Rapid, non-invasive, in vivo measurement of tissue mechanical properties using gravitational loading and a nonlinear virtual fields method

Figure 1.. The finite-element model of the brain: the right side of the model showing the ventricles and the fluid-filled subarachnoid space with spring elements representing the arachnoid trabeculae. The dura mater and CSF have been hidden for clarity.

Figure 2.. Displacement data from the MRI measurements [15]. The vectors show the displacements, scaled by a factor of 5 for clarity. Letters show anatomical directions: A = anterior, P = posterior, S = superior, I = inferior, L = left, R = right; transverse section (left), sagittal section (top right) and frontal section (lower right); the position of each section is indicated by the dashed lines in the other views which match the colours of the vectors in each view.

Figure 3.. (a) The first virtual field optimized to identify μ, plotted in the same way as figure 2 but with the vectors scaled by 105. (b). The second virtual field optimized to identify α, plotted in the same way as figure 2 but with the vectors scaled by 105. (c) The third virtual field optimized to identify K, plotted in the same way as figure 2 but with the vectors scaled by 105.

Figure 4.. Error function Perrtotal plotted as a function of the material parameters from multiple optimization runs using two different optimization algorithms: green stars show parameter sets with an error less than 1 × 10−8, yellow asterisks 1 × 10−8 < Perrtotal < 1 × 10−6, red + signs 1 × 10−6 < Perrtotal < 1 × 10−4 and blue cross symbols 1 × 10−4 < Perrtotal < 0.01. (a) the parameter space shown in three dimensions; (b) projection on to the μ–K plane. Extremely small errors were possible for any value of α but the other parameters were identified consistently and precisely.

Figure 5.. Errors for 2800 random parameter sets. Green stars show points with errors less than 1%, yellow asterisks 1% < Perrtotal < 2%, red plus signs 2% < Perrtotal < 5%, blue cross signs Perrtotal > 5%. Note the axis scales: K was identified with very little uncertainty, but a much wider range of values for μ result in only small errors.

Figure 6.. The effect of adding synthetic noise to the displacement data. The graph shows the values of the error function Perr plotted against the RMS noise level for each of the three virtual fields; this was repeated twice so that there are two sets of points. It is evident that there was no systematic bias in the error function Perrtotal as a result of adding noise and that the three virtual fields were affected independently and differently.