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Title: Gompertzian stochastic model with delay effect to cervical cancer growth

Abstract

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.

Authors:
;  [1];  [2]
  1. Faculty of Industrial Sciences and Technology, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300 Gambang, Pahang (Malaysia)
  2. Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor and UTM Centre for Industrial and Applied Mathematics (UTM-CIAM), Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor (Malaysia)
Publication Date:
OSTI Identifier:
22390950
Resource Type:
Journal Article
Resource Relation:
Journal Name: AIP Conference Proceedings; Journal Volume: 1643; Journal Issue: 1; Conference: 2. ISM International Statistical Conference 2014: Empowering the Applications of Statistical and Mathematical Sciences, Pahang (Malaysia), 12-14 Aug 2014; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 60 APPLIED LIFE SCIENCES; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; EFFICIENCY; ERRORS; MATHEMATICAL MODELS; NEOPLASMS; NONLINEAR PROBLEMS; OPTIMIZATION; STOCHASTIC PROCESSES; TIME DELAY

Citation Formats

Mazlan, Mazma Syahidatul Ayuni binti, Rosli, Norhayati binti, and Bahar, Arifah. Gompertzian stochastic model with delay effect to cervical cancer growth. United States: N. p., 2015. Web. doi:10.1063/1.4907496.
Mazlan, Mazma Syahidatul Ayuni binti, Rosli, Norhayati binti, & Bahar, Arifah. Gompertzian stochastic model with delay effect to cervical cancer growth. United States. doi:10.1063/1.4907496.
Mazlan, Mazma Syahidatul Ayuni binti, Rosli, Norhayati binti, and Bahar, Arifah. Tue . "Gompertzian stochastic model with delay effect to cervical cancer growth". United States. doi:10.1063/1.4907496.
@article{osti_22390950,
title = {Gompertzian stochastic model with delay effect to cervical cancer growth},
author = {Mazlan, Mazma Syahidatul Ayuni binti and Rosli, Norhayati binti and Bahar, Arifah},
abstractNote = {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.},
doi = {10.1063/1.4907496},
journal = {AIP Conference Proceedings},
number = 1,
volume = 1643,
place = {United States},
year = {Tue Feb 03 00:00:00 EST 2015},
month = {Tue Feb 03 00:00:00 EST 2015}
}
  • Purpose: To develop mathematical models to predict the evolution of tumor geometry in cervical cancer undergoing radiation therapy. Methods: The authors develop two mathematical models to estimate tumor geometry change: a Markov model and an isomorphic shrinkage model. The Markov model describes tumor evolution by investigating the change in state (either tumor or nontumor) of voxels on the tumor surface. It assumes that the evolution follows a Markov process. Transition probabilities are obtained using maximum likelihood estimation and depend on the states of neighboring voxels. The isomorphic shrinkage model describes tumor shrinkage or growth in terms of layers of voxelsmore » on the tumor surface, instead of modeling individual voxels. The two proposed models were applied to data from 29 cervical cancer patients treated at Princess Margaret Cancer Centre and then compared to a constant volume approach. Model performance was measured using sensitivity and specificity. Results: The Markov model outperformed both the isomorphic shrinkage and constant volume models in terms of the trade-off between sensitivity (target coverage) and specificity (normal tissue sparing). Generally, the Markov model achieved a few percentage points in improvement in either sensitivity or specificity compared to the other models. The isomorphic shrinkage model was comparable to the Markov approach under certain parameter settings. Convex tumor shapes were easier to predict. Conclusions: By modeling tumor geometry change at the voxel level using a probabilistic model, improvements in target coverage and normal tissue sparing are possible. Our Markov model is flexible and has tunable parameters to adjust model performance to meet a range of criteria. Such a model may support the development of an adaptive paradigm for radiation therapy of cervical cancer.« less
  • Purpose: To develop mathematical models to predict the evolution of tumor geometry in cervical cancer undergoing radiation therapy. Methods: The authors develop two mathematical models to estimate tumor geometry change: a Markov model and an isomorphic shrinkage model. The Markov model describes tumor evolution by investigating the change in state (either tumor or nontumor) of voxels on the tumor surface. It assumes that the evolution follows a Markov process. Transition probabilities are obtained using maximum likelihood estimation and depend on the states of neighboring voxels. The isomorphic shrinkage model describes tumor shrinkage or growth in terms of layers of voxelsmore » on the tumor surface, instead of modeling individual voxels. The two proposed models were applied to data from 29 cervical cancer patients treated at Princess Margaret Cancer Centre and then compared to a constant volume approach. Model performance was measured using sensitivity and specificity. Results: The Markov model outperformed both the isomorphic shrinkage and constant volume models in terms of the trade-off between sensitivity (target coverage) and specificity (normal tissue sparing). Generally, the Markov model achieved a few percentage points in improvement in either sensitivity or specificity compared to the other models. The isomorphic shrinkage model was comparable to the Markov approach under certain parameter settings. Convex tumor shapes were easier to predict. Conclusions: By modeling tumor geometry change at the voxel level using a probabilistic model, improvements in target coverage and normal tissue sparing are possible. Our Markov model is flexible and has tunable parameters to adjust model performance to meet a range of criteria. Such a model may support the development of an adaptive paradigm for radiation therapy of cervical cancer.« less
  • 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.
  • Purpose: To estimate and compare intensity-modulated radiotherapy (IMRT) with three-dimensional conformal radiotherapy (3DCRT) in terms of second cancer risk (SCR) for postoperative treatment of endometrial and cervical cancer. Methods and Materials: To estimate SCR, the organ equivalent dose concept with a linear-exponential, a plateau, and a linear dose-response model was applied to dose distributions, calculated in a planning computed tomography scan of a 68-year-old woman. Three plans were computed: four-field 18-MV 3DCRT and nine-field IMRT with 6- and 18-MV photons. SCR was estimated as a function of target dose (50.4 Gy/28 fractions) in organs of interest according to the Internationalmore » Commission on Radiological Protection Results: Cumulative SCR relative to 3DCRT was +6% (3% for a plateau model, -4% for a linear model) for 6-MV IMRT and +26% (25%, 4%) for the 18-MV IMRT plan. For an organ within the primary beam, SCR was +12% (0%, -12%) for 6-MV and +5% (-2%, -7%) for 18-MV IMRT. 18-MV IMRT increased SCR 6-7 times for organs away from the primary beam relative to 3DCRT and 6-MV IMRT. Skin SCR increased by 22-37% for 6-MV and 50-69% for 18-MV IMRT inasmuch as a larger volume of skin was exposed. Conclusion: Cancer risk after IMRT for cervical and endometrial cancer is dependent on treatment energy. 6-MV pelvic IMRT represents a safe alternative with respect to SCR relative to 3DCRT, independently of the dose-response model. 18-MV IMRT produces second neutrons that modestly increase the SCR.« less
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