Biology, Medicine, & Natural Product Chemistry ISSN 2089-6514 (paper) Volume 11, Number 1, April 2022 | Pages: 55-63 | DOI: 10.14421/biomedich.2022.111.55-63 ISSN 2540-9328 (online) A Mathematical Model of Cervical Cancer Treatment by Radiotherapy Followed by Chemotherapy Murtono1,*, Sugiyanto2, Mansoor Abdul Hamid3 1Department of Education Physics, 2Department of Mathematics; Universitas Islam Negeri Sunan Kalijaga Yogyakarta, Indonesia. 3Food Technology and Bioprocess, Universiti Malaysia Sabah, Malaysia. Corresponding author* hasnamur@yahoo.co.id Manuscript received: 15 April, 2022. Revision accepted: 10 May, 2022. Published: 21 May, 2022. Abstract Cervical cancer is the second most common type of malignant tumor found in women aged 15-44 years worldwide. Radiotherapy is one form of treatment that uses radiation that can eliminate or kill cancer cells and one way to treat cervical cancer is qui te popular. Chemotherapy is a cancer treatment using anticancer drugs designed to kill or slow the growth of cancer cells that divide rapidly in the body. The method used is to study mathematical model of cervical cancer with radiotherapy and radiotherapy treatment followed by chemotherapy. The results of this study are for the early stages of cervical cancer with radiotherapy is quite effective, while for the late stages of radiotherapy and chemotherapy is less effective. Therefore, the need for other treatments for end -stage cervical cancer in addition to radiotherapy and chemotherapy. Keywords: Cervical cancer; radiotherapy; chemotherapy. INTRODUCTION Cervical cancer is still the leading cause of cancer deaths among women in developing countries including Indonesia (Hoffman, et.al, 2008 and Woodman et al., 2007). Indonesia Cancer Foundation explained, the mortality rate of cervical cancer most among other types of cancer among women. It is estimated that 52 million Indonesian women are at risk for cervical cancer, while 36 percent of all women with cancer are cervical cancer patients. There are 15,000 new cases per year with the death of 8,000 people per year (Kompas, 2008). In Southeast Asia the incidence of cervical cancer becomes number two in women after breast cancer. Nevertheless, in Globocan (2012) the mortality rate due to cervical cancer is higher than the mortality rate due to breast cancer. Radiotherapy is one of treatments that can be done for a patient with cervical cancer. Radiotherapy is used as a therapy for cancer patients by using radiation that can eliminate and kill cancer cells, a way to treat cervical cancer which is quite popular. Treatment with radiation therapy is usually done with chemotherapy, or accompanied by other therapies and sometimes only with radiotherapy alone depending on the type of cervical cancer suffered and the level of cancer. MODEL FORMATION This model develops from Asih, et al. (2015) on the development of cervical cancer, Liu and Yang (2014) on general cancer treatment models with radiotherapy and Pillis et al. (2007) on general cancer treatment models with chemotherapy. Figure 1. Transfer Diagram of Cervical Cancer Treatment with Radiotherapy and followed by Chemotherapy. Caption Figure 1:  1 1SL S k MS S   , 2 2( ) IL I k MI I   , 3 3( ) PL P k MP P   , https://doi.org/10.14421/biomedich.2022.111.55-63 56 Biology, Medicine, & Natural Product Chemistry 11 (1), 2022: 55-63 4 4 ( ) C L C k MC C  is a function of chemotherapy and radiotherapy. Table 1. Subpopulations, Parameters and Units. Symbol Explanation Unit  S t Normal cells density cells/mm3  I t Infected cells density cells/mm3  P t Pre-cancerous cells density cells/mm3  C t Cancer cells density cells/mm3  V t Free virus density virus/mm3  M t Concentrations of chemotherapy drugs mg r Intrinsic