Multi-scale analysis of optic chiasmal compression by finite element modelling.

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Multi-scale analysis of optic chiasmal compression by finite element modelling Xiaofei Wang a,n, Andrew J. Neely a, Gawn G. McIlwaine b,c, Christian J. Lueck d,e a

School of Engineering and Information Technology, University of New South Wales Canberra, ACT, Australia Queen's University Belfast, Belfast, Northern Ireland, UK c Belfast Health and Social Care Trust, Belfast, Northern Ireland, UK d The Canberra Hospital, Canberra, ACT, Australia e Australian National University, Canberra, ACT, Australia b

art ic l e i nf o

a b s t r a c t

Article history: Accepted 23 April 2014

The precise mechanism of bitemporal hemianopia (a type of partial visual field defect) is still not clear. Previous work has investigated this problem by studying the biomechanics of chiasmal compression caused by a pituitary tumour growing up from below the optic chiasm. A multi-scale analysis was performed using finite element models to examine both the macro-scale behaviour of the chiasm and the micro-scale interactions of the nerve fibres within it using representative volume elements. Possible effects of large deflection and non-linear material properties were incorporated. Strain distributions in the optic chiasm and optic nerve fibres were obtained from these models. The results of the chiasmal model agreed well with the limited experimental results available, indicating that the finite element modelling can be a useful tool for analysing chiasmal compression. Simulation results showed that the strain distribution in nasal (crossed) nerve fibres was much more nonuniform and locally higher than in temporal (uncrossed) nerve fibres. This strain difference between nasal and temporal nerve fibres may account for the phenomenon of bitemporal hemianopia. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Bitemporal hemianopia Chiasmal compression Finite element modelling Multi-scale Nerve failure

1. Introduction In the human visual system, information from the right visual field (VF) is processed by the left side of the brain and vice versa. To achieve this, there is a partial crossing of optic nerve fibres in the optic chiasm (Fig. 1). Nerve fibres arising from the nasal hemiretinas, which receive input from the temporal VFs of the two eyes, cross over in the optic chiasm to enter the contralateral optic tracts, whereas nerve fibres arising from the temporal hemiretinas, which receive input from the nasal VFs, pass directly backwards through the chiasm to the ipsilateral optic tracts (Fig. 1). The crossed fibres lie in the central part of the chiasm while the uncrossed fibres pass directly back to the tracts in the lateral parts of the chiasm. Damage to the chiasm typically produces bitemporal visual field loss, or hemianopia, due to selective damage to the crossed nerve fibres arising from the nasal hemiretina.

n Correspondence to: School of Engineering & Information Technology, UNSW Canberra, Australian Defence Force Academy, Northcott Drive, Canberra, ACT 2612, Australia. Tel.: þ61 2 626 89474; fax: þ 61 2 626 88581. E-mail address: [email protected] (X. Wang).

Bitemporal hemianopia is a kind of partial blindness in which vision is impaired in the temporal halves of the VFs of both eyes (Fig. 1). The most common cause is a tumour of the pituitary gland that compresses the chiasm from below (Blumenfeld, 2010; Kosmorsky et al., 2008). Vision may recover rapidly after surgical decompression but this is not always the case (Anik et al., 2011; Kayan and Earl, 1975). Although this phenomenon is well-recognized, the precise mechanism whereby compressive lesions of the chiasm selectively damage nasal nerve fibres is still not clear. Several explanations have been offered over the last 50 years (Bergland et al., 1967; Hedges, 1969; Kosmorsky et al., 2008), principally relating to differential pressure or blood supply across the chiasm, but these explanations are unsatisfactory as they do not explain the abrupt change between temporal and nasal VFs. McIlwaine et al. (2005) proposed that the abrupt change between the nasal and temporal visual fields may result from the fact that the nasal nerve fibres are arranged roughly perpendicular to each other while the temporal nerve fibres are arranged approximately parallel. Therefore, when compressed, the crossed fibres will experience more severe stress concentration than the uncrossed fibres, resulting in greater damage to nerve function. This simple mechanical explanation assumed that the pressure across the chiasm was uniform and did not take account of the fact

http://dx.doi.org/10.1016/j.jbiomech.2014.04.040 0021-9290/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: Wang, X., et al., Multi-scale analysis of optic chiasmal compression by finite element modelling. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.04.040i

X. Wang et al. / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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that nerve fibres are 3D bodies which will deform in a nonuniform and non-linear manner when compressed. Finite element modelling (FEM) is a well-established engineering tool for structural analysis which has been used in the investigation of nerve failure in traumatic brain injury (Ho and Kleiven, 2009; Post et al., 2012). It has also been used in various fields of ophthalmology, such as modelling of the biomechanics of the optic nerve head in glaucoma and modelling of injury to the optic nerve from blunt trauma (Bellezza et al., 2000; Cirovic et al., 2006; Norman et al., 2011; Sigal et al., 2004; Sigal et al., 2007). In this paper, a multi-scale analysis framework was developed using FEM to help understand the relation between chiasmal compression and axonal injury at a cellular level with a view to helping to resolve the mechanism of bitemporal hemianopia.

