Multiscale modeling of cancellous bone considering full coupling of mechanical, electric and magnetic effects

Abstract

Modeling of cancellous bone has important applications in the detection and treatment of fatigue fractures and diseases like osteoporosis. In this paper, we present a fully coupled multiscale approach considering mechanical, electric and magnetic effects by using the multiscale finite element method and a two-phase material model on the microscale. We show numerical results for both scales, including calculations for a femur bone, comparing a healthy bone to ones affected by different stages of osteoporosis. Here, the magnetic field strength resulting from a small mechanical impact decreases drastically for later stages of the disease, confirming experimental research.

Keywords: Cancellous bone; Coupled problems; Maxwell equations; Multiscale modeling; Wave propagation.

Fig. 1

Bone phases depending on osteoporosis stage (cf. Laboratoires 2019) and corresponding RVEs

Fig. 2

Transition between macro- and microscale. State variables enter as boundary conditions of the RVE problem. Flux quantities at the macroscale are calculated by averaging the RVE quantities

Fig. 3

Program flow of the multiscale simulations

Fig. 4

Periodic RVE with cortical bone phase (gray) and bone marrow phase (transparent red) and lengths parameters

Fig. 5

Microscale simulation results of a coarse and fine mesh (left and right respectively) for all flux quantities. Top left: mechanical stress ${\sigma }_{\mathit{xy}}[\text{GPa}]$, top right: mechanical stress ${\sigma }_{\mathit{xy}}[\text{GPa}]$ in the xz-plane with $y=0$, bottom left: magnitude of the electric displacement field $\mathbf{D}[\text{As}/{\text{m}}^2]$, bottom right: magnitude of the magnetic field strength $\mathbf{H}[\text{A/m}]$

Fig. 6

Used meshes for the complete RVE (left) and only the cortical bone phase (right). a coarse hexahedron mesh, b fine hexahedron mesh, c coarse tetrahedron mesh, d fine tetrahedron mesh

Fig. 7

Effective Young’s modulus $E_{\mathrm{eff}}$ against cortical bone volume fraction ${\rho }_{\mathrm{b}}$ for different RVEs

Fig. 8

Cylinder mesh and displacement boundary conditions (red: all directions restricted, orange: only the x-direction restricted, blue-gray: no directions restricted)

Fig. 9

Cylinder front in the xy-plane for $z=0$ with grounded nodes in red

Fig. 10

Amplitude of the displacement function a against the time step t

Fig. 11

Magnitude of the average electric displacement field $\mathbf{D}[\text{As}/{\text{m}}^2]$, plotted against the time t

Fig. 12

Magnitude of the average magnetic field strength $\mathbf{H}[\text{A}/\text{m}]$, plotted against the time t

Fig. 13

Simulation results for RVE 1 (top) to 6 (bottom): stress ${\sigma }_{\mathit{xy}}[\text{GPa}]$, $t=25$

Fig. 14

Simulation results for RVE 1 (top) to 6 (bottom): magnitude of the electric displacement field $\mathbf{D}[\text{As}/{\text{m}}^2]$, $t=25$

Fig. 15

Simulation results for RVE 1 (top) to 6 (bottom): magnitude of the magnetic field strength $\mathbf{H}[\text{A}/\text{m}]$, $t=50$

Fig. 16

Simulation results for RVE 1 (top) to 6 (bottom): magnitude of the electric current density $\mathbf{J}[\text{A}/{\text{m}}^2]$, $t=50$

Fig. 17

Simulation results for RVE 1 for the magnetic field strength $\mathbf{H}[\text{A}/\text{m}]$ with ${\kappa }_1=1·{10}^2\text{S}/\text{m}$ (top), ${\kappa }_1=1·{10}^4\text{S}/\text{m}$ (in the middle) and ${\kappa }_1=1·{10}^6\text{S}/\text{m}$ (bottom), $t=50$

Fig. 18

Simulation results for RVE 1 for the electric current density $\mathbf{J}[\text{A}/{\text{m}}^2]$ with ${\kappa }_1=1·{10}^2\text{S}/\text{m}$ (top), ${\kappa }_1=1·{10}^4\text{S}/\text{m}$ (in the middle) and ${\kappa }_1=1·{10}^6\text{S}/\text{m}$ (bottom), $t=50$

Fig. 19

Anisotropic RVE with cortical bone phase (gray) and bone marrow phase (transparent red) and lengths parameters

Fig. 20

Simulation results for RVE 6 (top) and the anisotropic RVE (bottom) for the stress ${\sigma }_{\mathit{xy}}[\text{GPa}]$, $t=25$

Fig. 21

Simulation results for RVE 6 (top) and the anisotropic RVE (bottom) for the magnetic field strength $\mathbf{H}[\text{A}/\text{m}]$, $t=50$

Fig. 22

Femur bone mesh and displacement boundary conditions (red: all directions restricted, orange: only the x-direction restricted, blue-gray: no directions restricted)

Fig. 23

Femur bone front with grounded nodes in red

Fig. 24

Simulation results for RVE 1 (top) to 6 (bottom): stress ${\sigma }_{\mathit{xy}}[\text{GPa}]$, $t=25$

Fig. 25

Simulation results for RVE 1 (top) to 6 (bottom): magnitude of the electric displacement field $\mathbf{D}[\text{As}/{\text{m}}^2]$, $t=25$

Fig. 26

Simulation results for RVE 1 (top) to 6 (bottom): magnitude of the magnetic field strength $\mathbf{H}[\text{A}/\text{m}]$, $t=50$

Fig. 27

Simulation results for RVE 1 (top) to 6 (bottom): magnitude of the electric current density $\mathbf{J}[\text{A}/{\text{m}}^2]$, $t=50$

Fig. 28

Simulation results for RVE 1 (top) to 6 (bottom): magnitude of the magnetic field strength $\mathbf{H}[\text{A}/\text{m}]$, slice, $t=50$

Fig. 29

Simulation results for RVE 1 (top) to 6 (bottom): magnitude of the electric current density $\mathbf{J}[\text{A}/{\text{m}}^2]$, slice, $t=50$

Fig. 30

Average magnetic field strength for the different RVEs at $t=50$

Fig. 31

Maximum magnetic field strength for the different RVEs at $t=50$