Biology, Medicine, & Natural Product Chemistry  ISSN 2089-6514 (paper) 
Volume 8, Number 1, 2019 | Pages: 11-15 | DOI: 10.14421/biomedich.2019.81.11-15 ISSN 2540-9328 (online) 
 

 

 

 

Mathematical Model of Cervical Cancer Treatment 

Using Chemotherapy Drug 

 
Murtono1,*, Meksianis Zadrak Ndii2, Sugiyanto3 

1Department of Physics Education; 3Department of  Mathematics, Universitas Islam Negeri Sunan Kalijaga, 55281, Yogyakarta, Indonesia. 
2Department of Mathematics, Faculty of Science and Engineering, University of Nusa Cendana, Kupang-NTT, Indonesia. 

 

Author correspondency*:  
hasnamur@yahoo.co.id 

 

 

Abstract 

 

Cervical cancer is a malignant disease that causes problems in women's health, espe cially in developing countries such as Indonesia. 

Cervical cancer cells will develop quickly, uncontrollably, and will continue to divide and then infiltrate the surrounding t issue and 

continue to spread to connective tissue, blood, and attack important organs and spinal nerves. The aim of the research is to study the 

mathematical model of cervical cancer by chemotherapy treatment. The results of this study are that cervical cancer treatment using 

chemotherapy is effective enough to kill abnormal cells such as infected cells, pre-cancerous cells and cancer cells, although there are side 

effects, namely the killing of normal cells due to chemotherapy drugs. 

 

Keywords: Cervical cancer; infected cells; infected cells; cancer cells; chemotherapy 

 

 

INTRODUCTION 
 

Cervical cancer is an excessive and uncontrolled cell 

growth around the cervix (Walboomers et. al., 1999). 

Cervical cancer originated from cells in the cervix 

(Naganawa et al., 2005). Most cervical cancers begin in 

the transformation zone which is a shift from squamous 

cell type to cylindrical cell type. These cells do not 

directly turn into cervical cancer. Normal cervical cells 

due to the influence of carcinogenic substances can 

develop gradually into pre-cancerous cells and then 

become cancer cells (Sari et al., 2016).  

The main trigger for the emergence of cervical 

cancer is infection of several types of high-risk Human 

Papilloma Virus (HPV) which causes proliferation of the 

epidermal surface and cervical mucosa (Bosch, 1995). 

The types of HPV that are very common in cases of 

cervical cancer are types 16 and 18, which is more than 

70% of all cervical cancers reported. The results of a 

study of 1,000 samples from 22 countries proved the 

presence of HPV infection in 99.7% of cervical cancer 

cases (Wuryanti et. al., 2015). Cervical cancer is the 

second most common type of cancer in women 

worldwide to breast cancer (Boice et. al., 2002). 

Chemotherapy is a kind of cancer treatment that uses 

drugs to destroy cancer cells. Chemotherapy works by 

stopping or slowing the growth of cancer cells, which 

grow and divide rapidly. Chemotherapy can also harm 

healthy cells that divide rapidly, such as the lines of the 

mouth and intestines or cells that affect growth. Damage 

to healthy cells can cause side effects. Often, side effects 

will be disappear after chemotherapy is complete (Rose 

et. al., 1999). Reduction of the mass of cervical cancer 

can be used as used to measure the effectiveness of 

treatment because chemotherapy can cause shrinkage of 

the mass of cervical cancer. Cancer mass has an 

important role to detect the prognosis of a cervical 

cancer. 

 

 

FORMULATION OF MODEL 
 

This model developed frpm the past research (Asih, et. 

al., 2015) about the development of cervical cancer and 

Pillis et al. (2007) about the model of treating cancer in 

general with chemotherapy. 

 

 

 
 

 

Figure 1. Cervical Cancer Treatment Diagram by Chemotherapy. 

 

 

 

 

 

 

 

https://doi.org/10.14421/biomedich.2019.81.11-15


 

 

 

12 Biology, Medicine, & Natural Product Chemistry 8 (1), 2019: 11-15 
 

 

Table 1. Subpopulations, Parameters and units. 

