Computational modelling of the head is common in the literature across a broad range of themes.
These include but are not limited to:
• Neurosurgery
o Craniotomy and tumour resection (113, 145, 154-156) o Stereotactic procedures (59, 92, 157, 158)
o Infusion (159)
o Normal pressure hydrocephalus (162, 163) o Plagiocephaly (164)
A computational model that perfectly captured all geometry and material properties of the head, would in theory be able to model any loading scenario. Currently we, as a collective, are some way off this. Material properties, in particular, are often only applicable to the set of boundary
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conditions in which they were derived. To the best of the author’s knowledge, there is currently no computation model specifically designed to study PBS. With this in mind, we offer a brief review into methods used in adjacent modelling scenarios, with the hope of identifying modelling techniques which are equally relevant to the present.
High strain rate models
High strain rate injury is probably the oldest implementation of FE head modelling and given the applicability to the whole population, remains immensely popular. As such, the area is a subject of frequent review (107, 165, 166). Several FE models are the product of many years of development. For example, the well-known Kungliga Tekniska Hogskolan (KTH) model shown in Figure 22, was first presented in 2002 (7). Since then, improvements have included the addition of anisotropy within the brain (167) and most recently fluid structure-interactions (FSI) between the brain and cerebrospinal fluid (168). Whilst the load rate is very different to the case of PBS, this area of modelling is probably most similar, as it still concerns the intact cranium and no loss of CSF.
Figure 22 – The KTH model geometry as presented in (161).
With increasing computational power, geometric biofidelity of models is improving greatly.
Automated geometry generation methods are able to create FE meshes containing millions of elements (109, 160, 169). This improved geometry has led directly to some key findings, such as the involvement of the sulci in chronic traumatic encephalopathy (109). However, without appropriate smoothing, jagged edges can lead to numerical artefacts (169).
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Figure 23 – Examples of the high biofidelity finite element models for traumatic brain injury used in (109, 160) [top] and (169) [bottom].
One of the main risks with significant head impact is subdural haemorrhage, commonly caused by the rupture of bridging veins. As such, the bridging veins are a key component of many head impact finite element models (18). In some cases they are defined with no mechanical stiffness and only strain is assessed (170). More commonly they are defined as springs or beam elements with linear elastic stiffness (7, 161, 171, 172). An example of this linear representation can be seen in Figure 24 below.
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Figure 24 – Representation of the bridging vein pairs by means of spring elements with linear elastic stiffness (171). Reproduced from (18) for improved clarity.
More recently, Migeuis et al. (173) used a more realistic geometry and damage model to investigate the rupture of bridging veins. Vessels were defined with linear elastic properties and a plastic representation to capture vessel failure. In this case, these improvements led to more accurate prediction of kinematic parameters involved in vessel tearing.
Figure 25 – The finite element representation of the cranial sinuses and selected bridging veins, reproduced from (173), showing greatly improved biofidelity.
Even in some of the latest models, such as the Yet Another Head Model (YEAHM) (174) from which Figure 25 was based, there are still some significant simplifications and debate in the literature. Foremost of these is the PAC (133, 153), often referred to as the brain-skull interface.
There are a number of methods used in modelling this region (153):
1. Tied contact between the brain surface and skull (175).
2. Brain/skull sliding with a coefficient of friction of 0.2 (or similar) (171).
60 3. Brain/skull free sliding (176).
4. Connection of brain and skull surface nodes with two-dimensional connectors/springs (136, 153).
5. High bulk modulus/low shear stiffness solid element representation of CSF/PAC combined (7, 161, 167, 172, 174, 177).
6. Various combinations of the above.
In general, the CSF layer is well recognised to be important and most approaches use solid elements with varying degrees of sliding allowed between the CSF region and the skull. Some studies have compared these methods directly. Coats et al. (153) modelled five contact conditions separately: conditions 1, 2, 3 and 4 (with vasculature) from the above list, and solid elements with fluid properties. Impact testing on immature piglets was undertaken to validate the FE models.