cell growth rate is normal 1/ day N Homeostatic carrying capacity cells/mm3  The rate infection 1/(day x virus) 1 a The rate of proliferation of infected cell 1/day 1 d The rate of apoptosis of infected cell 1/day  The rate of progression, from being infected to pre-cancer 1/day 2 a The rate of proliferation of pre-cancerous cells 1/day 2 d The rate of apoptosis of pre-cancerous cells 1/day  The maximum invasion rate, from precancer to cancer 1/day K Half-saturation concentration cells/mm3 3a The rate of proliferation of cancer cells 1/day 3 d The rate of apoptosis of cancer cells 1/day n The average number of viruses produced one infected cell constant 4 d The death rate of free virus 1/day 1  The proportion of radiation to susceptible cells (normal cells) constant 2  The proportion of radiation to infected cells constant 3  The proportion of radiation to pre-cancerous cells constant  Strategy of radiotherapy 1/day S k Fractional susceptible cells killed by chemotherapy 1/day I k Fractional infected cells killed by chemotherapy 1/day P k Fractional pre-cancerous cells killed by chemotherapy 1/day C k Fractional cancer cells killed by chemotherapy 1/day  The rate of chemotherapy drug decay 1/day M v The rate of chemotherapy drug enter mg/day Murtono et al. – A Mathematical Model of Cervical Cancer Treatment by … 57 The dynamics of cervical cell changes from normal cells to cancer cells are given in the following differential equation Systems (1). 1 1 S dS S I rS SV k MS S dt N              (1a) 1 1 2I dI SV a I d I I k MI I dt          (1b) 2 2 2 32 2 P dP P I a P d P k MP P dt K P           (1c) 2 3 32 2 C dC P a C d C k MC C dt K P        (1d) 1 4 dV nd I d V dt   (1e) M dM M v dt    (1f) THE POINT OF EQUILIBRIUM The equilibrium point with radiotherapy alone, without chemotherapy System (1) without chemotherapy is 1 1 dS S I rS SV S dt N             (2a) 1 1 2 dI SV a I d I I I dt         (2b) 2 2 2 32 2 dP P I a P d P P dt K P          (2c) 2 3 32 2 dC P a C d C C dt K P       (2d) 1 4 . dV nd I d V dt   (2e) The requirement to determine the equilibrium point of System (2) is 0, 0, 0, 0, 0 dS dI dP dC dV dt dt dt dt dt      . So the point of equilibrium with radiotherapy in cervical cancer, without chemotherapy, is    * * * * *1 2 3 1 51 4, , , , , , , ,R REP S I P C V P       * * * * *2 2 3 2 52 4, , , , , , , ,R REP S I P C V P    (3)    * * * * *3 2 3 3 53 4, , , , , , , ,R REP S I P C V P    , where 1 1 4 nd d   , 1 2 1 2 1 d a         ,  1 2 3 1 r N r r N          , 4 1 3   , 3 1 2 2 3 A a d         , 2 2 2 2 2 3 1 2 2 3 a K d K K B a d          , 2 3 1 2 2 3 K C a d       , 3 1 1 1 1 1 2 9 27D A A B C   ,     2 3 3 2 1 1 1 1 1 1 1 2 9 27 4 3E A A B C A B     ,    1 3 31 1 1 1 1 1 1 1 1 3 3 2 3 2 R A P D E D E      ,    1 3 32 1 1 1 1 1 3 1 1 3 1 3 6 2 6 2 R A i i P D E D E         ,    1 3 33 1 1 1 1 1 3 1 1 3 1 3 6 2 6 2 R A i i P D E D E         ,    2 5 2 2 3 3 i i i P K P d a        . 58 Biology, Medicine, & Natural Product Chemistry 11 (1), 2022: 55-63 Existence of the equilibrium points with radiotherapy alone, without chemotherapy if it satisfies 1. 1 2 1 0d a      , 2.    1 2 1 4 1 1 0 d a d r N r nd                 , 3. 0, 1, 2, 3iP i  and 4. 