Due to uncertainty in the validity across the entire volume of the validation data, calibration was performed over the whole cerebrum and a subset where the exterior nodes were removed, leading to two parameter sets. In excluding such significant portions of the volumetric displacement data, the author concedes that this raises questions about the validity of using any of it. However, upon comparison to other literature, confidence was established that the second set was likely to represent the displacement pattern of a typical person. In this section, we consider the parameter identification method and compare the results obtained to the literature. To recap, the parameters found are re-presented in Table 6-1.
Table 6-1 β Material parameters identified after calibration with the entire cerebrum and reduced volume.
Parameter Calibration 1 Calibration 2
Bulk modulus /Pa 187,000 127,000
βShear modulusβ /Pa 1,580 1,250
Exponential coefficient -1 -16.9
Pia Youngβs modulus /Pa 254,000 250,000
Spring constant /N/m 10.4 0.5
Brain
The brain itself has received considerably more attention in the literature than any of the surrounding structures. Numerous constitutive models have been developed to capture its unique properties, each better or worse at doing so in specific loading scenarios. These can use as few as three, but as many as 20 material parameters (10). As mentioned in Chapter 3, the material parameters which can be identified for any model are normally only valid for the specifics of loading and constraint in which they were obtained. As such, when choosing a constitutive model, one should carefully consider the physiological process at hand (10).
When choosing the constitutive model to study PBS, the desired outcome was first considered.
This was namely to understand displacement throughout the brain with reorientation relative to a supine position. After some initial investigation, it was concluded that equilibrium would be reached in a relatively short time (~<10 minutes); although measurement could not be made in a shorter time than this due to practicalities of MR imaging. This settling time contradicted one previous report (4). The source of this discrepancy has not yet been identified, but it was felt that even if the brain had not completely settled over this time frame, any further displacement would be negligible. Within surgery it is certain that the time between final fix of head orientation and the beginning of the procedure would be greater than 10 minutes and therefore only the equilibrium shift is clinically relevant.
Time dependency does not need to be considered in the constitutive model when only concerned with the equilibrium displacement. This reduces the number of parameters needed to characterise the material response. This was important, as it is difficult to achieve a meaningful fit with a highly parameterised model when inverse modelling with equilibrium in-vivo displacement data
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alone. It was also assumed that displacements would be relatively small and other considerations such as tissue preconditioning would not apply.
The Ogden constitutive model is often suggested when considering the elastic response of brain tissue (10). Although there will always be debate, a lack of clear consensus about a better alternative makes it difficult to conclude that it is not a good option. Furthermore, the relatively simple formulation with as few as three material parameters and successful implementation in some of the prominent studies within the field (112, 117) provided sufficient grounds to use the Ogden model for this study.
Ogden coefficient - π1
The original formulation by Ogden has been modified by many groups over the years. Such additions considered the time dependant response (112) and material compressibility (115). Some small manipulation of these models is often required to compare the results across different studies. Equation 43 gives the strain energy function as is used in FEBio:
π = β ππ ππ2
π
π=1
(πΜ1ππ+ πΜ2ππ+ πΜ3ππβ 3) + π(π½) 43 Here, π allows up to 6 sets of parameters to be defined. This has been shown to improve the fit of experimental data, especially under complex loading regimes such as combined shear and compression/tension (11, 118). However, given the small strains expected, a one term was used here and compared to the literature.
Studies 1-3 of Table 6-2 all use the same formulation, whereas study 4 uses one more similar to the original Ogden version (116). For ease of manipulation, some parameters have had their symbol changed from the original manuscripts. In 1-3, π is defined to be equivalent to the classical shear modulus according to linear theory (98, 119). Considering first that πΌ = 2πΌ0, this shear modulus relates to πΆ0, according to π =πΆ02πΌ.
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Table 6-2 β Literature values for material parameters of the time-independent, one-term, Ogden strain energy density function. was referred to as the shear modulus, whereas π1 is actually two times this quantity. Therefore, calibrations 1 and 2 yielded shear modulus values of 790 and 625 Pa, respectively. The values found here show notable similarity to the results of mechanical testing in the literature. This is especially the case when considering the known levels of variation that exists both between subjects, and regionally over the brain volume.
The regional analysis by Budday et al. (119) considered samples from the corpus callosum, corona radiata, basal ganglia and the cortex. The model was fit to combined loading, yielding mean shear modulus values of 350, 660, 700 and 1,430 Pa, respectively. Without calculating the exact volumetric ratios within the brain, it can safely be said that the corpus callosum is very small in volume compared to the other regions. If these figures are representative of the rest of the brain, an average value would likely be towards the higher end.