2. Methods A chiasmal model was initially built for macro-scale analysis. Micro-scale models were subsequently generated using representative volume elements (RVE; described in detail below) to investigate the strain distributions in the nerve fibres. The models are schematically illustrated in Fig. 2. In the chiasmal model, the optic chiasm, optic nerves and optic tracts were considered to be composed of isotropic nerve tissue surrounded by a pial sheath. It is reasonable to assume that the chiasm tissues are macroscopically homogeneous (Post et al., 2012; Sigal et al., 2009). However, at the cellular level, the fibre arrangements vary in different locations. Therefore, RVE models were built to

Nasal visual field Temporal visual field

Temporal visual field

Nasal hemiretina Temporal hemiretina

Temporal hemiretina Eye Optic nerve Optic chiasm

Optic tract

Fig. 1. Illustration of the human visual pathway.

represent both the crossed and uncrossed configurations of the nerve fibres in the chiasm. In the chiasmal model, a spherical balloon representing a growing tumour was inflated and then translated inferior–superiorly so as to elevate and compress the chiasm from below (Fig. 2). This was in keeping with the experiment of Kosmorsky et al. (2008) in which a Foley catheter balloon was inflated beneath a cadaveric chiasm to represent a growing tumour. The use of a nominally spherical tumour was also consistent with observations from clinical MRI scans (Egger et al., 2010; Fahlbusch and Gerganov, 2012; Honegger et al., 2008). The deformation of the chiasmal model was then transformed to the RVE models as boundary conditions. In order to determine the difference between the strains experienced by crossed and uncrossed nerve fibres, the strain state of a point at the central part of the chiasmal model was extracted and applied on the RVE models, which will be introduced later. 2.1. Chiasmal model 2.1.1. Geometry The geometry of the chiasmal model was obtained from the 3D-reconstruction of human head slices available at the ‘US visible human project’ (Mikula et al., 2008). The images were segmented manually using 3D Slicer (Pieper et al., 2006) to obtain the boundaries of the optic nerves, chiasm and tracts. Then a 3D model containing only the surface information was saved as an STL file. This file was processed using CATIA V5 (Dassault Systemes, France) to form a solid model which was imported into the FEM package ANSYS (Ansys Inc., Canonsburg, PA, USA) for analysis. Because of the low image resolution, it was hard to detect the pial sheath. Therefore, the reconstructed slices were assumed to be comprised entirely of nerve tissue. A surrounding pial sheath of 0.06 mm thickness (Sigal et al., 2004) was added to the model in ANSYS. 2.1.2. Material properties For the large deformation problem in this study, the material properties of the pial sheath and the nerve tissue were modelled as nonlinear in keeping with the limited previous work available in the literature. Because we were not concerned with the stress levels in the tumour, the spherical balloon used to represent the tumour was simply modelled using linear elastic material properties of sclera. The parameters of these material models were all taken from the limited soft tissue data in the literature (Table 1). The densities of sheath, optic nerve and tumour were 1130 kg/m3, 1130 kg/m3 and 1000 kg/m3, respectively (Beckmann et al., 1999; Ho et al., 2009). 2.1.3. Boundary and loading conditions Anteriorly, the optic nerves are connected to the optic canal by fibrous adhesions (Hayreh, 1964) and posteriorly the optic tracts are embedded in the brain. The distal faces of the optic nerves and optic tracts in the model were therefore fixed. More complex arrangements, better representing the anatomy of these attachments, were tested but were found to have negligible influence on the stress levels in the regions of interest. Pressure was applied to the interior surface of the tumour (balloon) to inflate it and then a displacement in the superior direction was applied at its inferior surface to further elevate the chiasm. The contact between the balloon and chiasm was considered to be frictionless. The core tissues of the optic nerve, chiasm and tract were bonded to their respective pial sheaths.

