-H
(8 >
3665
S 3660
3655
3650
A =0.00184, A =0.00092, A =0.0015, A =0.0035,
1 I 4 £>
^=0.4
1.6
1.8
2.2
2.4
2.6
2.8
Transition rate (A3)
3.2
3.4
xlCP
Figure 6. Mean Survival Time (T0) versus transition rate (X3) for varying recovery rate (P)
3682
HI
5 v
1
3680 3678 , 3676 3674 3672 3670 3668 3666 3664
A^O.00184, Az=0.00092, A4=0.0015, A5=0.0035, p=0.7, q=0.3, p =0.6, q =0.4, /3 =0.7
/92=0.4 /3=0.6
3662
1.6
1.8
2.2
2.4
2.6
2.8
3.2
3.4
Transition rate (A3)
xlO"
3685
3680
<u
•I 3675 1
g 3670 aj
s
3665
3660
A^O.00184, A2=0.00092, A3=0.0025, As=0.0035, p=0.7, q=0.3, p^O.6, q^O.4, ^=0.9
^=0.4 ^=0.6 A- 0.8
0.6
0.8
1.2 1.4 1.6 1.8 Transition rate (A.)
4
2.2
x 10"'
Figure 8. Mean Survival Time (T0) versus transition rate (\) for varying recovery rate (Pt)
3678 r
3676
3674
o 3672 76
| 3670
3
CO
g 3668
3666
3664
=0.00184, A2=0.00092, A3=0.0025, A5=0.0035, p=0.7, q=0.3, Pj=0.6, qj=0.4, /^=0.7
-0.4 -0.6 =0.8
3662 L-0.6
1.4 1.6 Transition rate (A
2.4
x 10"'
3900 r
3850
3800
gj 3750
> 3700
>
g 3650 01
3600
3550
A =0.00184,A =0.00092, A =0.0025, A, =0.0015,
12 «3 4
p=0.7, q=0.3, p^O.6, qi=0.4, /32=0.9
3500 2.6
2.8
3.2
3.4
3.6
3.8
Transition rate (Ag)
/9, =0.4 ^=0.6 ^=0.8
4.2
4.4
x 10"'
Figure 10. Mean Survival Time (T0) versus transition rate (X5) for varying recovery rate (Pt)
3900
3850
3800
A^O.00184, A2=0.00092, A3=0.0025, A4=0.0015, p=0.7, q=0.3, Pl=0.6, qi=0.4, ¡3= 0.9
■/92=0.4 /32=0.6 ■ fl =0.8
g 3750
'-M
I 3700 &
P
Cfi
g 3650 <a 01
3600
3550
3500
2.6
2.8
3.2
3.4
3.6
3.8
4.2
4.4
Transition rate (Ar)
5
xl0";
H
№
3800
3750'
3700
3650
3600
3550
3500
A =0.00092, A =0.0025, A =0.0015, A =0.0035,
2 d 4 5
p=0.7, q=0.3, Pj=0.6, qj=0.4
/3 =0.9
=0.2
-A-/V =0.4
=0.6
=0.2
=0.4
-D-/V =0.6
1.76 1.78 1.8 1.82 1.84 1.86
Transition rate (A^)
1.9
1.92
x 10"
Figure 12. Mean Survival Time (T0) versus transition rate (X) for varying recovery rates P4 and P2
The following interpretations have been drawn from the plotted graphs from the figure 2 to figure12. It can be observed that mean survival time decreases as transition rates Xl, X2, X3, X4 and X5 increases and gives higher values with higher value of recovery rates P4 and P2. Further, the mean survival time (T0) in case of the treatment-2 is higher than that in case of treatment-1.
10. Sensitivity and Relative Sensitivity Analysis
Sensitivity analysis is performed to find out how the variation in involved parameters affect the specific mean survival time under certain specific conditions. Since, there is significance difference between the values of parameters, therefore to compare their effects on mean survival time (MST), relative sensitivity function is used. The sensitivity and relative sensitivity functions for mean survival time (MST) are formulated as under:
5 (MST) n, =- and o. = n,
k 5 k k k
MST
where k = Xl, X2, X3, X4, X5, P4 ,P2
Tablel: Sensitivity and relative sensitivity analysis of MST (T0) with transition rate (A^ for X2 — 0.00092, A3 — 0.0025,14 = 0.0015,AS = 0.0035,ft = 0.7,p2 — 0.9, p = 0.7, q — 0.3, p1 — 0.6, ft = 0.4
1
d(mst)
0.00176 0.00178 0.00180 0.00182 0.00184 0.00186 0.00188 0.00190 0.00192
- 2.4884
- 2.4678
- 2.4475
- 2.4274
- 2.4076
- 2.3880
- 2.3686
- 2.3495
- 2.3306
- 0.2953
- 0.2972
- 0.2991
- 0.3009
- 0.3027
- 0.3045
- 0.3063
- 0.3081
- 0.3098
5
Table 2: Sensitivity and relative sensitivity analysis of MST (70) with transition rate (12) for = 0.00184, A3 = 0.0025,14 = 0.0015,AS = 0.0035,ft = 0.7,^2 = 0.9, p = 0.7, qr = 0.3, p1 = 0.6, ft = 0.4
_ d(mst) / ^ \
^ 5a2 - ^ (mst)
0.00084 -1.1087 - 0.6019
0.00086 -1.0776 - 0.6075
0.00088 -1.0478 - 0.6130
0.00090 -1.0192 - 0.6183
0.00092 - 9.9172 - 0.6235
0.00094 - 9.6537 - 0.6285
0.00096 - 9.4006 - 0.6334
0.00098 - 9.1574 - 0.6382
0.00100 - 8.9234 - 0.6428
Table 3: Sensitivity and relative sensitivity analysis of MST (70) with transition rate (A3) for A1 — 0.00184, A2 — 0.00092, A4 — 0.0015,As — 0.0035,ft — 0.7,^2 — 0.9, p — 0.7, qr — 0.3, p1 — 0.6, ft — 0.4
d(mst) 5*3 — (ms3t)
0.0017 -1.0919 - 0.0013
0.0019 - 1.0910 - 0.0014
0.0021 - 1.090 - 0.0016
0.0023 - 1.0891 - 0.0017