 

Symbol Symbol Explanation Unit Unit 

S(t) Normal cell density cell/mm2 

I(t) Infected cell density cell/mm2 

P(t) Pre-cancerous cell density cell/mm2 

C(t) Cancer cell density cell/mm2 

V(t) Virus density virus/mm2 

M(t)
 

Concentration of chemotherapy 

drugs 
mg/m2 

r Growth rate of normal cell 1/day 

N  Homeostatic carrying capacity  cell/mm3 

  The rate of infection 1/(day.virus) 

1a  
The rate of proliferation of 

infected cells 
1/day 

1d
 

The rate of infected cell apoptosis 1/day 


 

The rate of progression, from 

infection to pre-cancer 
1/day 

2a
 

The rate of proliferation of pre-

cancerous cell 
1/day 

2d
 

The rate of pre-cancerous cell 

apoptosis 
1/day 


 

The maximum invasion rate, from 

precancerous to cancer 
1/day 

K
 

Half-saturation consentration  cell/mm3 

3a
 

Cancer cell proliferation rate 
 

1/day 

3d
 

Summing the rate of apoptosis and 

the rate of cancer cell metastasis 
1/day 

n
 

The average number of viruses 

produced by an infected cell
 

constant 

4d
 

The rate of virus death 1/day 

Sk
 

Fractional susceptible cells kill by 

chemotherapy 
1/day 

Ik
 

Fractional infected cells kill by 

chemotherapy 
1/day 

Pk
 

Fractional pre-cancerous cells kill 

by chemotherapy 
1/day 

Ck
 

Fractional cancer cells kill by 

chemotherapy 
1/day 


 

The rate of chemotherapy drug 

decay 
1/day 

Mv
 

The rate of chemotherapy drug 

intake 
mg/m2.day 

 

The dynamics of changes in cervical cells from 

normal cells to cancer cells are given in the following 

system of differential equations: 

 

1 S
dS S I

rS SV k MS
dt N


 

    
 

 

(1a) 

 

1 1 I

dI
SV a I d I I k MI

dt
     

 

(1b) 

 

1 1 4

dV
n d I d V

dt
 

 

(1c) 

2

2 2 2 2 P

dP P
I a P d P k MP

dt K P


    

  

(1d) 

 

2

3 32 2 C

dC P
a C d C k MC

dt K P


   


 

(1e) 

 

M

dM
M v

dt
  

  

(1f) 

 

Let  

1 1 2 2 3 3 1 1

1 1 1 1 4

, , , , ,

, , , , ,

S I
a d a b a d k d a S I

N N

P C N
P C n n d N c d p

K K K K




       

       

 

By non-conventionalizing the System (1a) - (1f) to 

be 

 

  1 1 1 1 1 11 S
dSdS

rS S I S V k MS
dt dt

     
 

(2a) 

 

1
I

dIdI
SV aI I k MI

dt dt
     

 

(2b) 

 

dV
nI cV

dt
 

 

(2c) 

 

2
1

12
1

p

dP P
pI bP k MP

dt P
    


 

(2d) 

 

2

2
1

C

dC P
kC k MC

dt P
  


 

(2e) 

 

M

dM
M v

dt
  

 

(2f) 

 

By eliminating the variable index, System (2a) - (2f) 

which is non-professional becomes 

 

  1 S
dS

rS S I SV k MS
dt

      (3a) 

 

I

dI
SV aI I k MI

dt
      (3b) 

 

dV
nI cV

dt
 

 

(3c) 

 

2

2
1

P

dP P
pI bP k MP

dt P


   


 (3d) 

 

2

2
1

C

dC P
kC k MC

dt P


  


 (3e) 

 

M

dM
M v

dt
    (3f) 

 

 



 

 

 

 Murtono et al. – Mathematical Model of Cervical Cancer Treatment … 13 
 

 

EQUILIBRIUM POINT 
 

Theorem 1. Let 3 22 9 27 , 3u x xy z v x y     .  

The equilibrium point with chemotherapy in cervical 

cancer from System (3a) - (3f) is 

 
 

* * * * * *

3 4 5 5 2 5 1

, , , , ,

, , , , ,i i

EP S I P C V M

p c      



   

1 2 3i , , , where 1 0
Mv


  , 
2 0

n

c
   , 

1
3 1

Sk

r


   , 

2
4 1 0

r




 
    

 
,  

1 2 3
5

2 4

Ia k   
 

  
 , 5

1

,
P

p
x

k b

  







 

51

1 1

, .P

P P

pk b
y z

k b k b

 

 


  

 

 

 

Proof. 

