The correct engineering model to use in the current study would depend on the type of boundary condition at the attachment points i.e. fixed, roller or pinned . The use of a fixed condition would assume that the angle at the endpoint is zero with respect to the horizontal  (i.e. the attachment does not bend). As the angle at the attachment endpoints are non-zero, the most suitable model would require the equivalent of pinned endpoints i.e. the simply supported beam. The use of a roller condition would yield the same result as the pinned condition in this instance, as the same mathematical assumptions with regard to the beam deflection are used .
The modelling requires an estimate of the force applied across the sinus walls in the controls. Thus, we need to know the CSF and venous pressures. The normal ICP and sinus pressure and transmural pressures were obtained from the literature.
The modelling also requires an accurate estimate of the deflection of the walls. It became obvious that direct measurement of the deflection was too inaccurate. The resolution of the post contrast 3D T1 images is 0.85 mm with each pixel being 0.72 mm2. The average deflection in the controls was 0.48 mm (half a pixel) but the combined reduction in the sinus area was 6.6 mm2 and this represented approximately 9 pixels. Given the attachment points are relatively fixed, the deflection could be estimated from the change in the cross-sectional area. In a cadaver study of adults mean age 39 years, the sagittal sinus just above the Torcular was found to be an isosceles triangle of mean width 11.6 mm and height 8.3 mm with a 48 mm2 cross-sectional area and a calculated average free wall length of 10.1 mm. . These measurements are very similar to the mean width, height, triangle area and free wall length found in the present study, with the difference between the cadavers and control patients being 8%, 5%, 7% and 1% respectively. Sensitivity analysis suggests an 8% error in either the area or length measurement would have a minimal effect on the final outcome of this study. There is no transmural pressure gradient in a cadaver, meaning the stress free state for the normal sinus walls is for them to be straight. Thus, they were compared to the calculated triangle sinus area. The length measurements in MS were not significantly different to the controls, indicating multiple sclerosis is unlikely to alter the fixed point positions. As a pressure gradient is equivalent to a force which is equally applied in all directions, the expected deflection when compared to the original straight wall position should approximate a circle segment. According to Laplace’s law, when a pressure is applied to a thin membrane, the volume change will be accommodated by the smallest possible change in membrane area because this would give the minimum energy state. The smallest surface area for a given volume is always a circle segment. The same mechanism underlies why cerebral aneurysms develop as circle segments . The deflection as seen in Fig. 2 was calculated from the area between the free wall or chord and the curved line using the circle Eq. (1).
In the controls, a small smooth deflection averaging 0.48 mm develops from a 4 mmHg pressure gradient, giving a Young’s modulus averaging 6.1 MPa. The dural elastic modulus has been measured in humans using fresh samples with the values varying from 29.4 MPa , 44 MPa , up to 61 MPa . The dura from these studies comes from various regions of the skull but to our knowledge the wall of the sinus has never been sampled in humans and it is possible that the differing regions may be optimised for varying degrees of absolute strength rather than flexibility. In pigs, the dura over the inner table of the skull has been measured to be between 8 and 16 MPa  i.e. the stiffness is somewhat less than that in humans. In pigs, the longitudinal and circumferential SSS stiffness within the occipital region has been measured and found to be 58.1 ± 17.2 MPa longitudinally with the circumferential figure being 3.0 ± 0.7 MPa . The circumferential stiffness in pigs compares to the circumferential stiffness of 6.1 MPa we estimated, again suggesting human dura is somewhat stiffer than porcine. The difference between the longitudinal and circumferential stiffness in the SSS is due to the collagen alignment. The fraction of collagen within the pig sinus wall is 84%. The collagen is randomly directed in that portion of the sinus adjacent to the bone but is longitudinally directed in those portions adjacent to the subarachnoid space . This suggests the free walls of the sinus are designed to be stiff in the longitudinal direction but flexible at 90 degrees to this direction.