Right

Left coronal plane

x3 ON Crossed nerve fibres

0.0

5.0

10.0 (mm)

x2

Uncrossed nerve fibres

1 μm Macroscale (whole optic chiasm)

x1

Microscale (nerve fibres)

1 μm RVEs

Fig. 2. The macroscopic model of the optic chiasm and tumour and the microscopic RVE models. (OC: optic chiasm; ON: optic nerve; OT: optic tract).



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Table 1 Material properties used for the chiasmal model. Tissue

Tissue on which model Material model and reference based

Material constants

Sheath

Pia mater

Hyperelastic-2nd polynomial (Ho and Kleiven, 2009)

C10¼ 1.5269 105 Pa; C01¼ 7.1920 104 Pa; C20 ¼ 1.2554 107 Pa; C11¼ 1.7873 107 Pa; C02¼ 7.27 106 Pa

Optic nerve

Brain white mater

Hyperelastic-2nd ogden (Franceschini, 2006)

μ1 ¼ 1044 Pa; α1 ¼ 4.309; μ2 ¼1183 Pa; α2 ¼ 7.736

Tumour (balloon)

Sclera

Linear elastic (Bellezza et al., 2000)

Elastic modulus¼ 5.5 MPa; Poisson's ratio¼ 0.47

2.2. RVE model An RVE should be able to capture the major features of the underlying microstructure of the macroscopic model (Ghanbari and Naghdabadi, 2009). The RVEs for the crossed and uncrossed nerve fibres in this work are shown in Fig. 2. The strain vector of a point in the central part of the chiasm, when elevated inferiorly–superiorly to 5.6 mm, was applied to the RVEs. 2.2.1. Geometry At nerve fibre scale, it was assumed that the temporal uncrossed nerve fibres were aligned in parallel, whereas the nasal nerve fibres crossed each other perpendicularly (Fig. 2). The micro-scale RVE models were developed based on these two types of nerve fibre packing. An individual nerve fibre was considered to be circular in cross-section and to consist of an axon surrounded by a myelin sheath. In the optic nerve, microscopic images reveal that nerve fibres contact each other directly (Sandell and Peters, 2002). The space between the nerve fibres is the extracellular space. We will hereafter use Material of the Extra-Cellular Space (MECS) to refer to the material occupying this space. For simplicity, the diameter of nerve fibres was taken to be 1.35 μm based on reported measurements of nerve fibre area (Jonas et al., 1990). The g-ratio (axon diameter/total fibre diameter) was assumed to be 0.8 (Guy et al., 1989; Sherman and Brophy, 2005). 2.2.2. Material properties Myelin sheaths are formed by oligodendrocytes, one of the several types of glial cell in the central nervous system (CNS). Astrocytes are another type of glial cell. Previous studies have shown that glial cells are softer than axons by a factor of 2–3 (Arbogast and Margulies, 1999; Lu et al., 2006; Shreiber et al., 2009). Again, for simplicity, the material properties of axons were considered to be the same as those for the nerve tissue in the macroscopic chiasmal model, and the myelin sheath was considered to be three times softer (Table 2). The glial matrix, comprising the cell bodies of astrocytes and oligodendrocytes, is the main substance in the optic nerve tissue other than axons and myelin sheaths. However, the cell bodies of astrocytes and oligodendrocytes are large compared to axons so they tend to occupy the space outside the bundles of nerve fibres, i.e., they would be unlikely to be present in the extracellular space between nerve fibres in the RVE models. Therefore, we assumed that the extracellular space is mostly filled by fluid-like material, specifically modelling it as a nearly incompressible solid material with a very low shear modulus. For simplicity, the material constants of MECS were directly modified from those of the axons to generate a material approximately as soft as the cerebrospinal fluid (Chafi et al., 2009). The material constants are reported in Table 2. Based on previous studies (Arbogast and Margulies, 1999; Cloots et al., 2012), it was assumed that there was no slip or separation at the interfaces of different materials, i.e., all the materials were assumed to be fully compatible at their interfaces. 2.2.3. Boundary conditions applied to RVEs RVEs can be stacked together in rows and columns to form the actual structure at the macro level. Therefore, periodic boundary conditions, which incorporate the strain loads from the macro-scale model, were applied to insure that there was no overlap or separation between neighbouring RVEs after deformation. The node at one corner of the model was fixed to avoid rigid body motion. As large deflection analysis was used, to help convergence, dummy nodes in ANSYS were used to allow the deformation applied to the RVE to increase gradually. These boundary conditions allowed the transfer of the actual strains from the macroscopic model to the micro-level. Interested readers are referred to Xia et al. (2003), Wang et al. (2007) and Cloots et al. (2012) for detailed procedures of this method. 2.2.4. Parametric study of the RVE model Geometry and material data for the microscopic nerve fibre model are scarce in the literature and, even when reported, the values can vary considerably between studies. Therefore, design of experiments (DOE) was used to investigate the impact

