Sains Malaysiana 44(8)(2015): 1153–1157

Modelling the Cervical Cancer Growth Process by Stochastic Delay Differential Equations

(Pemodelan Proses Pertumbuhan Kanser Serviks oleh Persamaan Pembezaan Stokastik Lengahan)

MAZMA SYAHIDATUL AYUNI MAZLAN¹, NORHAYATI ROSLI¹*, NINA SUHAITY AZMI¹

 & ARIFAH BAHAR²

¹Faculty of Industrial Sciences and Technology, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300 Gambang, Pahang Darul Makmur, Malaysia

²UTM Centre for Industrial & Applied Mathematics (UTM-CIAM), Universiti Teknologi Malaysia

81310 Johor Bahru, Johor Darul Takzim Malaysia

Received: 28 November 2014/Accepted: 11 March 2015

ABSTRACT

In this paper, the uncontrolled environmental factors are perturbed into the growth rate deceleration factor of the Gompertzian deterministic model. The growth process under Gompertz’s law is considered, thus lead to stochastic differential equations of Gompertzian with time delay. The Gompertzian deterministic model has proven to fit well with the clinical data of cancerous growth, however the performance of stochastic model towards clinical data is yet to be confirmed. The prediction quality of stochastic model is evaluated by comparing the simulated results with the clinical data of cervical cancer growth. The parameter estimation of stochastic models is computed by using simulated maximum likelihood method. 4-stage stochastic Runge-Kutta is applied to simulate the solution of stochastic model. Low values of root mean-square error (RMSE) of Gompertzian model with random effect indicate good fits.

Keywords: Gompertzian model; simulated maximum likelihood; stochastic delay differential equation; 4-stage stochastic Runge Kutta

ABSTRAK

Kertas kerja ini mempertimbangkan faktor persekitaran tidak terkawal yang diganggu ke atas faktor penurunan kadar pertumbuhan model Gompertzian berketentuan. Proses pertumbuhan di bawah hukum Gompertz dipertimbangkan, seterusnya membawa kepada persamaan pembezaan stokastik Gompertzian dengan masa lengahan. Model Gompertzian berketentuan telah terbukti sesuai untuk data klinikal pertumbuhan kanser, walau bagaimanapun keberkesanan model stokastik terhadap data klinikal masih belum disahkan. Kualiti ramalan model stokastik dinilai dengan membandingkan keputusan simulasi dengan data klinikal pertumbuhan kanser pangkal rahim. Anggaran parameter model stokastik dikira dengan menggunakan kaedah simulasi kebolehjadian maksimum. Stokastik Runge-Kutta peringkat 4 diaplikasikan untuk mencari penyelesaian simulasi model stokastik. Nilai punca min ralat kuasa dua (RMSE) yang rendah bagi model Gompertzian dengan kesan rawak menunjukkan keputusan yang baik.

Kata kunci: Model Gompertzian; persamaan pembezaan stokastik dengan masa lengahan; simulasi kebolehjadian maksimum; stokastik Runge Kutta peringkat 4

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*Corresponding author; email: norhayati@ump.edu.my