The simplest explanation for the 20% larger sinus area in MS is an increase in sinus pressure compared to the subarachnoid pressure i.e. the transmural pressure. As discussed in the introduction, a decrease in the sagittal sinus transmural pressure would require an alternate route for CSF to be absorbed other than through the arachnoid granulations. The capillary bed drained by the deep venous system could possibility account for the CSF drainage but as noted, the straight sinus is increased in size in MS by 13% suggesting that both the superficial and deep drainage systems are similarly affected. In favour of the elevated pressure hypothesis, the transverse sinuses in MS patients have been found to have an average 39% effective stenosis in area and there was a 62% increase in jugular bulb height , with both of these findings suggested to increase venous pressure. However, this pressure increase is probably modest at best. Mathematical modelling of a 7.5 mm diameter cerebral vessel suggests a pressure drop across a stenosis of 40–50% by area would be only 1–2 mmHg . A 38% area stenosis (almost identical to the MS stenosis) in the sagittal sinus was estimated to increase the venous pressure by only 0.7 mmHg . Given the jugular bulb pressure between MS and controls was noted to be unchanged , the sagittal sinus pressure may be possibly increased by 1–2 mmHg at most in MS but not the 2.8 mmHg increase as required by the modelling. However, even a 1–2 mmHg increase would be unlikely to affect the transmural pressure because although the normal ICP is 11.5 mmHg , the ICP measured at lumbar puncture in 32 MS patients was 12.9 ± 3.3 mmHg (1.4 mmHg higher), with the MS ICP being identical to normal pressure hydrocephalus patients . Normal pressure hydrocephalus patients are known to have mildly elevated ICP which is still within normal range . Thus, any mild increase in venous pressure in MS would only match the mild increase in ICP, giving a normal transmural pressure overall. Thus, the venous pressure theory causing sinus size change is probably not feasible.
The other possibility is a change in the structure of the sinus wall. The modelling suggests either a 3.3 times increase in wall stiffness or a 49% increase in wall thickness or perhaps a combination of both. By solving Eq. (3) for both the elasticity and wall thickness (using a normal pressure) in both the MS patients and controls and calculating the combined change in these two variables by division, it can be shown that the change in elasticity multiplied by the change in wall thickness cubed is equal to 3.3. Indicating which combinations of each variable would be possible.
ΔE x ΔD3 = 3.3 (5)
In order to clarify the possibilities further we require an independent measurement. In MS the time taken for the arterial pulsation to pass into the SSS was reduced by 35% compared to controls . This represents a measurement of the pulse wave velocity between the arteries and the venous system via the subarachnoid space including the spinal canal. The square of the pulse wave velocity within a vessel is equal to the elastic modulus multiplied by the wall thickness divided by two times the fluid density multiplied by the radius  i.e.
PWV2 = E x D / ρ x R (6)
The blood density is a constant and the difference in the sagittal sinus hydraulic diameter (proportional to the radius) in this cohort has been previously measured , which would allow the radial difference to be estimated. If the dura mater of the entire system (spinal canal and sinus walls) were similarly affected, then we can solve Eq. (6) for both the controls and MS patients and by division show that.
ΔE x ΔD = 2.7 (7)
The mechanical response of the spinal canal and sinus walls have been shown to be similarly affected by MS with spinal canal pulsation propagation reduced by 40% and the venous sinus propagation by 50% . Solving Eqs. (5) and (7) simultaneously gives a change in wall stiffness of 3 times normal and wall thickness of 1.11 times normal. Is a 3 fold increase in circumferential wall stiffness feasible? As previously discussed, the longitudinal stiffness is much higher (approximately 20 fold) than the circumferential stiffness in the SSS due to the longitudinal orientation of the fibers. Reorientation of the fibers into a random distribution could increase the circumferential stiffness by enough to make the findings feasible. There is chronic inflammation of the vein walls in MS. Forty seven percent of MS patients showed evidence of dural inflammation, which was equally distributed across all age groups i.e. it is likely chronic . Chronic inflammation could cause the fibers to be reoriented secondary to remodeling and scarring. There is a change in type 1 collagen in the jugular veins in MS with microcalcification deposition , similarly suggesting structural wall changes. Finally, there is a 10 fold increased risk of MS in Ehlers-Danlos syndrome (EDS) patients . EDS is characterized by altered collagen synthesis and enzyme dysfunction . EDS is usually thought to be associated with decreased vascular stiffness, however, one paper has suggested an increased arterial stiffness occurs compared to controls . Cell culture of fibroblasts indicates that there is a significant reduction in directional fiber orientation in EDS compared to normal , suggesting a random distribution of fibers in the SSS could increase circumferential stiffness but decrease the longitudinal stiffness.