Abstract
Stimulation through a surface electrode installed on the eyelid can activate the visual anterior pathways. In order to study the underlying physiological phenomena, a macroscopic axisymmetric model of the orbit has been elaborated. By building an appropriate mesh on this model and under some boundary conditions, the Finite Element Method can be applied to compute the electric field generated by the electrode. A model of a retinal ganglion cell axon starting from the retina and travelling through the optic nerve has been used to illustrate the fiber reaction to such a potential distribution. Under those simulation conditions, the optic nerve head seems to be the first structure to be activated by the stimulation.
Introduction
The last decade has seen several attempts to restore visual
perceptions in late blind patients through functional electrical stimulation.
Whether the system is based on retinal implants [1]-[2] or optic nerve direct
stimulation [3], functionality of retinal ganglion cells must be assessed.
Optic nerve stimulation through a surface eyelid electrode has been used as a
screening test for candidates to an optic nerve visual prosthesis implantation
[4]. This approach has proven harmless, efficient and easy to implement. Moreover,
in conjunction with more classical techniques such as evoked potentials, this
method could turn out as a useful tool to fundamental studies about the visual
system.
However, such use of this method requires a good knowledge
of the involved phenomena, the main issue being the location of the initial
activation. In order to investigate this, a modeling approach is essential.
The classical modeling procedure known as the two-step
approach can be applied to this problem. As a first step the macroscopic
electric potential distribution is computed, while step 2 involves calculation
of the ensuing fiber response. The purpose of the present work is to achieve
step 1. From then we develop a macroscopic model of the orbit and automatic
methods to build a finite element mesh of the domain in order to obtain the
electric field distribution generated by the electrode.
To briefly illustrate the possible fiber response to the extracellular potential, we set up an elementary model
including a representative retinal ganglion cell (RGC) axon. Thus the studied
excitable structure is the axon in its unmyelinated
path upon the retina, the sharp turn at the optic nerve head and the
rectilinear trajectory through the optic nerve trunk where the axon
becomes myelinated.
The location of the initial action potential resulting from the stimulation
will be determined on this simplified model.
Methods
1.
The multi-layer axisymmetrical
orbit model
The
human orbit displays a rough rotational symmetry about the visual axis. The main
exceptions reside at the optic nerve whose axis forms a small but non-zero
(about 10°) angle with the visual axis, and the six extra-ocular eye muscles
which are not continuous along the azimuthal
direction. Neglecting these irregularities, we model the domain in an axisymmetrical way, as shown in Fig. 1 (left part). We
account for the general tissue topology by considering several layers defined
by their frontiers.
Figure 1: Left: axisymmetrical
multi-layer
model of the orbit. Right: mesh with 500 points.
2.
Mesh generation
In order
to solve the differential equations describing the electric potential
distribution within the domain by a Finite-Elements technique, we dicretize the domain by adopting triangles as finite
elements.
The mesh is generated according to an automatic method
which dynamically generates nodes on segments in order to achieve a preset
level of local size for the element. This size distribution directly depends on
the accuracy requirement of the FEM method.
3.
Computation of the electric field
Each subdomain of the model is considered as purely resistive. The conductivity values can be found in Table 1.
The effect of the electrode on the domain is modeled as a proper boundary condition (Dirichlet condition) where the metal is in contact with the skin; so the electrode is seen as a voltage source external to the computation domain. A second constant Dirichlet condition is applied on the lower part of the domain and homogeneous Neumann conditions on the rest of the boundary.
The electric potential distribution is calculated by a mixed FEM-Fourier technique developed by Parrini et al [5].
|
tissue |
conductivity [Wm]-1 |
Remark |
|
aqueous |
1.6 |
as for the CSF |
|
choroid |
0.38 |
|
|
cornea |
0.26 |
|
|
ciliar body |
0.26 |
as for the cornea |
|
vitreous |
1.6 |
as for the CSF |
|
lens |
0.11 |
|
|
fat |
0.04 |
|
|
iris |
0.29 |
mean betw. choroid & lens |
|
tears |
1.6 |
as for the CSF |
|
CSF |
1.6 |
|
|
muscle |
0.2 |
|
|
optic nerve |
0.34 |
|
|
bone |
0.015 |
|
|
retina |
0.34 |
as for the nerve |
|
sclera |
0.03 |
|
|
tarsa |
0.26 |
as for the cornea |
Table 1: Conductivity values for
the orbit subdomains.
4.
Model of a representative RGC axon
As we want
to model a typical RGC axon, we need a suitable model for both myelinated and unmyelinated optic
nerve fibers and an adequate fiber trajectory within the domain. Starting with
a model for human peripheral nerve fibers, Parrini et
al. [6] propose a model adapted to experimental results from electrical
stimulation of the human optic nerve [3] and from propagation velocity measured
on primates [7]. We adopt this model to represent our myelinated
optic nerve fiber.
Concerning the unmyelinated
portion of the fiber, we consider the same membrane dynamics, but as suggested
by Rattay [8], we discretize
spatially the partial differential equation describing the relationship between
current and voltage along the fiber. This leads to represent the spatially
distributed fiber as a set of equal size connected cylinders. The length of
these cylinders has been fixed to 50mm, which yields a rather good compromise between
computational time and precision in the determination of the fiber activation
threshold.
The axon is considered as activated when an induced
action potential propagates until its extremity in the optic nerve subdomain.
Figure 2: Potential distribution
within the domain.
Results
Fig. 1 (right part) shows an example of the triangulation generated with 500 points. A potential distribution within the orbit computed for a mesh of 4000 points is shown at Figure 2.
Figure 3 gives the corresponding potential distribution along the fiber. The asterisk indicates the initial location of the propagating action potential. It corresponds to the optic nerve head.
Figure 3: Extracellular
potential distribution along the fiber. The asterisk indicates the location of
the initial activation.
Discussion
The automatic triangulation technique can produce high quality meshes allowing to compute electric potential distribution.
Concerning the retinal cell response to the electric potential distribution, many potential activation sites should be considered. A simulation of this elementary optic nerve fiber model shows the excitation occurring at the optic nerve head, rather than in the portion located in the retina, closer to the electrode.
The effect of different conductivities within the domain and various boundary conditions on the fiber response should be investigated as an assessment of the model robustness. Several fibers trajectories should be studied as well.
At last, the excitable structures model could be rather completed by including dendrites and soma of the RGC, each of these portions being described by a suitable model (e.g. Fohlmeister et al. [9]). The description of the neural network between photoreceptors and RGC could be added too.
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Acknowledgments: Support: CEU ‘Esprit’ grant # 22 527 and FMSR grant #
3.4584.98.