Embracing uncertainty in cerebrospinal fluid dynamics: A Bayesian approach to analysing infusion studies

Abstract

Introduction: Cerebrospinal fluid (CSF) infusion test analysis allows recognizing and appropriately evaluating CSF dynamics in the context of normal pressure hydrocephalus (NPH), which is crucial for effective diagnosis and treatment. However, existing methodology possesses drawbacks that may compromise the precision and interpretation of CSF dynamics parameters.

Research question: This study aims to circumvent these constraints by introducing an innovative analysis method grounded in Bayesian inference.

Material and methods: A single-centre retrospective cohort study was conducted on 858 patients who underwent a computerized CSF infusion test between 2004 and 2020. We developed a Bayesian framework-based method for parameter estimation and compared the results to the current, gradient descent-based approach. We evaluated the accuracy and reliability of both methods by analysing erroneous prediction rates and curve fitting errors.

Results: The Bayesian method surpasses the gradient descent approach, reflected in reduced inaccurate prediction rates and an improved goodness of model fit. On whole cohort level both techniques produced comparable results. However, the Bayesian method holds an added advantage by providing uncertainty intervals for each parameter. Sensitivity analysis revealed significance of the CSF production rate parameter and its interplay with other variables. The resistance to CSF outflow demonstrated excellent robustness.

Discussion and conclusion: The proposed Bayesian approach offers a promising solution for improving robustness of CSF dynamics assessment in NPH, based on CSF infusion tests. Additional provision of the uncertainty measure for each diagnostic metric may perhaps help to explain occasional poor diagnostic performance of the test, offering a robust framework for improved understanding and management of NPH.

Keywords: Bayesian inference; Brain physics; Cerebrospinal fluid dynamics; Infusion studies; Intracranial pressure; Normal pressure hydrocephalus.

Fig. 1

A diagram showing the curve fitting approach. Data from infusion study (top left) is combined with a model of CSF dynamics (bottom left) and the discrepancy between the two curves can be evaluated (right). P_m is the measured intracranial pressure [mmHg], $\hat{P}\left(t\right)$ is the intracranial pressure predicted by the model [mmHg], I_inf is the infusion rate [mL/min], I_f is the CSF formation rate [mL/min], R_out is the resistance to CSF outflow [mmHg/mL/min], P_ss is the pressure in sagittal sinus [mmHg], C is the cerebrospinal compliance [mL/mmHg], and P₀ is the reference pressure [mmHg]. ICP – intracranial pressure.

Fig. 2

A schematic representation of the Bayesian approach. Apart from input data and model, this method utilises information about the parameter value distributions. The main advantage is that it produces value distributions instead of point estimates. Therefore, one can evaluate uncertainty (here shown as standard deviation) for each parameter prediction. For simplicity, only two parameters (E, R_out) are shown. Posterior ∝ Likelihood × Prior. R_out – resistance to CSF outflow, E – elasticity coefficient.

Fig. 3

Histograms showing parameter value distributions for each method. Resistance to CSF outflow (top left), elasticity coefficient (top right), CSF production rate (bottom left), and reference pressure (bottom right). The results from each method have been overlaid. Bayesian method is depicted in solid black, and the gradient descent method is depicted in semi-transparent orange.

Fig. 4

The figure presents a scatter plot that juxtaposes the pressure-volume curves derived from two distinct methodologies for a single patient, with the y-axis being depicted in a semi-logarithmic scale. In this illustrative instance, the Bayesian method exhibits a coefficient of determination (R²) of 0.99, while the gradient descent method yields an R² value of 0.93. This disparity indicates a marginally superior fit when employing the parameters obtained using the Bayesian approach. It is important to note that the volume represented here does not pertain to the infused fluid volume; rather, it characterizes the change in the volume of the craniospinal space in response to the infusion. P – measured intracranial pressure, Pss – sagittal sinus pressure, Pb – baseline pressure.

Fig. 5

Histograms showing goodness of fit diagnostics for both methods: Root-mean-square error of the model fit to data (left), and coefficient of determination of the pressure-volume curve (right). The results from each method have been overlaid. Bayesian method is depicted in solid black, and the gradient descent method is depicted in semi-transparent orange.

Fig. 6

Histograms showing standard deviations for parameter value distributions for the Bayesian method. Resistance to CSF outflow (top left), elasticity coefficient (top right), CSF production rate (bottom left), and reference pressure (bottom right).