Open Access

ARTICLE

Fractional Order Nonlinear Bone Remodeling Dynamics Using the Supervised Neural Network

Narongsak Yotha¹, Qusain Hiader², Zulqurnain Sabir³, Muhammad Asif Zahoor Raja⁴, Salem Ben Said⁵, Qasem Al-Mdallal⁵, Thongchai Botmart⁶, Wajaree Weera^6,*

1 Department of Applied Mathematics and Statistics, Rajamangala University of Technology Isan, Nakhon Ratchasima, 30000, Thailand
2 Department of Mathematics, University of Gujrat, Gujrat, 50700, Pakistan
3 Department of Mathematics, Hazara University, Mansehra, 21120, Pakistan
4 Future Technology Research Center, National Yunlin University of Science and Technology, Douliou, Yunlin, 64002, Taiwan
5 Department of Mathematical Science, College of Science, United Arab Emirates University, Al Ain, Abu Dhabi, UAE
6 Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand

* Corresponding Author: Wajaree Weera. Email:

Computers, Materials & Continua 2023, 74(2), 2415-2430. https://doi.org/10.32604/cmc.2023.031352

Received 15 April 2022; Accepted 17 May 2022; Issue published 31 October 2022

Abstract

This study aims to solve the nonlinear fractional-order mathematical model (FOMM) by using the normal and dysregulated bone remodeling of the myeloma bone disease (MBD). For the more precise performance of the model, fractional-order derivatives have been used to solve the disease model numerically. The FOMM is preliminarily designed to focus on the critical interactions between bone resorption or osteoclasts (OC) and bone formation or osteoblasts (OB). The connections of OC and OB are represented by a nonlinear differential system based on the cellular components, which depict stable fluctuation in the usual bone case and unstable fluctuation through the MBD. Untreated myeloma causes by increasing the OC and reducing the osteoblasts, resulting in net bone waste the tumor growth. The solutions of the FOMM will be provided by using the stochastic framework based on the Levenberg-Marquardt backpropagation (LVMBP) neural networks (NN), i.e., LVMBPNN. The mathematical performances of three variations of the fractional-order derivative based on the nonlinear disease model using the LVMPNN. The static structural performances are 82% for investigation and 9% for both learning and certification. The performances of the LVMBPNN are authenticated by using the results of the Adams-Bashforth-Moulton mechanism. To accomplish the capability, steadiness, accuracy, and ability of the LVMBPNN, the performances of the error histograms (EHs), mean square error (MSE), recurrence, and state transitions (STs) will be provided.

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Keywords

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