Table 2 Ogden hyperelastic constants used in the RVE models. Tissue

μ1 (Pa)

μ2 (Pa)

α1

α2

a

1044.0 348.0 52.2

1183.0 394.3 59.2

4.309 4.309 4.309

7.736 7.736 7.736

Axon Myelin sheathb MECSc a b c

Franceschini (2006). Modified to be 1/3 the stiffness of the axon material model. Modified to be 1/20 the stiffness of the axon material model.

of uncertainty of these factors on the output of the RVE model. DOE is an established tool which permits better understanding and control of variability in engineering. Interest has increased in using DOE for sensitivity analysis in biomechanics (Malandrino et al., 2009; Shen et al., 2008; Yao et al., 2008). In particular, DOE provides information about the effects of interactions between factors, which is not obtainable when testing one factor at a time. A five-factor two-level full factorial analysis was designed to investigate the effect of five parameters in the RVE model systematically. Full factorial analysis was used to allow examination of the effects of individual factors and all possible interactions of those factors. The five factors and their levels are listed in Table 3. The high and low levels of a given factor were based on a reasonable range derived from data in the literature. A total of 32 (25) simulations (see the Supplementary material), which reflect all possible combinations of high and low levels for each factor, were performed and the DOE results were processed using Minitab (State College, PA).

2.3. Output measurements In the macroscopic model, we have reported output in terms of the commonlyused value of von Mises strain, although there is ongoing discussion concerning the best mechanical measurement of nerve damage (Sigal et al., 2007). Pressure (Ueno et al., 1995) was also determined to permit validation based on experimental measurements. In the RVE models, both the von Mises strain and the strain along the axonal direction (axonal strain) were reported as the latter has been argued to be a better measure of axonal injury in brain white matter (Cloots et al., 2011; Wright and Ramesh, 2012) although whether this is the case in the chiasm is unclear. As the myelin sheath plays an important role in the failure of conduction of nerve fibres (LaPlaca et al., 2007; Shi and Pryor, 2002), the strain in axons and myelin sheaths were reported separately.

3. Results 3.1. The macroscopic model The chiasm was elevated inferior–superiorly to 5.6 mm, a level expected to result in bitemporal hemianopia (Ikeda and Yoshimoto, 1995). Fig. 3a shows a coronal section of the deformed chiasm in our simulation, along with a clinical MRI scan of a patient experiencing chiasmal compression. Fig. 3b–d shows the von Mises strain distribution in a coronal section of the chiasm which roughly cuts it in half. As illustrated in Fig. 3b, 41 lines were defined in the section. The average von Mises strain of each line was plotted as a function of the location in the left–right direction. It can be seen that the strain at the central part of the chiasm was generally higher than that at the peripheral



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Table 3 Factors and their levels of the DOE study. Factor name

A

Cross-sectional shape

B C D E

g-Ratio Sheath stiffness MECS stiffness Crossing angle

Low level

High level

References Potts et al. (1972)

Round-cornered square 0.6 1/3 (Of the axon) 1/20 (Of the axon) 01 (Parallel)

Circle 0.8 1/2 (Of the axon) 1/10 (Of the axon) 901 (Perpendicular)

Guy et al. (1989), Sherman and Brophy (2005) Arbogast and Margulies (1999), Lu et al. (2006), Shreiber et al. (2009) Chafi et al. (2009) McIlwaine et al. (2005)

0.15

Von Mises Strain (mm/mm)

Factor

0.12 0.09 0.06 0.03 0 -6.4 -4.8 -3.2 -1.6 0 1.6 Location (mm)

3.2

4.8

6.4

0.0371 0.0778 0.1185 0.1592 0.1976 0.0168 0.0575

0.0982 0.1389 0.1795

Fig. 3. The deformation and von Mises strain distribution of a coronal section of optic chiasm elevated inferior–superiorly to 5.6 mm. (a) Comparison of the deformation predicted by the FEM model with a coronal MRI scan of a patient with optic chiasmal compression (Evanson, 2009). (b) Lines used to calculate the strain values in (c). (c) The average cross-sectional von Mises strain along the left–right orientation. (d) The contour plot of the von Mises strain.

