# Predicting the Aqueductal Cerebrospinal Fluid Pulse: A Statistical Approach
## 4 Discussion  
The principal finding of the study is that the motion of the aCSF pulse can be accurately predicted using the arterial, venous and cCSF flow rates in the neck and a simple linear regression model (i.e., GLM). An exemplificative typical subject (subject 5005) is shown in Figures 1 and 2, where the arterial, venous, cCSF (Figure 2G), aCSF (Figure 2H) and predicted aCSF (Figure 2H) flow rate curves are shown. The venous morphology is also shown, with the subject's coronal TOF MIP (Figure 1C), where the \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & & **Arterial Flow Rate** & **Venous Flow Rate** & **cCSF Flow Rate** & **aCSF Flow Rate** & **aCSF Flow Rate** \\  & **Mean r Value (SD)** & **Mean r Value (SD)** & **Mean r Value (SD)** & **Mean r Value (SD)** & **Mean r Value (SD)** \\ \hline Arterial flow rate & - & - & - & - & - & - \\ Venous flow rate & -0.601 (0.293) & - & - & - & - \\ cCSF flow rate & -0.837 (0.126) & 0.621 (0.178) & - & - & - \\ aCSF flow rate & -0.450 (0.406) & 0.763 (0.266) & 0.539 (0.384) & - \\ \hline cCSF—cervical cerebrospinal fluid (CSF) flow rate; aCSF—aqueductal CSF flow rate (i.e., CSF in the aqueduct of Sylvius). & & & & & & \\ \end{tabular}
\end{table} 
Table 3: Mean signal correlation r values for all the subjects aggregated together. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & & **Arterial** & **Venous** & **cCSF** & **Model** & **Model** & **Model** & **Arterial** & **Venous** & **cCSF** \\ \hline
**Subject** & **Intercept** & **Coef. ** & **Coef. ** & **Coef. ** & **p-value** & **MSE** & **Adj. ** & **r2** & **RIW** & **RIW** & **RIW** \\ \hline
5002 & 0.1941 * & 0.0444 & 0.0596 & 0.0805 & \(<\)0.001 & 0.002 & 0.787 & 21.905 & 45.036 & 33.059 \\
5004 & 0.2413 & \(-\)0.0006 * & 0.0204 & \(-\)0.0169 * & \(<\)0.001 & 0.002 & 0.656 & 31.312 & 57.687 & 11.001 \\
5005 & 0.1716 & 0.0039 & 0.0179 & 0.0206 & \(<\)0.001 & \(<\)0.001 & 0.884 & 11.208 & 72.520 & 16.272 \\
5006 & 0.5092 & \(-\)0.0150 & 0.0287 & \(-\)0.0178 & \(<\)0.001 & \(<\)0.001 & 0.894 & 33.611 & 51.167 & 15.221 \\
5007 & 0.1438 * & 0.0090 * & 0.0247 & 0.0200 * & \(<\)0.001 & \(<\)0.001 & 0.876 & 6.841 & 84.902 & 8.257 \\
5010 & \(-\)0.0944 * & 0.0150 & 0.0078 * & 0.0455 & \(<\)0.001 & 0.006 & 0.587 & 10.934 & 25.313 & 63.753 \\
5011 & 0.16 symmetric IJVs are well visible. Given that the cranium is a rigid container filled with incompressible gels (i.e., the brain parenchyma) [18] and fluids (i.e., blood, CSF and interstitial fluid), it is perhaps not unexpected that the crani spinal fluid system should behave in a linear manner. Like all hydraulic systems involving rigid vessels and incompressible fluids, one would expect expansion in one fluid component within the system to have an instantaneous effect on the other fluid components in the system. Therefore, it is to be expected that motion of the aCSF will be influenced by the characteristics of the fluids entering and leaving the cranium via the neck. Indeed, it has been shown that the volumetric changes in the various fluid compartments of the intracranial space can be computed using fluid flow rates in the vessels of the neck [20]. As such, our finding that the GLM can be used to predict aCSF appears to support the opinion that the intracranial fluidic system behaves in a broadly linear manner. In our study, we found that the median adjusted \(r^{2}\) value for the model was 0.857. This implies that, on average, approximately 86% of the variance in the aCSF pulse could be explained by the cervical arterial, venous and cCSF flow rates. While this demonstrates that the behaviour of the system is broadly linear, it is noticeable that about 14% of the variance in the cCSF remained unexplained, suggesting that other, possibly non-linear, factors may influence the dynamics of the cCSF pulse.