Stochastic model of cancer growth with the effect of glycosaminoglycans (GAGs) as anticancer therapeutics

Ordinary differential equations (ODEs) and stochastic differential equations (SDEs) have been widely used to describe the biological process of cancer growth. In natural phenomena, the untreated cancer growth is influenced by random effects as the result of uncontrolled factors in the human body. In addition, the dynamical behaviour of cancer cell growth rate depends on not only its structure at the present time, but also on its structure at a previous time. Thus, ODEs and SDEs are not capable to model the uncontrolled fluctuation and delay feedback for the untreated cancer growth process. It is necessary to model the untreated cancer growth process via stochastic delay differential equations (SDDEs). Nowadays, cancers are traditionally treated with surgery, chemotherapy, and radiotherapy. However, the most advanced treatment for cancer is targeted therapies. In this research, the drug that is used is Glycosaminoglycans (GAGs). The content of GAGs indicates the presence of Chondroitin Sulphate (CS). The effects of CS on the processes related to biological systems is crucial to promote apoptosis. For this research, CS is extracted from Blue-spotted stingrays, taken from Wild Life Handler Resource, Selangor. For the treated cancer growth, CS can be employed for targeted cancer therapy. The untreated cervical and breast cancer data for this research is taken from Hospital Sultanah Nur Zahirah (HSNZ), Kuala Terengganu. Laboratory experiments of CS for treated cancer cell is done in International Islamic University Malaysia (IIUM) on HeLa (cervical cancer) and MCF-7 (breast cancer) cell lines. The experimental treated data is used to validate the stochastic model. CS has significantly reduced cell viability of Hela and MCF-7 cell lines. Futhermore, the mRNA gene expression of HeLa and MCF-7 cells are studied using the RT-qPCR technique. The apoptotic gene, activation caspase3 is only presented in HeLa cell line. Understanding the quantitative dynamics of the protective anticancer, CS to cancer cell proliferation is required in designing an effective treatment. Mathematical model can be used as a tool in promoting knowledge about the effects of CS in cancer growth. The cell growth of cancer is influenced by uncontrolled factors, which is referred to as noise. To date, no deterministic or stochastic models have been formulated to represent the growth of cancer affected by CS. Thus, this research is carried out to formulate the deterministic and stochastic models for cancerous growth affected by CS as anticancer therapeutics. A new stochastic system for the treated cancer growth affected by anticancer therapeutics of CS is formulated via SDEs. The kinetic parameters is estimated via non-parametric stimulated maximum likelihood function. Numerical method of 4-stage stochastic Runge-Kutta (SRK4) is employed to simulate the solution. The algorithms of simulating the numerical solution are then developed. Numerical solution of stochastic model is adequately described the effects of CS on HeLa and MCF-7 cell lines compare than its deterministic counterpart. The newly developed of stochastic model is expected to be appropriate for other types of cancer cells as well. The findings of this research may help physicians and biologists planning better strategies for treatment of cancer

Effects of Chondroitin Sulfate (CS) on (HeLa) Cervical Cancer and Breast Cancer (MCF-7) cell lines

Chondroitin Sulfate (CS) is the main constituent of Blue-spotted Stingray which shows promising in vitro anticancer activities in cancer cell lines. However, the effects of CS on human breast cancer and cervical cancer cell lines remain to be explored. Here we report that CS induced different degree of cytotoxicity in two human cancer cell lines, cervical cancer HeLa and breast cancer MCF-7 cell lines. We found that MCF-7 was more resistant to CS exposure than HeLa cell line. Moreover, CS induced apoptosis in HeLa but not MCF-7 cell line as shown by caspase-3 activity assay. The CS-induced caspase-3 activation in HeLa cells was also confirmed by using quantititative RT-PCR. Our findings show that the caspase-3 activation induced by CS in HeLa cells was transcriptional. These results indicate that as an anticancer candidate, CS is more potent on cervical cancer than the breast cancer cell line

Modelling the Cervical Cancer Growth Process by Stochastic Delay Differential Equations

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

Modelling the cancer growth process by Stochastic Differential Equations with the effect of Chondroitin Sulfate (CS) as anticancer therapeutics

