AMERICAN SOCIETY OF BIOMECHANICS
Presented at the Twenty-First Annual Meeting
of the American Society of Biomechanics
Clemson University, South Carolina
September 24-27, 1997
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| AMERICAN SOCIETY OF BIOMECHANICS
Presented at the Twenty-First Annual Meeting |
Remodeling of bone serves both to eliminate bone in a disuse state and repair fatigue damage. These effects were investigated simultaneously in an adaptive finite element model for the human femur. This model accounted for the temporal and porosity effects caused by the fact that resorption precedes formation in remodeling by basic multicellular units (BMUs).
A modified version of the model of Martin (1995) was used. The rate at which damage was formed in the bone (D'F) was assumed to be related to the loading rate (RL) and the resulting strain range (s) from a mixture of loads applied over the course of a day:
D'F=kD*[(s1**q)RL1+(s2**q)RL2+...+(si**q)RLi]
where kD is the damage coefficient found from the equilibrium state and q was taken to be 2.89. The damage for the equilibrium condition was determined to be 0.085mm/mm2 from the average crack density for 50 year old females and males (Schaffler et al., 1995) and an assumed crack length of 75mm. This gave kD=2.18, which would produce an equilibrium condition for 1800me applied for 3000 cycles per day (cpd). Minimum principal strain was used as the predictor (s) for damage formation.
The rate at which damage was repaired (D'R) was:
D'R=D*fa*pi*(rc**2)*Fs
where D is the existing damage, fa is the BMU activation frequency, rc is the osteonal cement line radius (assumed to be 0.095mm), and Fs is the damage repair specificity factor (set to 5 for this study).
Based on the data by Mori et al. (1993), Martin (1995) hypothesized a relationship between the BMU activation frequency and fatigue damage. Evidence also suggests that BMU activation frequency increases in disuse (Schaffler et al., 1990). The total daily activation frequency was assumed to be a function of the existing damage (Figure 1), strains below an equilibrium value (Figure 2), and the internal surface area available for the remodeling of bone in the region being investigated (Martin, 1984).
The remodeling period was assumed to be 90 days; 21 days for resorption (TR), 9 days for reversal (TI), and 60 days for refilling (TF). The history of the activation frequency was used to determine how many BMUs were in the resorption and refilling stages on a given day. The total number resorbing for the current day (NR) was found by integrating fa from TR days ago to the present time. The number refilling for the current day (NF) was found by integrating from (TR+TI+TF) to (TR+TI) days ago.
The net amount of bone removed or added per day was estimated as
QC=(pi*rc**2)/TR and QB=(pi*rc**2)/TF
for each resorbing and refilling BMU, respectively. The daily change in porosity (p), expressed as a percentage, was then calculated to be
delta(p)=QC*NR-QB*NF.
A minimum cortical bone porosity of 5% was allowed in this simulation. Young's modulus was determined from the relationship
E=23440*(1-p)**5.74 MPa (Currey, 1988).
A two-dimensional finite element model (linearly elastic, isotropic) of the proximal femur, consisting of 1117 4-node quadrilateral elements, was created from radiographs. The structural contribution of the out-of- plane cortical bone was accounted for by the addition of a bony side plate (Weinans et al., 1992). An initial porosity of 5% and Poisson's ratio of 0.3 were assumed for all elements and the porosity of the bony side plate was kept constant throughout the simulation. Three load cases, each consisting of joint reaction and abductor muscle forces, were used to simulate the daily loading history for normal activity (Carter et al., 1989). For this simulation, the first load case was applied for 3000cpd while the second and third were applied for 1000cpd.
The simulation was run for the equivalent of 1600 days using ABAQUS 5.5 (HKS, Pawtucket, RI), with the bone adaptation algorithm integrated into the analysis through a UMAT subroutine. Daily porosity changes were calculate at the integration points for each element.
Porosity distribution contour plots (Figure 3) for (a) 500, (b) 1100, and (c) 1600 days of the simulation show predictions similar to observed femoral morphology: distinct cortices surrounding a porous medullary canal, dense bone in the calcar region of the neck, and a dense region in the head aligned with the primary load case and surrounded by bone of higher porosity. Activation frequencies were found to be higher and more variable in trabecular areas than in cortical regions. Results for trabecular bone, however, appear to simulate porosities lower than those expected.
Most of the porosity and cortical thinning resulted from low strain, but damage repair also introduced significant porosity. The distribution of porosity from these two factors was similar. These preliminary results show that the internal structure of the femur can be predicted on the basis of remodeling provoked by both disuse and damage repair.
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