Three-Dimensional Finite Element Models of the Human Pubic Symphysis with Viscohyperelastic Soft Tissues

Three-dimensional finite element (FE) models of human pubic symphyses were constructed from computed tomography image data of one male and one female cadaver pelvis. The pubic bones, interpubic fibrocartilaginous disc and four pubic ligaments were segmented semi-automatically and meshed with hexahedral elements using automatic mesh generation schemes. A two-term viscoelastic Prony series, determined by curve fitting results of compressive creep experiments, was used to model the rate-dependent effects of the interpubic disc and the pubic ligaments. Three-parameter Mooney-Rivlin material coefficients were calculated for the discs using a heuristic FE approach based on average experimental joint compression data. Similarly, a transversely isotropic hyperelastic material model was applied to the ligaments to capture average tensile responses. Linear elastic isotropic properties were assigned to bone. The applicability of the resulting models was tested in bending simulations in four directions and in tensile tests of varying load rates. The model-predicted results correlated reasonably with the joint bending stiffnesses and rate-dependent tensile responses measured in experiments, supporting the validity of the estimated material coefficients and overall modeling approach. This study represents an important and necessary step in the eventual development of biofidelic pelvis models to investigate symphysis response under high-energy impact conditions, such as motor vehicle collisions.

APPENDIX

Implementation of Material Model for the Interpubic Disc

The material model (Type 77) was used to simulate the three-parameter Mooney-Rivlin model for the interpubic disc with a viscoelastic option. The physical meaning of the constants in the card can be found in LS-DYNA Keywords User's Manual (Version 970). $G = 2(C_{10} + C_{01} ) = 0.5\hbox{\it MPa}$, $G_i = \alpha _i G_{}$ and $\beta _i = {1 \mathord{\left/ {\vphantom {1 {\tau _i }}} \right. \kern-\nulldelimiterspace} {\tau _i }}$. Card values input in the material model are provided in the Tables below:

Card 1

 

Variable	MID	RO (kg/m3)	PR	N	NV	G (MPa)
Value	3	1200	0.495	0	2	0.5

Card 2 for material constants of interpubic disc (male)

 

Variable	C10 (MPa)	C01 (MPa)	C11 (MPa)
Value	0.05	0.2	0.25

Card 3 for viscoelastic constants of two-term Prony series

 

Variable	G1 (MPa)	β1	G2 (MPa)	β2
Value	0.016	0.54	0.07	0.06

Implementation of Material Model for the Pubic Ligaments

The material model (Type 91) in LS-DYNA was used for the ligament model with a viscoelastic option. The material axis was assumed globally orthotropic (AOPT=2) and the transverse x-axis was considered for collagen fiber direction of the ligaments (normal to the midline plane of the interpubic disc). The spectral strengths for Prony series relaxation kernel were calculated as $S_i = \alpha _i G_0$, where $G_0 = 2(C_1 + C_2 ) = 2.88\,\hbox{MPa}$ and T_i = τ_i. The card values input in the FE model are provided in the Tables below.

Card 1

 

		RO	C1	C2	C3	C4	C5
Variable	MID	(kg/m3)	(MPa)	(MPa)	(MPa)	(no unit)	(MPa)
Value	4	1200	1.44	0.0	0.19	35.5	155.0

Card 2

 

Variable	XK(MPa)	XLAM	FANG	XLAM0
Value	1440	1.055	0.0	0.0

Card 3

 

Variable	AOPT	AX	AY	AZ	BX	BY	BZ
Value	2	1.0	0.0	0.0	0.0	1.0	0.0

Card 4

 

Variable	LA1	LA2	LA3
Value	0.0	0.0	0.0

The following two cards were used for the viscoelastic option. (S_i, T_i) are spectral strengths and characteristic times for Prony series relaxation kernel.

Card 5 (male)

 

Variable	S1 (MPa)	S2 (MPa)
Value	0.092	0.403

Card 6

 

Variable	T1	T2
Value	1.85	16.72