Abstract
This article focuses on modeling bone formation process using a fractional differential approach, named bones remodeling process. The first goal of the work is to investigate existence and uniqueness of the proposed fractional differential model. The next goal is to investigate how similar is the proposed approach to the method based on system classical differential equations. The dynamical system of equations used is built upon three main parameters. These are chemical substances, namely, calcitonin secretion, osteoclastic and osteoblastic, which are involved in the bone’s formation process. We implement some numerical simulations to graphically show the impact of an arbitrary fractional order of derivative. We finally obtained that modeling bone formation process using fractional differential equations yielded comparable results with those obtained through a system of classical differential equations. Flexibility in the choice of the fractional order of derivative is an advantage as it helps in selecting the best fractional order of derivative.
Highlights
Modeling natural phenomena through differential equation has long been used by scientists
Conclusion and Comments e goal of this paper was to build a system of differential equations for bone formation process based on various fractional differential equations
Letnikov and Caputo–Fabrizio fractional derivatives were used alongside classical differential equation method for comparison purposes. e work does not follow traditional quantitative modeling approach in which historical data exist, various models including a proposed model are used to fit the data; and their error rates are compared to pick the best model
Summary
Modeling natural phenomena through differential equation has long been used by scientists. In earlier days of fractional differential equations, researchers mainly focused on theoretical concepts, investigating existence and uniqueness of solution of built models. Researchers build models based on fractional differential by analogy to the approach that would be used in the case of classical differential [6,7,8]. Following the aims to prove the use of fractional differential equations, we decided to investigate how efficient it will be in modeling bone formation process. We build a system of fractional differential equations to modeling bone formation, which we call the bone remodeling process. Detailed information on the biological and chemical processes involved in the study as well as the chemical elements involved in bone formation are found in [12,13,14,15,16,17] and references therein. In the setting of this work, there is no comprehensive way to compare performances of the proposed method with those of existing methods
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