From equation (3f) is obtained 

1 0
MvM 


   . (4) 

From equation (3c) is obtained  

2

nI
V I

c
  , (5) 

where 
2 0

n

c
   . 

If equation (4) and equation (5) are substituted into 

equation (3a), then we obtain 

 

3 4S I   , (6) 

 

where 13 1
Sk

r


    and  24 1

r




 
   

 
. 

 

If equations (4), (5) and (6) are substituted in equation 

(3b), they are obtained 

 

0I   or 1 2 3 5
2 4

Ia kI
   


 

  
  ,

 

(7) 

 

where 1 2 35
2 4

Ia k   
 

  
 .

   

  

 

For 𝐼 = 0 impossible, because the virus is the cause of 
cervical cancer, so it was chosen 𝐼 = 𝜉5. If equation (4) 
and equation (7) are substituted into equation (1d), then 

they are obtained 𝑃3 + 𝑥𝑃2 + 𝑦𝑃 + 𝑧 = 0, where 
 

5 51

1 1 1

, , .P

P P P

p pk b
x y z

k b k b k b

    

  

 
   

  
 

 

 

The roots are 

2 33
1

2 33

1 1
4

3 3 2

1 1
4 ,

3 2

x
p u u v

u u v

      
  

  
  

, 

2 33
2

2 33

1 3 1
4

3 6 2

1 3 1
4

6 2

x i
p u u v

i
u u v

       
  

   
  

2 3
3

3

2 3
3

1 3 1
4

3 6 2

1 3 1
4 .

6 2

x i
p u u v

i
u u v

       
 

   
 

  (8) 

 

If equation (4) and equation (8) are substituted into 

equation (3e), then they are obtained 𝐶 =
𝜃𝑝𝑖

2

(𝑘+𝑘𝐶𝜉1)(1+𝑝𝑖
2)
= 𝑐𝑖, where 𝑖 = 1,2,3.■ 

 

Theorem 2. If 1 2 3 0Ia k       , 3 4 5 0     

and if one 𝑝𝑖 > 0 and real, then the equilibrium point 
with chemotherapy or value 𝐸𝑃 exist. 
 

Proof. 

From Theorem 1 is obtained 𝑀 =
𝑣𝑀

𝛾
= 𝜉1 > 0. Because 

𝑎 + 𝛿 + 𝑘𝐼𝜉1 − 𝑎𝜉2𝜉3 > 0, then 𝐼 =
𝑎+𝛿+𝑘𝐼𝜉1−𝑎𝜉2𝜉3

𝑎𝜉2𝜉4
=

𝜉5 > 0. Because 𝐼 = 𝜉5 > 0, then 𝑉 =
𝑛𝐼

𝑐
= 𝜉2𝐼 > 0. 

Because 𝜉3 + 𝜉4𝜉5 > 0, then 𝑆 = 𝜉3 + 𝜉4𝜉5 > 0. One is 
𝑝𝑖 > 0, where 𝑖 = 1,2,3 and real. Value is 

𝜃𝑝𝑖
2

(𝑘+𝑘𝐶𝜉1)(1+𝑝𝑖
2)
= 𝑐𝑖 > 0. So, the equilibrium point with 

chemotherapy or value 𝐸𝑃 exist.■ 
 

 

STABILITY 
 

Theorem 3. Let 𝑢 = 2𝑥3 − 9𝑥𝑦 + 27𝑧, 𝑣 = 𝑥2 − 3𝑦. If 

−𝑘 − 𝑘𝐶𝜉1 < 0, 𝑏 −
2𝜃𝑝𝑖

(1+𝑝𝑖
2)
− 𝑘𝑃𝜉1 < 0, where 𝑖 =

1,2,3, 𝑥 > 0, 𝑢 > 0, 𝑢2 = 4𝑣3, and 𝑢 < 2𝑥3, then 
equilibrium point 𝐸𝑃 = (𝜉3 + 𝜉4𝜉5,𝜉5,𝜉2𝜉5,𝑝𝑖, 𝑐𝑖,𝜉1) 
asymptotic stable.

 

 

Proof.  