aspect. This could contribute to the selective injury of nasal fibres but this factor itself cannot explain the vertical cutoff of VFs in bitemporal hemianopia. The pressure values predicted by our simulations when the chiasm was elevated to 5.6 mm are compared to the experimental pressure measurements made by Kosmorsky et al. (2008) in Table 4. Kosmorsky et al. used only two pressure transducers, one located at the centre of the chiasm and the other at the periphery. Table 4 shows that our model's results broadly agree with the experimental data for cadaver no. 2. Kosmorsky et al. did not report the precise locations of their measurements, which leads to uncertainty in the appropriate peripheral values to use for the comparison. 3.2. The RVE model Fig. 4a shows the von Mises strain field of the axons and sheaths in both crossed and uncrossed RVE models. It can be seen that the strain distributions are very different. The maximum von Mises strain in the crossed model is higher than the uniform level in the uncrossed model. By performing an in vivo study of guinea pig optic nerve, Bain (1998) defined an axonal strain threshold of 0.06 to cause functional nerve injury at an axonal level. The axonal strains in our

Table 4 Comparison of the hydrostatic pressures predicted by the simulation and the averaged experimental measurements from Kosmorsky et al. (2008). (Unit: kPa). Location

Cadaver no. 1

Cadaver no. 2

Simulation

Central Peripheral

2.63 0.80

1.23 0

1.32 0.33

simulations were compared with this threshold and the injury and non-injury regions in the nerve fibres are plotted in different colours in Fig. 4b. In general, as expected, some part of the crossed nerve fibres were functionally injured while the uncrossed nerve fibres were relatively unaffected. In Fig. 5, the average von Mises strains and axonal strains of the cross-sections (with their normal along the nerve fibre length) of the axons and the myelin sheaths were plotted along the nerve fibre length. Under the same loading conditions, the strain distribution in crossed nerve fibres was more non-uniform and some parts of the crossed nerve fibres experienced higher strains than the uncrossed fibres. Fig. 6 shows the statistically significant factors (95% confidence level) for the four outputs in the parametric study, which were determined by the method of Lenth (1989) in Minitab.



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0.1828 0.1676 0.1523 0.1371 0.1218 0.1066 0.0913 0.0761 0.0608 0.0456

injuried uninjured Fig. 4. (a) The von Mises strain distribution in RVEs and (b) the injury and non-injury regions in the nerve fibres based on strain in the axonal direction. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

b

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

Crossed_Axon_1 Crossed_Axon_2 Uncrossed_Axon_1 0

0.5

Von Misesstrain (mm/mm)

Von Misesstrain (mm/mm)

a

0.14 0.12 0.1 0.08 0.06 0.02 0

1

0

The location along the nerve fibre segment (µm)

d

0.07 0.06 0.05 0.04 0.03 Crossed_Axon_1 Crossed_Axon_2 Uncrossed_Axon_1

0.02 0.01 0

0.5

1


Axonal strain (mm/mm)

Axonal strain (mm/mm)

c

Crossed_Sheath_1 Crossed_Sheath_2 Uncrossed_Sheath_1

0.04

0.07 0.06 0.05 0.04 0.03 Crossed_Sheath_1 Crossed_Sheath_2 Uncrossed_Sheath_1

0.02 0.01 0

0

0.5

1


0

0.5

1


Fig. 5. The average cross-sectional (a, b) von Mises strain and (c, d) axonal strain distribution of the axons and myelin sheaths as a function of the location along the nerve fibre. ‘Crossed_sheath_1’ in the legend means the ‘myelin sheath’ in the ‘crossed model’ at the ‘x1 direction’ according to Fig. 2.

The horizontal bars represent the effects of individual factors or interaction of these factors. Because one numerical simulation yields only one set of results, there are no replicates which are necessary to evaluate error in physical experiments. Lenth's method is specially designed for the study of numerical

simulations to overcome this issue. Crossing angle was ranked as either the first or the second most significant factor for these four outputs. Crossing angle was 0.21% and 4.52% less significant than the g-ratio for two outputs. However, for an individual person, the g-ratio will vary only slightly, or even remain constant (Sherman



