The architecture of the optic nerve SAS has been shown to have numerous trabeculae, septae and pillars.
11,45,46 Based on the work of Killer et al.
46 as well as using the closed form solutions for permeability given by Westhuizen and Du Plessis,
47 the lumped resistance (
Display Formula\({R_{58}}\)) offered to the CSF flow by this meshwork is estimated as follows. Since the mean flow rate of the CSF flowing into and out of the optic nerve SAS is not known,
Display Formula\({R_{58}}\), is estimated assuming the flow can be approximated as flow through a porous medium described by Darcy's law:
\begin{equation}\tag{14}R = {{\mu L} \over {kA}}\end{equation}
where
Display Formula\(\mu \) is the viscosity of the CSF,
L is the length over which the pressure drop occurs,
k is the permeability and
A is the area of cross-section of the optic nerve SAS. The viscosity of the CSF is assumed to be equal to that of water.
48,49 The length of the optic nerve SAS is reported to be 40 to 50 mm and is divided into four main regions: intraocular, midorbital, intracanalicular, and intracranial.
50–52 The cross-sectional area of each region is estimated from the optic nerve sheath diameter values reported at distances of 3, 9, and 15 mm behind the globe.
53 The optic nerve diameter and optic nerve sheath diameter values reported for the control group in Ref.
53 are used to estimate the area of cross-section within each region. Since the structure and density of the arachnoid trabeculae in the three segments varies,
45,46,54,55 the resistances for each segment were calculated separately based on their respective permeabilities. The permeability is estimated by assuming an idealized geometry wherein the trabeculae were considered to be straight cylindrical pillars, extending normally from the arachnoid layer to the pia layer.
48 This idealized representation of the SAS can be compared to a bed with unidirectional fibers, for which the analytical solutions for longitudinal and transverse permeability exist.
47 The longitudinal and transverse permeabilities for each of the three segments are calculated using
Equations 15 and 16 and the corresponding resistance is calculated using
Equation 14.
\begin{equation}\tag{15}{k_l} = {{\left[ {\pi + 2.157\left( {1 - \varphi } \right)} \right]{\varphi ^2}{r^2}} \over {48{{\left( {1 - \varphi } \right)}^2}}}\end{equation}
\begin{equation}\tag{16}{k_t} = {{\pi \varphi {{\left( {1 - \sqrt \varphi } \right)}^2}{r^2}} \over {24{{\left( {1 - \varphi } \right)}^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-1.2pt}\!\lower0.7ex\hbox{$2$}}}}}}\end{equation}
where
Display Formula\({k_l}\) and
Display Formula\({k_t}\) are the longitudinal and transverse permeabilities, respectively,
Display Formula\(\varphi \) is the porosity and
r is the cross-sectional radius of the fibers. The porosity for the intracranial SAS is reported to be 0.99 in some studies.
48,56 However, since an exact value for the porosity of the optic nerve SAS is not known, the porosity is varied from 0.5–0.9. Based on the work of Killer et al.,
46 the bulbar region is characterized by round trabeculae with their profiles varying from 5 to 7 μm. An average diameter of 6 μm is assumed to estimate the resistance of this region. The midorbital region has large perforations interspersed with broad septae with an average diameter of 20 μm, while the intracanalicular region has mostly continuous SAS with pillars of approximately 25 μm diameter. Using the values for the anatomy of the optic nerve SAS and the radii of the trabeculae, septae and pillars, the resistance values for the three segments were estimated. Since the porosity values for each segment are not currently known, the lumped resistance (
Display Formula\({R_{58}}\)) would not be a simple sum of the individual resistances of each segment at one particular porosity. Based on the histologic study carried out by Killer et al.
46 the porosity of the optic nerve SAS increases from the bulbar segment towards the intracanalicular segment. Due to the difficulty associated with characterizing the porosity of each segment,
Display Formula\({R_{58}}\) is varied from its possible estimated minimum value to its maximum value in order to see its effect on the resulting RSAS pressure. The minimum value of 6.752 × 10
9 Pa s/m
3 (0.844 mm Hg min/mL) is obtained by adding the longitudinal resistances at a porosity of 0.9 in all segments and similarly, the maximum value of 2.409 × 10
12 Pa s/m
3 (301.162 mm Hg min/mL) is obtained by adding the transverse resistances at a porosity of 0.5.