Fractal Operators and Fractional-Order Mechanics of Bone

Abstract

In recent years, the concept of physical fractal space has been abstracted from muscle/ligament fibers, nerve fibers and blood flows. In the physical fractal space, bio-fractal mechanics may be set up. Surprisingly, the concepts and methods developed in the above bio-fractal mechanics are of universalities, i.e., the physical fractal spaces, fractal operators and fractional-order mechanics universally exist in various biological systems such as bones. This paper will focus on the bones in which the physical fractal space is abstracted, the fractal operators are derived and the fractional-order mechanics is established.

Convolution Kernel Function and Its Invariance Properties of Bone Fractal Operators

“…The previous article studied the multi-level structure in bone and abstracted the physical fractal space from the compact bone fibers. From "physical fractal space", the fractal cells and fractal components have been abstracted, as shown in Figure 1 [12,15]. The constitutive response expression in physical fractal space is…”

“…For example, common biological composite materials (such as bones, tendons, and shells) use an integrated "strategy" of alternating softness and hardness, where protein and mineral components each play their respective functions, allowing these natural micro-nano composite structures to achieve good mechanical properties while also possessing various biological functions (such as volume growth) [7][8][9]. In recent years, we have abstracted the multi-level chainlike topology from muscle/ligament fibers [10], nerve fibers [11], and compact bone fibers [12,13] and set up a multi-level micro elastic cavity topology from arterial blood flows [14]. We have built the biological fractal and fractional-order mechanics based on physical components.…”

“…Based on the staggered arrangement pattern of hydroxyapatite in the collagen matrix, a spring-dashpot fractal network with a self-similar topology, named "fractal cell", is abstracted from the micro/nano-structure of chain-like fibers [10,11]. Based on operational calculus, we built a fractional-order constitutive model of compact bone [12].…”

“…Figure 1.Fractal operators and fractional-order mechanics of bone: (a) long bone (femur)[15]; (b) the bone tissue and mineralized bone fibers surrounding osteocytes ("OC")[15]; (c) the arrangement of hydroxyapatite crystals and collagen fibers in mineralized bone fibers[15]; (d) fractal cell abstraction from collagen fibrils[12].…”