CSF dynamic analysis of a predictive pulsatility-based infusion test for normal pressure hydrocephalus

Abstract

Disturbed cerebrospinal fluid (CSF) dynamics are part of the pathophysiology of normal pressure hydrocephalus (NPH) and can be modified and treated with shunt surgery. This study investigated the contribution of established CSF dynamic parameters to AMP_mean, a prognostic variable defined as mean amplitude of cardiac-related intracranial pressure pulsations during 10 min of lumbar constant infusion, with the aim of clarifying the physiological interpretation of the variable. AMP_mean and CSF dynamic parameters were determined from infusion tests performed on 18 patients with suspected NPH. Using a mathematical model of CSF dynamics, an expression for AMP_mean was derived and the influence of the different parameters was assessed. There was high correlation between modelled and measured AMP_mean (r = 0.98, p < 0.01). Outflow resistance and three parameters relating to compliance were identified from the model. Correlation analysis of patient data confirmed the effect of the parameters on AMP_mean (Spearman’s ρ = 0.58–0.88, p < 0.05). Simulated variations of ±1 standard deviation (SD) of the parameters resulted in AMP_mean changes of 0.6–2.9 SD, with the elastance coefficient showing the strongest influence. Parameters relating to compliance showed the largest contribution to AMP_mean, which supports the importance of the compliance aspect of CSF dynamics for the understanding of the pathophysiology of NPH.

Appendix

The volume change of CSF over time can be written as:

$$\frac{{{\text{d}}V}}{{{\text{d}}t}} = I_{\text{f}} - I_{\text{a}} + I_{\text{ext}} .$$

(15)

I _f and I _a are equal at ICP_r, and thus, with the assumption that formation is pressure independent [4], I _f can be described using Eq. (4):

$$I_{\text{f}} = \frac{{\left( {{\text{ICP}}_{\text{r}} - P_{\text{d}} } \right)}}{{R_{\text{out}} }}.$$

(16)

Equation (15) can then be rewritten as:

$$\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{{\left( {{\text{ICP}}_{\text{r}} - P_{\text{d}} } \right)}}{{R_{\text{out}} }} - \frac{{\left( {{\text{ICP}} - P_{\text{d}} } \right)}}{{R_{\text{out}} }} + I_{\text{ext}} = \frac{{\left( {{\text{ICP}}_{\text{r}} - {\text{ICP}}} \right)}}{{R_{\text{out}} }} + I_{\text{ext}} .$$

(17)

The derivative of ICP with respect to time [Eq. (3)] can thus be expressed as:

$$\frac{\text{dICP}}{{{\text{d}}t}} = \frac{{k\left( {{\text{ICP}} - P_{0} } \right)}}{{R_{\text{out}} }}\left( {{\text{ICP}}_{\text{r}} - {\text{ICP}} + R_{\text{out}} I_{\text{ext}} } \right).$$

(18)

The variable substitution x = 1/(ICP − P ₀), or ICP = 1/x + P ₀, gives:

$$\frac{\text{dICP}}{{{\text{d}}t}} = \frac{\text{dICP}}{{{\text{d}}x}}\frac{{{\text{d}}x}}{{{\text{d}}t}} = \frac{ - 1}{{x^{2} }}\frac{{{\text{d}}x}}{{{\text{d}}t}}.$$

(19)

In Eq. (18) this results in:

$$\frac{ - 1}{{x^{2} }}\frac{{{\text{d}}x}}{{{\text{d}}t}} = \frac{k}{{xR_{\text{out}} }}\left( {{\text{ICP}}_{\text{r}} - \left( {\frac{1}{x} + P_{0} } \right) + R_{\text{out}} I_{\text{ext}} } \right).$$

(20)

Rearranging this equation gives:

$$\frac{{{\text{d}}x}}{{{\text{d}}t}} + \frac{kx}{{R_{\text{out}} }}\left( {{\text{ICP}}_{\text{r}} - P_{0} + R_{\text{out}} I_{\text{ext}} } \right) = \frac{k}{{R_{\text{out}} }}.$$

(21)

Equation (21) can be solved using the integrating factor (IF) method, where

$${\text{IF}}\frac{{{\text{d}}x}}{{{\text{d}}t}}{\text{ + IF}}\frac{kx}{{R_{\text{out}} }}\left( {{\text{ICP}}_{\text{r}} - P_{0} + R_{\text{out}} I_{\text{ext}} } \right) = {\text{IF}}\frac{k}{{R_{\text{out}} }}$$

(22)

with the appropriate IF results in:

$$\frac{\text{d}}{{{\text{d}}t}}\left( {{\text{IF}}x} \right) = {\text{IF}}\frac{k}{{R_{\text{out}} }}$$

(23)

and thus:

$$x = \frac{{\int {{\text{IF}}\frac{k}{{R_{\text{out}} }}{\text{d}}t + A} }}{\text{IF}}$$

(24)

where A is a constant. In this case, the appropriate IF is:

$${\text{IF}} = \exp \left( {\int {\frac{{k\left( {{\text{ICP}}_{\text{r}} - P_{0} + R_{\text{out}} I_{\text{ext}} } \right)}}{{R_{\text{out}} }}} {\text{d}}t} \right) = {\text{e}}^{{tk\left( {I_{\text{ext}} + {{\left( {{\text{ICP}}_{\text{r}} - P_{0} } \right)} \mathord{\left/ {\vphantom {{\left( {{\text{ICP}}_{\text{r}} - P_{0} } \right)} {R_{\text{out}} }}} \right. \kern-0pt} {R_{\text{out}} }}} \right)}} .$$

(25)

For constant infusion (I _ext = constant) starting at ICP_start (i.e., ICP(0) = ICP_start), the following equation for ICP as a function of time is then derived:

$${\text{ICP}}(t) = \frac{{\left( {R_{\text{out}} I_{\text{ext}} + {\text{ICP}}_{\text{r}} - P_{0} } \right)\left( {{\text{ICP}}_{\text{start}} - P_{0} } \right)}}{{{\text{ICP}}_{\text{start}} - P_{0} + \left( {R_{\text{out}} I_{\text{ext}} + {\text{ICP}}_{\text{r}} - {\text{ICP}}_{\text{start}} } \right)e^{{ - tk\left( {{{I_{\text{ext}} + \left( {{\text{ICP}}_{\text{r}} - P_{0} } \right)} \mathord{\left/ {\vphantom {{I_{\text{ext}} + \left( {{\text{ICP}}_{\text{r}} - P_{0} } \right)} {R_{\text{out}} }}} \right. \kern-0pt} {R_{\text{out}} }}} \right)}} }} + P_{0} .$$

(26)