0.0025 - 1.0881 - 0.0019
0.0027 - 1.0872 - 0.0020
0.0029 - 1.0863 - 0.0022
0.0031 - 1.0853 - 0.0023
0.0033 -1.0844 - 0.0024
Suman, Rajeev Kumar
STOCHASTIC MODEL ON EARLY-STAGE BREAST CANCER WITH 10t ' „(nJ
TWO TYI'15 OFTREATME NTS_Volume 18, December 2023
Table 4: Sensitivity and relative sensitivity analysis of MST(T0) with transition rate (X4) for X1 — 0.00184, X2 — 0.00092, X3 — 0.0025,XS — 0.0035, fa — 0.7,fa — 0.9,p — 0.7,q — 0.3, p1 — 0.6,— 0.4
_ d(MST) p _ {14\
X4 ^ — [MST)
0.0007 - 567.6037 - 2.7142
0.0009 - 567.2638 - 3.4879
0.0011 - 566.9242 - 4.2608
0.0013 - 566.5849 - 5.0329
0.0015 - 566.246 - 5.8041
0.0017 - 565.9073 - 6.5746
0.0019 - 565.5689 - 7.3442
0.0021 - 565.2309 -8.1131
0.0023 - 564.8931 - 8.0011
Table 5: Sensitivity and relative sensitivity analysis of MST(T0) with transition rate (15) for X1 — 0.00184, X2 — 0.00092, — 0.0025, X4 — 0.0015, fa — 0.7, fa — 0.9,p — 0.7,q — 0.3, p1 — 0.6, q1 — 0.4
a,
— ■
d(MST)
Si.
— (
MST)
0.0027 0.0029 0.0031 0.0033 0.0035 0.0037 0.0039 0.0041 0.0043
- 5.1318
- 4.4481
- 3.8926
- 3.4351
- 3.0537
- 2.7325
- 2.4595
- 2.2254
- 2.0232
- 0.0927
- 0.0868
- 0.0817
- 0.0771
- 0.0730
- 0.0694
- 0.0660
- 0.0630
- 0.0603
Table 6: Sensitivity and relative sensitivity analysis of MST(T0) with recovery rate (fa) X1 — 0.0018, X2 — 0.00092,X3 — 0.0025,X4 — 0.0015,AS — 0.0035, fa — 0.9,p — 0.7,q — 0.3, p1 — 0.6, q1 — 0.4
d(MST) U(v1)— d/h SPi — {MST)
0.4 8.5702 0.0023
0.6 3.8331 0.0016
0.8 2.1629 0.0012
1.0 1.3869 9.4735
1.2 0.9643 7.9033
1.4 0.7091 6.7796
1.6 0.5433 5.9357
1.8 0.4295 5.2786
Suman, Rajeev Kumar
STOCHASTIC MODEL ON EARLY-STAGE BREAST CANCER WITH 10„ ' ,(„J
TWOTYPES OFTREATME NJTS)_Volume 18, December 2023
Table 7: Sensitivity and relative sensitivity analysis of MST (T0) with recovery rate (S2) for — 0.0018, A2 — 0.00092,A3 — 0.0025,14 — 0.0015, As — 0.0035, ft — 0.7,p — 0.7, q — 0.3, p1 — 0.6, ft — 0.4
d(msr)
^2
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
2.5847 1.1526 0.6494 0.4161 0.2891 0.2125 0.1628 0.1286
7.0676 4.7265 3.5504 2.8430 2.3707 2.0329 1.7794 1.5821
The sensitivity and relative sensitivity analyses of mean survival time are carried out with involved parameters (Xx, X2, X3, X4, X5, Px, P2). The sensitivity and relative sensitivity analyses of mean survival time with these parameters are tabulated in table 1 to table 7. Tables 1-5 show that signs of the sensitivity of mean survival time with parameters X, X2, X, X and X5 are negative which implies that increase in these parameters decline the value of MST. Tables 6-7 show that signs of the sensitivity of mean survival time with parameters P: and P2 are positive which lead to the conclusion that increase in these parameters improve the value of mean survival time. As transition rates X , X , X , X and X increase, sensitivity function increases whereas relative
sensitivity function decreases and whenever recovery rates P and P increase, sensitivity function decreases.
11. Conclusion
Breast cancer is indeed one of the most common cancers diagnosed in women globally. It is a major public health concern and significant impact on women's health. In the paper, the evaluated expressions for mean sojourn time in the different states of the model gives estimates of the times for patient remains in a particular stage. The investigation through the stochastic analysis of the model on breast cancer considering two types of treatments in various progression stages concludes that the mean survival time lessens with the rise in the rates of transition. It has been observed that mean survival time from tamoxifen plus radiation is higher than tamoxifen only. It is concluded that tamoxifen plus radiation is more effective and useful than only tamoxifen for treatment of early-stage breast cancer.
References
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[2] Bayer, P., Brown, J.S., Dubbeldam, J. and Broom, M. (2021). A Markov chain model of cancer treatment. doi: https://doi.org/10.1101/2021.06.16.448669.
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