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output: Axonal-A

Term

Term

output: Von-A B E D A BD DE AC AE AB C CD CE ADE ABE BE 0

0.002

0.004 Effect

E A AE DE D C CE ADE AD BE B BDE BD ABE AB BCE BC 0

0.006

output: Axonal-S

Term

Term

output: Von-S B E D A AE C BD BE ADE AC CE AB DE 0

0.004

0.008 Effect

0.012

0.002 0.004 0.006 0.008 0.01 Effect

0.016

E AE A B BE C CE ADE AD DE D ABDE ABD AC ACE ACD ACDE 0

0.005

0.01

0.015

Effect

Fig. 6. Comparison of the relative sensitivities of the strain predictions of the model to the input parameters and their combinations. (A): cross-sectional shape, (B): g-ratio, (C): sheath stiffness, (D): MECS stiffness and (E): crossing angle. Von-A and Von-S denote the maximum cross-sectional von Mises strain in the axon and sheath, respectively. Axonal-A and Axonal-S denote the maximum cross-sectional axonal strain in the axon and sheath, respectively. Only the statistically significant effects are shown.

and Brophy, 2005). We can conclude that, for any given g-ratio, the crossing angle is the most significant factor that affects a wide range of outputs of the model.

4. Discussion This study is the first to look into the biomechanics of chiasmal compression in different length scales. The simulation results were found to be within the scatter of the limited experimental results available, indicating that finite element modelling can be useful for analysing chiasmal compression. The numerical results showed that the strain distributions in the crossed nerve fibres were much more non-uniform than those in the uncrossed nerve fibres. Furthermore, the crossed nerve fibres experienced a higher local strain than the uncrossed fibres. This may explain the selective failure of crossed nerve fibres while uncrossed nerve fibres still function. Parametric studies showed that crossing angle is the leading factor that influences the strain differences in crossed and uncrossed nerve fibres. The different strain distributions in crossed and uncrossed nerve fibres neatly explain the vertical cutoff of the visual field defect in bitemporal hemianopia. It should be noted that the strain applied on the crossed and uncrossed RVE models was the same, using a value which was extracted from the central part of the chiasm. Given that this region generally experiences higher strains than the peripheral part, even greater differences of strain distribution should exist in some of the nasal and temporal nerve fibres as the crossing nasal fibres tend to pass through the central region of the chiasm while the uncrossed temporal fibres pass further to the periphery. Further developments of this model will incorporate

more detailed nerve fibre distributions as relevant histological data becomes available. In the simulation, the chiasm was elevated to 5.6 mm as this degree of elevation has been shown to cause bitemporal hemianopia (Ikeda and Yoshimoto, 1995). Unfortunately, the degree of elevation of the chiasm was not measured in the experiment performed by Kosmorsky et al. (2008) (personal communication with the authors). From the volume data reported, cadaver no. 1 appears to have had a much larger elevation compared to cadaver no. 2. This may explain why the results of our simulation agree closely with the measurements from one cadaver but not so closely with those of the other (see Table 4). Nevertheless, the pattern of pressure distribution was nominally the same, i.e. the central aspect of the chiasm bore significantly higher pressure than the peripheral aspect. As material properties and the geometry of human chiasms vary between individuals and, considering the difficulties of performing experiments in vivo, we believe our simulation is a reasonable approximation to what happens in vivo. Ideally, the RVE models should be validated by experiments. However, the paucity of experimental data of nerve compression on axon scale has made the validation of our model infeasible. The techniques used in the RVE models were tested by the authors via application to another analogous problem, albeit not in soft tissue. Specifically, the software and procedure in this research were applied to model a composite woven structure reported in the literature (Wang et al., 2007) and was found to successfully reproduce the results reported in the reference. Although the exact mechanism of conduction failure of axons is still not well understood, the von Mises strain and axonal strain reported in this paper have been shown to be highly relevant to functional nerve injury (Chatelin et al., 2011; Cloots et al., 2011;



Hui et al., 2008). The absolute values of axonal strain obtained in our simulations were similar to the strain threshold of axonal dysfunction found by Bain (1998). However, given the complexity of this biological system, the actual threshold for axonal functional injury may vary widely among individuals. Therefore, the threshold used here is only indicative of the potential consequences of the strain differences in crossed and uncrossed nerve fibres. Higher fidelity modelling, incorporating more accurate material properties and chiasmal histology, will be required to draw more accurate quantitative conclusions on this injury mechanism. In conclusion, we believe that the use of multi-scale FEM sheds significant light on the basic mechanism of bitemporal hemianopia and has the potential to allow clinicians to modify their clinical management of patients with temporal visual field defects. This methodology can also be used to investigate other types of disease caused by nerve compression.

Conflict of interest statement None of the authors has a conflict of interest to disclose.

Acknowledgements The authors gratefully acknowledge Dr. Murat Tahtali and Thomas P. Lillicrap for the valuable discussions.

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