ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

Figure 3 exemplifies for intradiscal pressure and the contact force in the left facet joint,
the response can indeed vary widely, depending on the chosen model output and the
load case. Common point estimates like sample means and sample standard deviations
(given normally distributed results) describe the scatter in the model response. Yet, the
computed point estimates are random variables themselves, whose reliability depends
on the chosen sample size, 500 in our case.
Figure 3: Model responses for all 500 variations for all load cases (Cmp: compression, Ex: Extension,
Flex: flexion, LatL/LatR: lateral bending left/right, RotL/RotR: rotation left/right)
One way of estimating the distribution (and therefore the reliability) of the means would
be to repeat the whole aforementioned procedure times, i.e. for 100 this would
require solving ⋅ 50,000 FE models. Fortunately bootstrapping the computed
500 solutions provides reasonable precise distribution estimates at a fraction of the cost
(with 1000 bootstrap runs). 11
We can use this approach to estimate the reliability of the computed mean model responses
depending on the sample size by sub-sampling before drawing each bootstrap
sample. We found that a sample size of 100 already provides a reasonable stable
approximation of the population mean (Fig. 4).
7. SENSITIVITY ANALYSIS
As a byproduct of the uncertainty analysis, we can use the results to also investigate the
relative importance of each individual parameter for the model response. Scatter plots of
input vs. output variables give first hints of possible relationships (Fig. 5); the correlation
matrix (PPMCC) quantifies the degree of linear relationship between all parameters.
By decomposing the total variance of the model response, we can compute the relative
contribution of each parameter to the model response and hence its relative importance.
12–14 We found that disc height and the position of the facet joints as well as the
distance between articulating facet joint surfaces are especially important for determining
the mechanics of the spine segment.