3 3 0d a   . The equilibrium points with radiotherapy and followed by chemotherapy In this system is the same as System (1). To find the equilibrium point in a way 0 dS dt  , 0 dI dt  , 0 dP dt  , 0 dC dt  , 0 dV dt  , 0 dM dt  . So the equilibrium point of radiotherapy followed by chemotherapy in cervical cancer is    * * * * * *1 7 9 8 9 1 101 1 9 6, , , , , , , , , ,RC RC RCEP S I P C V M P            * * * * * *2 7 9 8 9 2 102 1 9 6, , , , , , , , , ,RC RC RCEP S I P C V M P         (4)    * * * * * *3 7 9 8 9 3 103 1 9 6, , , , , , , , , ,R RC RCEP S I P C V M P         where 6 M v    , 1 7 1 N r           , 6 1 8 S Nk N N r r        , 1 8 1 1 6 2 9 1 7 I a d k                , 11 2 2 2 6 3 , P A a d k             22 2 6 3 2 2 2 6 3 P P a d k K B a d k              , 2 11 2 2 2 6 3P K C a d k         , 3 2 2 2 2 2 2 9 27D A A B C   ,     2 3 3 2 2 2 2 2 2 2 2 2 9 27 4 3E A A B C A B        2 3 31 2 2 2 2 1 1 1 1 3 3 2 3 2 RC A P D E D E      ,    2 3 32 2 2 2 2 1 3 1 1 3 1 3 6 2 6 2 RC A i i P D E D E         ,    2 3 33 2 2 2 2 1 3 1 1 3 1 3 6 2 6 2 RC A i i P D E D E         ,    2 10 2 2 3 3 6 4 iRC iRC iRC C P K P a d k           . Existence of the equilibrium points with radiotherapy and followed by chemotherapy, if it satisfies 1. 6 1 7 9 0S Nk N N r r         , 2. 1 8 1 1 6 2 0Ia d k            and 3. 0iRCP  . STABILITY OF THE EQUILIBRIUM POINT Stability of the equilibrium points with radiotherapy alone, without chemotherapy Theorem 1. If 3 3 0a d    ,   2 2 3 32 2 2 2 0i i PK a d K P        , 0 i A  , 3 2 9 27 0 i i i i A A B C   , 2 3 0 i i A B  where 3, 4, 5i  then the equilibrium point Murtono et al. – A Mathematical Model of Cervical Cancer Treatment by … 59    * * * * * 2 3 6 5, , , , , , , ,iR i iEP S I P C V P    asymptotically stable. Proof. From System (2) is obtained characteristic equation in Equation (5).            2 1 3 3 2 3 32 2 2 1 3 232 51 1 1 1 2 4 32 51 1 1 1 2 4 2 1 1 2 4 2 1 51 32 51 1 2 2 2 2 P K a d a d K P rr r a d d N N rr r a d d N N r a d d nd N rr r N N                                                                                                     1 1 2 4 2 1 2 51 4 2 1 0 a d d nd r d nd N                                               (5) Stability of the equilibrium points with radiotherapy and followed by chemotherapy Theorem 2. If 3 3 4 0 M C v a d k            ,   2 2 2 32 2 2 2 0iRC M P iRC P K v a d k K P               , where 3, 4, 5i  and 6 0A  , 3 6 6 6 6 2 9 27 0A A B C   , 2 6 6 3 0A B  then the equilibrium point    * * * * * * 7 9 8 9 10 1 9 6, , , , , , , , , ,iRC iRC iRCEP S I P C V M P         asymptotically stable. Proof. From System (1) is obtained characteristic equation in Equation (6).            2 2 3 3 2 3 32 2 2 2 3 232 52 1 1 1 2 4 32 52 1 1 1 2 4 2 1 1 2 4 2 1 52 32 52 1 2 2 2 2 P K a d a d K P rr r a d d N N rr r a d d N N r a d d nd N rr r N N                                                                                                     1 1 2 4 2 1 2 52 4 2 1 0 a d d nd r d nd N                                               (6) 60 Biology, Medicine, & Natural Product Chemistry 11 (1), 2022: 55-63 SIMULATION In this section will be discussed about the numerical simulation and medical interpretation of the mathematical model of cervical cancer in the presence of radiotherapy and chemotherapy effects are divided into 2 cases. 1. The influence of radiotherapy 2. The influence of radiotherapy and followed by chemotherapy Simulation of radiotherapy effect The parameter values for this case are given in the following Table 2. Table 2. Values of parameters for cases of radiotherapy effects. Symbol Value Reference r 0.0255 Asih et al. (2015) N 9900.99 Asih et al. (2015)  0.00001 Asih et al. (2015) 1  0.004 Liu and Yang (2014)  0.05 Liu and Yang (2014) 1 a 100 Asih et al. (2015) 1 d 100.01 Asih et al. (2015)  0.0082 Asih et al. (2015) 2  0.03 Liu and Yang (2014) 2 a 101 Asih et al. (2015) 2 d 100 Asih et al. (2015)  20.03 Asih et al. (2015) K 2 Asih et al. (2015) 3a 0.03 Asih et al. (2015) 3 d 101.01 Asih et al. (2015) n 1 Asih et al. (2015) 4 d 500 Asih et al. (2015) Figure 2 shows the first 30 days or the first month after radiotherapy is showed a decrease in the number of cells from 13 cell/mm3 to 12.78 cell/mm3. This means that the decrease in cell number