Results showed the use of spring connectors or solid elements gave the greatest agreement with experimental data. Saboori et al. (138) compared soft solid, viscous fluid and porous elastic material representations of the subarachnoid space. The results of all three material representations were very similar, meaning one method was not advocated over any others. Most recently, Wang et al (178) considered four representations:
1. The approach used in the THUMS model (179) where CSF, dura, arachnoid and pia are all modelled separately as solid elements.
2. Rigid connection of the pia and skull through tied contacts.
3. Frictionless sliding between the brain and skull.
4. A cohesive layer between the brain and skull connected by spring elements.
Once again, the original solid element representation was found to best match validation data.
However, it is acknowledged that a better understanding of the cohesive material properties may improve the results.
Multiscale models are the next logical step in investigating the role of the PAC more closely.
Zoghi-Moghadam et al. (180) calculated loads in head impact using separate global solid and global fluid models. These were then applied to a detailed local model (Figure 26 [right]) to assess the chance of vessel rupture. Scott et al. (133) assessed the local geometry of the PAC and also regional changes in volume fraction (VF) of solid components. Figure 26 [left] shows the result of random generation of chords (blue), narrow sheets (green), broad sheets (red) and combined sheets and vessels (grey).
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Figure 26 – [left] Example of the randomly generated PAC geometry reproduced from (133) - [right] local model geometry reproduced from (180).
This technique suggested significantly different cortical stress compared to a uniform representation of the PAC, with excellent regional prediction of haemorrhage. From these studies it can be concluded that on a global scale, a biofidelic representation of the PAC is essential for the continued improvement of computational models, although implementation is still challenging.
Fluid structure-interactions
One of the main limitations of many previous investigations of traumatic brain injury (TBI) has been the absence of FSI in the model (178). CSF has generally been modelled with incompressibility and low shear stiffness; crucially not considering potential fluid redistribution.
Zhou et al. (168) developed the KTH model to overcome this problem, using a Lagrangian-Eularian multi-material formulation to capture FSI. Firstly, the cortical boundary was defined as a void mesh, where CSF could flow under deformation. Then, in a two-step process, the Lagrangian deformation of the FE mesh was computed followed by transportation of element state variables to the reference frame. Bridging veins were included in the model, which was found to achieve improved validation performance on solid CSF representations. However, even with FSI considered, this is still an over-simplification of the brain-skull interface. A better solution would be a model which incorporates fluid CSF with FSI and the tethering arachnoid trabeculae together (168).
Neurosurgery
Computational modelling of neurosurgery is also prolific in the literature. Focus is mainly on the model-driven update of navigational neuroimages in craniotomy and tumour resection (156), where computational speed is the main concern (96, 181). As such, non-commercial computational software is often developed for this purpose, and used with significant geometric simplification. For example, some authors advocate the generation of FE models with a physical gap, no PAC and a sliding interface between the brain and skull (108). Craniotomy causes the loss of large volumes of CSF with subsequent displacements in the order of 10 mm (182). This method may be acceptable in such cases; however, in PBS this is not likely to be the case.
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Loading of other neurosurgical models is somewhat unique in that it is often displacement driven, with measurements taken from the cortical surface of the patient during surgery (9, 108, 183). In these cases, it is suggested that the displacements throughout the remaining domain are only weakly dependant on the mechanical properties (9, 184). However, this property is mathematically driven and again is almost certainly not applicable in PBS.
Some studies have investigated brain shift in the context of stereotactic neurosurgery (59, 92, 157, 158, 185). In the initial works, it was considered that CSF loss was both unavoidable and unpredictable. Instead, the authors employed worse case boundary conditions to generate a risk volume that must be avoided to ensure safe insertion of the electrode. Subsequent focus turned to automation of this process and intra-operative registration of the computational model to a deformed image. These works utilised a physics-based approach, in which the loss in CSF was accounted for by a reduction in force applied directly to the surface of brain itself, calculated as a function of the fluid head above each point. Fixation of the brain stem, and a non-penetrating contact algorithm between the brain and skull/falx cerebri contained the brain within the intracranial cavity when equilibrium was lost. Whilst potentially improving surgical safety, these works do not address PBS directly.