When comparing the results of this study to others, it must be remembered that the brain was modelled as an isotropic, homogeneous body, with no inclusion of the cerebral vasculature.
Therefore, the stiffness found is likely to be a slight overestimate compared to mechanical testing on samples with no vasculature. Interestingly, calibration 2 did not consider the cortex, which is known to be twice as stiff as the other structures (119), with the obtained value dropping by over 150 Pa compared to calibration 1. Much future work would be required to determine if this is just a coincidence. If not, this result supports the mechanical testing data and also shows the method to be potentially sensitive to regional property change.
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The final sensitivity plots of Figure 64 show variation in the order of 0.01 mm due to full range (0 to 1500 Pa) variation of the shear modulus. This suggests that the shear modulus is not a highly influential variable in PBS, and that the impact of regional variation or anisotropy might be limited. On the other hand, the combination of a lack of sensitivity and a result which is very much in line with the literature is extremely encouraging evidence towards the efficacy of the experimental method developed here.
Ogden coefficient - π1
The exponential coefficient π1, often referred to as πΌ, was identified in calibration 2. This parameter functions to scale the stresses as the stretch changes, allowing models to capture the complex non-linear behaviour that is often found experimentally. The effect of this is illustrated in Figure 88, which shows the impact of variation in the value of πΌ on the second derivative of the strain energy function (instantaneous deviatoric stiffness), with π at a value of 750 Pa.
Figure 88 β Plot demonstrating the effect of exponential coefficient variation on the instantaneous deviatoric stiffness, using a realistic corresponding shear modulus.
There are two key things to note from this. Firstly, at small stretches, variation in the value of this coefficient has a small impact on the deviatoric stiffness. This effect is almost negligible when considering variation over a range of -5 to -15, for example. Secondly, it illustrates the reason most groups have identified the parameter to be negative for the brain, as testing has shown the tissue to have increasingly greater stiffness in compression and reducing stiffness in tension (119).
In calibration 1, this coefficient was found to have no effect on the result, probably due to the very small average strains involved. In the calibration 2, a value of -16.9 was identified, falling very much in the range of commonly reported values in the literature. Sensitivity was significant after the initial iteration obtained more realistic parameter ranges for the bulk modulus.
0
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Bearing in mind that π and πΌ should not be interpreted alone, but together, the identified combination of 625 Pa and -16.9, respectively, represent a realistic approximation of whole brain behaviour.
Bulk Modulus
It is now well established that the incompressibility assumption is only valid in certain circumstances (10, 115). During mechanical testing, Budday et al. (119) found a clear pre-conditioning effect over three consecutive cycles. After soaking the sample with phosphate-buffered saline solution for 60 minutes, they found the impact of pre-conditioning to effectively disappear. This suggests that the effect is down to fluid loss, rather than damage to the tissue itself, further emphasising the influence of relative volume change.
When not interested in the time-dependant modelling of consolidation theory, the bulk modulus and Poissonβs ratio can be used to account for volume change within a material. Conversion between the quantities is only valid with certain measures of stiffness and at small strains, but this is the case with PBS. The bulk modulus was determined to be 127 kPa in the final model. With the shear modulus of 625 Pa, the Poissonβs ratio Ξ½ can be found according to Equation 44:
π = 3π β 2π 2(3π + π)
44
yielding a result of 0.497. Assessment of the bulk modulus directly in the literature has been limited. Values for the bulk modulus in the order of 40,000 Pa (Ξ½ ~ 0.4) were initially reported (127, 131, 132), but are thought to be incorrect due to methodological assumptions. In direct consolidation testing of real human tissue, Franceschini et al. determined a drained Poissonβs ratio of 0.496 (115), with full consolidation occurring with volumetric strain in the order of 3%.
The lack of attention given to the bulk modulus in most modelling implementations initially gave the suggestion that it was an unimportant factor purely required for mathematical stability in modelling. It quickly became clear that it is arguably the most influential factor in PBS, at least when varied over commonly used ranges. With a bulk modulus in the order of 500,000 Pa (Ξ½ ~ 0.499) and up, brain shift is limited to small scale, rigid body displacement. On the other hand, much lower values lead to unrealistically large displacements. The obtained value of 0.497 strongly supports the findings of Franceschini et al. (115); especially when considering that the volumetric strains found here peaked at 0.7%, suggesting full consolidation was not achieved.
Although the difference between 0.499 and 0.497/0.496 is small in terms of Poissonβs ratio, it is very large in terms of bulk modulus and this study confirms that in confined loading scenarios such as PBS it is very influential.
Whilst is unlikely that local anisotropy would impact PBS significantly (119), regional variation of bulk properties might offer interesting insight into local displacement patterns, especially when considering the non-equilibrium state at a similar strain rate to that studied here.
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