PSI - Issue 28

Juan Du et al. / Procedia Structural Integrity 28 (2020) 577–583 J. Du et al./ Structural Integrity Procedia 00 (2019) 000–000

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4

The corresponding local remodelling reaction C was linearly proportional to Λ i , where A is a remodelling rate constant for density change set to 1 × 10 8 [17]: i C A   . (3) The rate of changing density was predicted from the calculated stimuli:

i 

d

,

(4)

( C S S   i

)



s

ref

dt

where μ s is the strain-sensitivity factor and S ref is the reference value of resorption and formation (Table 2). The current density � � is calculated using the forward Euler method: 1 i i i t t d t dt        , (5) where Δ t is the time step in the simulation process. The elastic modulus was also updated at each step, according to the newly predicted density based on Eq. (1). By utilizing this remodelling formulation, the local values of bone density can be calculated, which, in turn, affect the magnitude of the elastic modulus based on Eq. (1) for the next step. This iterative process continued until the prescribed time limit was reached.

Table 2: Model parameters used in algorithm to simulate trabecular bone adaptation. Values were given according to bone physiology and chosen based on previous studies. Parameters Symbol Unit Value Signal limit of resorption S R με 3000 [12] Signal limit of formation S F με 5000 [12] Strain-sensitivity factor μ s - 2.5×10 -4 Remodelling rate constant A - 1×10 8

3. Results Trabecular structures with a density distribution were generated for both multi- and single-material models (Fig. 2). The FE simulations produced trabecular-like morphologies and density distributions. In all cases, newly formed and resorbed bone with density changes can be observed. However, by visualising the differences, it is apparent that the bone volume fraction was higher in the multi-material model. The effect of using single-material distribution can also be observed from single trabeculae. In the single-material model, trabeculae were intended to be disconnected with declining density; however, the same trabecular structure was maintained in the multi-material model (indicated by black circles in Fig. 2). The volume of the elements with a density in a given range was summarized and demonstrated in Fig. 3. Different distributions of density were obtained with the multi-material model and, to a lesser extent, in the single-material one. The highest fractions- 43.92% and 45.93% of the elements- were in the density range between 1.25-1.35 g/cm 3 in both material models. The character of change in a volume fraction is illustrated in Fig. 4: initially the volume-fraction rises sharply, overshooting and eventually stabilizing at 41%.