A stochastic model is introduced to describe the growth of cancer affected by anti- cancer therapeutics of Chondroitin Sulfate (CS). The parameters values of the stochastic model are estimated via maximum likelihood function. The numerical method of Euler-Maruyama will be employed to solve the model numerically. The efficiency of the stochastic model is measured by comparing the simulated result with the experimental data

Performance of 5-stage, 4-stage and specific stochastic Runge-Kutta methods in approximating the solution of stochastic biological model

In recent years, the transition on modelling physical systems via stochastic differential equations (SDEs) has attracted great interest among researchers. This is due to the limitations of ordinary differential equations in presenting the real phenomenon. To the fact that the stochastic models incorporate the random effects that may influence the behaviour of physical systems, SDEs seems to be the best model that can be used i n assessing those systems. The growing interest among researchers in modelling the systems via SDEs comes with the rise in the need of numerical methods to approximate the solutions for SDEs. This is because by taking into account the random fluctuations in SDEs resulting to the complexity of finding the exact solution of SDEs. Therefore, it contribute to the increasing number of research to decide on the best numerical approach to solve the systems of SDEs. This paper is devoted to investigate the performance of 5-stage stochastic Runge-Kutta ( SRK5) with order 2.0, 4-stage stochastic Runge-Kutta ( SRK4), specific stochastic Runge-Kutta with order 1.5 ( SRKS1.5) and commutative specific stochastic Runge-Kutta with order 1.5 (SRKST2) in approximating the solution of stochastic model in biological system. A comparative study of SRK5, SRK4, SRKS1.5 and SRKST2 methods will be presented in this paper. The linear SDE model and the stochastic model of C. Acetobutylicum cell growth will be used to examine the performance of those methods and the numerical experiment will be conducted. The numerical solutions obtained will be discussed

A Gompertzian Model With Random Effects To Cervical Cancer Growth

In this paper, a Gompertzian model with random effects is introduced to describe the cervical cancer growth. The parameters values of the mathematical model are estimated via maximum likehood estimation. We apply 4-stage Runge- Kutta (SRK4) for solving the stochastic model numerically. The efficiency of mathematical model is measured by comparing the simulated result and the clinical data of the cervical cancer growth. Low values of root mean-square error (RMSE) of Gompertzian model with random effect indicate good fits

Stochastic Gompertzian model for breast cancer growth process

In this paper, a stochastic Gompertzian model is developed to describe the growth process of a breast cancer by incorporating the noisy behavior into a deterministic Gompertzian model. The prediction quality of the stochastic Gompertzian model is measured by comparing the simulated result with the clinical data of breast cancer growth. The kinetic parameters of the model are estimated via maximum likelihood procedure. 4-stage stochastic Runge-Kutta (SRK4) is used to simulate the sample path of the model. Low values of mean-square error (MSE) of stochastic model indicate good fits. It is shown that the stochastic Gompertzian model is adequate in explaining the breast cancer growth process compared to the deterministic model counterpar

Modelling The Cancer Growth Process By Stochastic Delay Diffferential Equations Under Verhults And Gompertz's Law

In this paper, the uncontrolled environmental factors are perturbed into the intrinsic growth rate factor of deterministic equations of the growth process. The growth process under two different laws which are Verhults and Gompertz’s law are considered, thus leading to stochastic delay differential equations (SDDEs) of logistic and Gompertzian, respectively. Gompertzian deterministic model has been proved to fit well the clinical data of cancerous growth, however the performance of stochastic model towards clinical data is yet to be confirmed. The prediction quality of logistic and Gompertzian SDDEs are evaluating 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. We adopt 4-stage stochastic Runge-Kutta to simulate the solution of stochastic models

Gompertzian stochastic model with delay effect to cervical cancer growth

In this paper, a Gompertzian stochastic model with time delay is introduced to describe the cervical cancer growth. The parameters values of the mathematical model are estimated via Levenberg-Marquardt optimization method of non-linear least squares. We apply Milstein scheme for solving the stochastic model numerically. The efficiency of mathematical model is measured by comparing the simulated result and the clinical data of cervical cancer growth. Low values of Mean-Square Error (MSE) of Gompertzian stochastic model with delay effect indicate good fits

Modelling the cervical cancer growth process by stochastic delay differential equations

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