Rotational Effect on Waves Propagation Modeling in a Poroelastic Bone

In this work, the dynamic behavior of a wet long bone that has been modeled as transversely isotropic hollow cylinder of crystal class 6 subjected to rotation is investigated. The solution wave for the wave propagation problem is expressed in term of a potential function which satisfies an eight-order partial differential equation, whose solutions lead to the derivation of the explicit solution of the wave equation. The mechanical boundary conditions corresponds to those of stress free lateral surface, while the fluid boundary condition correspond to of fluid stress free lateral surfaces. The satisfaction of the boundary conditions leads to the dispersion relation which is solved numerically. These frequencies are computed for poroelastic bone in terms of several values of the rotation and bone porosity.The results can benefit the theoretical development of orthopedic study projects connected to cylindrical poroelastic long bones. A comparison was made between the theoretical findings and the in vitro experimental values reported by a previously developed noncontacting device.

Keywords: Poroelastic Medium; Rotation; Wave Propagation; Wet Bone; Porous Media; Natural

The tissues of the bones are categorized into two types. The first type is the cancellous bone whose volume fraction of solid is low (less than 70%), whereas the second type is known as the cortical bone that has more than 70% solid field. Cancellous bone consists of two components, namely fatty marrow in the pores and calcified bone matrix. The study of the propagation of waves over a continuous medium is important practically in bio-engineering, medicine, and engineering. Moreover, applying poroelastic materials in some fields of medicine, e.g., orthopedics, dentistry, and cardiovascular medicine are common. In orthopedics, the propagation of waves through bone shelpsmonitor the rate of fracture curing. Theoretical wave propagation problems in wet bone are also considered [1-4]. Biot [5] developed the fundamental relation and consolidation for poroelastic media. The authors of [6] explored the stresses in human long bones. A micro-level 3D computational study of Wolff's law via the remodeling of trabecular bone in the human proximal femur using design space topology optimization was investigated by Boyle and Kim [7]. Elastic wave propagation for the surface bone of alveolar bone remodeling was studied by Mengoni and Ponthot [8]. Internal remodeling of poroelastic bone was investigated by Papathanasopoulou et al. [9]. The surface of the remodeling of bone under electromagnetic loads was studied by Qu et al. [10]. The surface waves of the remodeling of bone show several dynamic responses in disuse and overload,as studied by Hazelwood et al. [11]. Additionally, the authors of [12] examined the mechanical adaptation of trabecular bone combining cellular accommodation and the impacts of micro damage and disuse. Thenumerical calculations ofthe remodeling of bones sensitive to harmonic load over a poroelastic performance were studied by Malachanne et al. [13]. General analysis of mathematical models of the remodeling of bones was investigated by Zumsande et al. [14]. Ramtani and He [15] investigated the surface of wet of the remodeling of bone driven by metallic pin fitted into the medulla of a long bone. Ganghoffer [20] studied the surface growth's mechanics and thermodynamics. The authors of [16] explored the surface of the remodeling of bones of diaphysial surfaces under frequency load. Tsili [27] obtained the phase velocity for the internal remodeling of bones of diaphyseal shafts using Lame’s potential.Jang and Kim [18] discussed the numerical simulations of simultaneous cortical and trabecular bone changesin the human proximal femur during the remodeling of bones. Martínez et al. [19] examined the propagation of waves in wet bone in damage mechanics and boundary elements. The authors of [20] solved analytically the thermo-electro-elastic for the remodeling of surface bones subjected to axial and transverse loads. Kameo et al. [21] explored the functional adaptation of trabeculae predicted by the remodeling of bones subjected to loading frequency. The dynamic behavior method such as wave propagation and vibration of bone is necessary in measuring in vivo properties of bone by the above non-invasive method [22-29].

In the present paper, we explore the propagation of waves in poroelastic bone subjected to a rotation. The analytical solution for the propagation of waves in a wet bone subjected to a rotation was obtained, which satisfies the fundamental equation whose solutions lead to the derivation of the frequency equations for certain boundary circumstances. The numerical solution of the equations of frequency was obtained using the bisection method.The numerical findings showed the applicability of the proposed solutions and the impact of the rotation on the surface of the bones. The coefficients of the poroelastic bone were concluded for the different values of the rotation and the porosity of bones.

The system of solid plus fluid is assumed to be a system with conservation properties. The solid part is considered to have compressibility and shearing rigidity and the fluid to be compressible. The deformation of a unit cube is assumed is assumed to be completely reversible. The system geometry is defined by providing cylindrical coordinates r,θ,z, as shown in Figure 1.

Modelling on the concept of the Biot [5] the constitutive equations for a transversely isotropic case with z as the axis of the symmetry are taken in polar coordinates as

The σij (i,j=1,2,3) and σ are the average stresses of solid and fluid, respectively , with elastic constants C_ij, M, Q, R and c₆₆=(c₁₁c₁₂)/2

The equation of motion of the flow is

where is a Laplacian operator in polar coordinates, μ is the viscosity, f is the porosity and krr, kzz are the permeability of the medium. The average displacements of solid and velocity of fluid phases are taken as ui and vi respectively.The strains are expressed as

and dilation of the phases as e =u_i,j, and ε= v_i,i

For a material of crystal class 6 the equations of motion in cylindrical coordinates are given as

Consider a homogeneous, transversely isotropic, infinite hollow poroelastic cylinder with inner and outer radii a and b respectively, having a thickness h whose axis is in the direction of the z-axis.

where u_r,u₀,u_z,v_r,v₀,v_z are mechanical displacements and velocities, k is the wavenumber, p is the frequency and h is the thickness of the cylinder h = b − a , Ω is the rotation and φ,ψ ,w,η are secular potentials and it functions of r andθ .