From System (3a) - (3f) the Jacobian matrix with 

eigenvalue is obtained 
 

 1   , 2 1Ck k    , 

 
3 12

2

2

1

i
P

i

p
b k

p


   



,
 



 

 

 

14 Biology, Medicine, & Natural Product Chemistry 8 (1), 2019: 11-15 
 

 

2 33
4

2 33

1 1
4

3 3 2

1 1
4 ,

3 2

x
u u v

u u v

       
  

  
  

 

2 33
5

2 33

1 3 1
4

3 6 2

1 3 1
4

6 2

x i
u u v

i
u u v


       

  

   
  

2 33
6

2 33

1 3 1
4

3 6 2

1 3 1
4 .

6 2

x i
u u v

i
u u v


       

  

   
  

 
 

Value is 𝜆1 = −𝛾 < 0. Because −𝑘 − 𝑘𝐶𝜉1 < 0, then 

𝜆2 = −𝑘 − 𝑘𝐶𝜉1 < 0. Because 𝑏 −
2𝜃𝑝𝑖

(1+𝑝𝑖
2)2

− 𝑘𝑃𝜉1 < 0, 

then 𝜆2 = 𝑏 −
2𝜃𝑝𝑖

(1+𝑝𝑖
2)2

− 𝑘𝑃𝜉1 < 0. Because 𝑣 = 0, then 

𝜆4 = −
𝑥

3
−

1

3
√𝑢
3

, 𝜆5 = −
𝑥

3
−

1

3
√
1

2
𝑢

3
, 𝜆6 = −

𝑥

3
+

1

3
√
1

2
𝑢

3
. Because 𝑢 > 0, then 𝜆4 < 0. Because 𝑢 < 𝑥

3, 

then 𝜆5 = 𝜆6 < 0. Because 𝜆1,𝜆2,𝜆3,𝜆4,𝜆5 and 𝜆6, then 
the equilibrium point 𝐸𝑃 asymptotic stable.■ 

 

 

SIMULATION 
 

In this section we will discuss numerical simulations and 

medical interpretations of the mathematical model of 

cervical cancer affected by chemotherapy. First, the 

parameter values used and the initial values for each 

variable are given first. Taking parameter values for 

numerical simulations is still in the form of assumptions 

based on the rate of growth of cancer cells in general. 

Sources and interpretations of parameter values can be 

seen in the reference. The parameter values for this case 

are given in Table 2. 

 
Table 2. Case Parameter Value of the Effects of Chemotherapy. 

 

Symbol Value Reference 

𝑟 0.02 Asih et al. (2015) 
𝛼 0.0001 Asih et al. (2015) 
𝑎 0.01  Asih et al. (2015) 
𝛿

 
0.0082  Asih et al. (2015) 

𝑛
 

10000 Asih et al. (2015) 

𝑐
 

50 Asih et al. (2015) 

𝑝
 

13.44 Asih et al. (2015) 

𝑏
 

1 Asih et al. (2015) 

𝜃
 

2.03 
 

Asih et al. (2015) 

𝑘
 

1.01  Asih et al. (2015) 

𝑘𝑆
 

0.0006 Estimation 

𝑘𝐼
 

0.6 Pillis et al. (2007) 

𝑘𝑃
 

0.6 Pillis et al. (2007) 

𝑘𝐶
 

0.8 Pillis et al. (2007) 

𝛾
 

0.9 Pillis et al. (2007) 

𝑣𝑀
 

1 Pillis et al. (2007) 

 

 

 

  
 

Figure 2. System Simulation (3a) - (3f) with parameter values 𝑟 = 0, 𝛼 =
0.0001, 𝑎 = 0.01, 𝛿 = 0.0082, 𝑛 = 10000, 𝑐 = 50, 𝑝 = 13.44, 𝑏 = 1, 
𝜃 = 2.03,  𝑘 = 1.01, 𝑘𝑆 = 0.0006,  𝑘𝐼 = 0.6,  𝑘𝑃 = 0.6,  𝑘𝐶 = 0.8, 𝛾 =
0.9,  𝛾 = 1, and initial conditions (13, 13, 13, 13, 13, 13). The dynamics 
due to normal cell chemotherapy drugs, infected cells, precancerous cells, 

cancer cells, cancer cells, and viruses declined in three days. (A) normal 

cells. (B) Infected cells, precancerous cells, and cancer cells. (C) Viruses. 

(D) Chemotherapy drug concentration. 