is 0.22 cell/mm3. At six months and one year of radiotherapy if it continues to show a successive decrease to 11.8 cell/mm3 and 10.73 cell/mm3. This shows the ill effects of radiotherapy because of healthy cells. For infected cells in the first and third months after radiotherapy successively from 13 cell/mm3 to 4,156 cell/mm3 and 0.4364 cell/mm3. This is good for curing cancer, because the infected cells will eventually become pre-cancerous cells. While the cancer cells on the fourth day after radiotherapy immediately fell rapidly, from 13 cell/mm3 to 0.3526 cell/mm3. For pre-cancerous cells almost the same as cancer cells, that is down very quickly. For the virus goes down quickly, because the infected cell goes down as well. This is because the virus replicates itself from infected cells. Murtono et al. – A Mathematical Model of Cervical Cancer Treatment by … 61 Figure 2. Trajectory diagram the influence of radiotherapy. Simulation of the Influence of Radiotherapy and Followed by Chemotherapy The parameter values for this case are given in the following Table 3. Table 3. Value of case parameters the influence of radiotherapy and followed by chemotherapy. Symbol Value Reference r 0.0255 Asih et al. (2015) N 9900.99 Asih et al. (2015)  0.00001 Asih et al. (2015) 1  0.004 Liu and Yang (2014)  0.05 Liu and Yang (2014) 1 a 100 Asih et al. (2015) 1 d 100.01 Asih et al. (2015)  0.0082 Asih et al. (2015) 2  0.03 Liu and Yang (2014) 2 a 101 Asih et al. (2015) 2 d 100 Asih et al. (2015) Symbol Value Reference  20.03 Asih et al. (2015) K 2 Asih et al. (2015) 3a 0.03 Asih et al. (2015) 3 d 101.01 Asih et al. (2015) n 1 Asih et al. (2015) 4 d 500 Asih et al. (2015) S k 0.01 Pillis et al. (2007) I k 0.01 Pillis et al. (2007) P k 0.01 Pillis et al. (2007) C k 0.01 Pillis et al. (2007)  0.1 Pillis et al. (2007) M v 0.01 Pillis et al. (2007) Figure 3 shows a trajectory with the effects of radiotherapy and followed chemotherapy, which is an advanced patient of cervical cancer. Normal cells during the first 30 days showed a rapid decline, from 13 cell/mm3 to 3,638 cell/mm3. This makes patients with 62 Biology, Medicine, & Natural Product Chemistry 11 (1), 2022: 55-63 advanced cervical cancer in this condition will get worse. Therefore, in this study advised not to use this treatment. Cancer cells also drop very rapidly, in the first 4 days of radiotherapy and chemotherapy, from 13 cell/mm3 to 0.2061 cell/mm3. Figure 3. Trajectory diagram the influence of radiotherapy and followed by chemotherapy. CONCLUSION Cervical cancer is one type of cancer and second highest incidence of cancer in women compared to other cancers. Treatment of cervical cancer with radiotherapy at an early stage is quite effective. This mathematical model showed that treatment of cervical cancer at an advanced stage with radiotherapy and followed by chemotherapy is not effective. Because normal cells die very quickly, and will aggravate the patient's condition of cervical cancer. This research needs to be continued with other more effective treatments, especially for advanced stages of cervical cancer. Conflict of Interest: The authors declare that there is no conflict of interest. REFERENCES Asih T. S. N., Suzanne Lenhart, Steven Wise, Lina Aryati, F. Adi- Kusumo, Mardiah S. Hardianti, Jonathan Forde, (2015), The Dynamics of HPV Infection and Cervical Cancer Cells, Bull Math Biol, DOI 10.1007/s11538-015-0124-2. Globocan, (2012), Download February 5, 2018. http://globocan.iarc.fr/old/FactSheets/cancers/cervix-new.asp Hoffman B., Schorge J., Schaffer J., Halvorson L., Bradshaw K., Cunningham F., (2008), Williams Gynecology, Second Edition, MC GrawHill’s, USA. Kompas.com, July 10, (2017). 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