Substituting (1),(3),into the equations(2),(4),and using (5), the following equations are obtained.

By defining the dimensionless, coordinate x =r/h and ε = kh the above equations are written in dimensionless parameter x and ε as

The reason for ξ being defined as above and not been solved for the variation is that the flow of fluid through the boundaries of bone does not take place during the study of the propagation of waves. However can be calculated if the flow of the boundaries is prescribed.

Writing an equation (7) in the determinants from

Evaluating the determinant form, the following equations are obtained

Using Eq. (10), the proposed solutions of equation(5) can be written as

Where, α_i² are the non-zero roots of the equation

Where, d_i and e_i are given by

Solving equation (7a) we have

Where

For traction-free boundary conditions in the present study, stresses must vanish on the inner and outer surfaces of the hollow cylinder, i.e.,

where a^′=a/h and b^′=b/h

Using equations (5),(12),(15) in (1), we obtain frequency equation in the form

The expressions of a_ij coefficients are given in Appendix A.

We calculated the numerical findings of the frequency equation considering the Matlab package for the hydrated bone. The frequency equation has an infinite number of roots because it is transcendental. The frequencies (i.e., the roots of Eq. (20)), n = 0, the axisymmetric mode, and flexural n = 1, 2 modes are provided. The results are assessed in the range 0 < ε1 < 4 and 0 < ch < 4, with the ratio of b/a =3.0 The values of the elastic constant of the bone are derived from [8], and the poroelastic constant is assessed using the expression given by

The expressions of γ, δ, χ are given by

where, c equals zero for the incompressibility of the fluid

The porosity of the bones of the people aged 35-40 years equals 0.24. To assess one more poroelastic constant,M/Q ≅ c₁₂/c₁₃ as the value M is not provided. Because the fluid is isotropic, brr = bzz. The density of the fluid in the porospace, the permeability of the medium, and mass density of the bone can be assessed. For different frequency values, the wave numbers, the wave velocity, and the attenuation coefficients are derived from the frequency equation (Table 1).

Figure 1: shows that the schematic of the problem.

Figure 2: shows that the variation of thesecular determinant ,|a_ij| , wave velocity Re(|a_ij| )and attenuation coefficient Im(|a_ij| ) with respect to thickness h for different values of wave number k. It is observed that the,secular determinant, wave velocity and attenuation coefficient increase with increases of the wave number, while it increases with increasing of the thickness in the interval 0 ≤ h ≤ 0.08, as well as there is no significant variation on the secular determinant, wave velocity and attenuation coefficient in the interval 0.08 ≤ h ≤ 1.

Figure 3: shows that the variation of the secular determinant |a_ij| , wave velocity Re(|a_ij| ) and attenuation coefficient Im(|a_ij| ) with respect to thickness h for different values of porosity f. It is obvious that the secular determinant, wave velocity and attenuation coefficient increase with increases of the porosity, while it increases with increasing of the thickness in the interval 0 ≤ h ≤ 0.08, as well, there is no significant variation on the secular determinant,wave velocity and attenuation coefficient in the interval 0.08 ≤ h ≤ 1.

Figure 4: shows that the variation of the secular determinant |a_ij| , wave velocity Re(|a_ij| ) and attenuation coefficient Im(|a_ij| ) with respect to thickness for different values of rotation while the attenuation coefficient increases with increasing of rotation. It is observed that the secular determinant and wave velocity increase with increases of the rotation, while it increases with increasing of the thickness in the interval 0 ≤ h ≤ 0.08, as well, there is no significant variation on the secular determinant, wave velocity and attenuation coefficient in the interval 0.08 ≤ h ≤ 1. This behavior was projected by other models [28,29] and experimental studies [30,31].

The case of the complex conjugate roots has not been handled. These cases possibly happen for specific combinations of bone property values. Consequently, much more complex computation is required since for each step both the real and imaginary parts of the determinant components must be calculated. We handled those cases successfully (Figures 2,3 and 4).

After examining the propagation of waves in hollow poroelastic bone with a circular cylindrical cavity of infinite extent, the authors conclude that

1. Bones are heterogeneous and anisotropic. The solid part is perfectly elastic and the fluid part is Newtonian viscous and compressible. The pores are interrelated and the flow of fluid resulting from bone deformation is controlled by Darcy’s law.

2. We explored the propagation of waves in an infinite hollow rotating cylinder of crystal class 6. We did the analysis, and the solution to the problem was described concerning the potential function. The resulting frequency equation was solved numerically.

3. The numerical results illustrated that, except for the mechanical conditions, the rotation, wave number and the porosity can influence the propagation of waves on the bone. This feature can be taken into account and employed in controlling the healing process of the injured bones. All results are obtained based on the numerical model that may differ from those of the individual bone materials. Thus, more experimental validation is essential before using the present results for the clinical practice.

4. We made observations of the effect of the rotation, wave number and porosity in wave propagation in wet bone’s surface.

5. In short, we can have a theoretical simulation of the in vivo case by taking thme properties of the muscle and the skin that cover bones in the limbs