 

 

 

 Murtono et al. – Mathematical Model of Cervical Cancer Treatment … 15 
 

 

Figure 2 shows a trajectory with the influence of 

chemotherapy drugs, on cervical cancer patients. 

Normal cells in the first 3 days showed a decrease, from 

13 cells/mm2 to 7,254 cells/mm2. This makes a severe 

effect for cervical cancer patients with this 

chemotherapy drug. Infected cells, precancerous cells 

and cancer cells drop very quickly, in the first 3 days of 

chemotherapy, from 13 cells/mm2 to 0.01308 cells/mm2. 

The initial virus rose would derease after chemotherapy. 

The virus in 3 days is from 13 viruses/mm2 to 0.3771 

viruses/mm2. This is because, many infected cells die 

because of the effects of chemotherapy drugs. The 

chemotherapy drug on the third day still contained 

1.91mg/m2. It means that chemotherapy drugs still have 

an effect on normal cells and other abnormal cells, 

including cancer cells. 

 

 

CONCLUSION 
 

Cervical cancer is one type of malignant cancer and is 

most prevalent in women compared to other cancers. 

Treatment of cervical cancer with chemotherapy is quite 

effective. Abnormal cancer cells such as infected cells, 

precancerous cells, and many cancer cells die from this 

treatment. Because normal cells are dead, this is the side 

effect of the of this treatment. This research needs to be 

continued with other treatments that are more effective, 

especially to reduce the adverse effects to the normal 

cell. 

 

 

REFERENCES 
 
Asih, T. S. N., Lenhart, S., Wise, S., Aryati, L., Adi-Kusumo, F., 

Hardianti, M. S., & Forde, J. (2016). The dynamics of HPV 

infection and cervical cancer cells. Bulletin of mathematical 

biology, 78(1), 4-20. 

Boice Jr, J. D., Day, N. E., Andersen, A., Brinton, L. A., Brown, 

R., Choi, N. W., ... & Hakama, M. (1985). Second cancers 

following radiation treatment for cervical cancer. An 

international collaboration among cancer registries. Journal of 

the National Cancer Institute, 74(5), 955-975. 

Bosch, F. X., Manos, M. M., Muñoz, N., Sherman, M., Jansen, A. 

M., Peto, J., ... & Shan, K. V. (1995). Prevalence of human 

papillomavirus in cervical cancer: a worldwide 

perspective. JNCI: Journal of the National Cancer 

Institute, 87(11), 796-802.  

de Pillis, L. G., Gu, W., Fister, K. R., Head, T. A., Maples, K., 

Murugan, A., ... & Yoshida, K. (2007). Chemotherapy for 

tumors: An analysis of the dynamics and a study of quadratic 

and linear optimal controls. Mathematical 

Biosciences, 209(1), 292-315. 

Naganawa, S., Sato, C., Kumada, H., Ishigaki, T., Miura, S., & 

Takizawa, O. (2005). Apparent diffusion coefficient in 

cervical cancer of the uterus: comparison with the normal 

uterine cervix. European radiology, 15(1), 71-78. 

Rose, P. G., Bundy, B. N., Watkins, E. B., Thigpen, J. T., Deppe, 

G., Maiman, M. A., ... & Insalaco, S. (1999). Concurrent 

cisplatin-based radiotherapy and chemotherapy for locally 

advanced cervical cancer. New England Journal of 

Medicine, 340(15), 1144-1153. 

Sari, H. E., Mudigdo, A., & Demartoto, A. (2016). Multilevel 

Analysis on the Social Determinants of Cervical Cancer in 

Yogyakarta. Journal of Epidemiology and Public 

Health, 1(2), 100-107. 

Walboomers, J. M., Jacobs, M. V., Manos, M. M., Bosch, F. X., 

Kummer, J. A., Shah, K. V., ... & Muñoz, N. (1999). Human 

papillomavirus is a necessary cause of invasive cervical 

cancer worldwide. The Journal of pathology, 189(1), 12-19. 

Wuryanti, S., Andrijono, A., Susworo, S., & Witjaksono, F. 

(2015). The effect of high poly unsaturated fatty acid (PUFA) 

dietary supplementation on inflammatory status of patients 

with advanced cervical cancer on radiation treatment. Acta 

Medica Indonesiana, 47(1). 

 

 

 



 

THIS PAGE INTENTIONALLY LEFT BLANK