Image analysis techniques for in vivo quantification of cerebrospinal fluid flow

Abstract

Over the past decade, there has been a tremendously increased interest in understanding the neurophysiology of cerebrospinal fluid (CSF) flow, which plays a crucial role in clearing metabolic waste from the brain. This growing interest was largely initiated by two significant discoveries: the glymphatic system (a pathway for solute exchange between interstitial fluid deep within the brain and the CSF surrounding the brain) and meningeal lymphatic vessels (lymphatic vessels in the layer of tissue surrounding the brain that drains CSF). These two CSF systems work in unison, and their disruption has been implicated in several neurological disorders including Alzheimer’s disease, stroke, and traumatic brain injury. Here, we present experimental techniques for in vivo quantification of CSF flow via direct imaging of fluorescent microspheres injected into the CSF. We discuss detailed image processing methods, including registration and masking of stagnant particles, to improve the quality of measurements. We provide guidance for quantifying CSF flow through particle tracking and offer tips for optimizing the process. Additionally, we describe techniques for measuring changes in arterial diameter, which is an hypothesized CSF pum** mechanism. Finally, we outline how these same techniques can be applied to cervical lymphatic vessels, which collect fluid downstream from meningeal lymphatic vessels. We anticipate that these fluid mechanical techniques will prove valuable for future quantitative studies aimed at understanding mechanisms of CSF transport and disruption, as well as for other complex biophysical systems.

Appendix A Particle tracking uncertainty analysis

In this Appendix, we analytically calculate the approximate uncertainty in particle tracking velocity measurements using propagation of error based on estimated uncertainties in particle position and image acquisition time. The numbers used here are specific for one particular data set (Kim et al. 2023), serving as a concrete example, but the reader could readily repeat these calculations for different imaging parameters. This derivation applies to two-photon microscopy, wherein each 2D image is assembled over a finite window in time during which each pixel is individually rastered. Hence, for images acquired at 29.6 Hz, each acquired image has an uncertainty of approximately $\Delta t=33.8$ ms. Note this value is an upper bound for the temporal uncertainty, this upper bound will generally be larger/smaller when a greater/lesser number of pixels are recorded, and more precise (smaller) values of $\Delta t$ could be obtained by accounting for details of the TPM rastering technique.

The velocity of a tracked particle in frame n can be estimated, to the first order, as $U_n = \frac{\Delta x}{\Delta t}$, where $\Delta x$ is the measured displacement of the particle since frame $n-1$, and $\Delta t$ is the measured time elapsed since frame $n-1$. Denoting the error in $\Delta x$ as $u_x$ and the error in $\Delta t$ as $u_t$, the error in $U_n$ is

$$\begin{aligned} \epsilon _U= & {} \left( \left( u_t \frac{\partial U_n}{\partial \Delta t} \right) ^2 + \left( u_x \frac{\partial U_n}{\partial \Delta x} \right) ^2 \right) ^{1/2} \nonumber \\= & {} \left( \left( u_t \frac{\Delta x}{\Delta t^2} \right) ^2 + \left( u_x \frac{1}{\Delta t} \right) ^2 \right) ^{1/2}. \end{aligned}$$

(A1)

Our image processing algorithm locates each particle by finding the centroid of a contiguous bright region above a given threshold, typically achieving error on the order of 0.1 pixel. In these data, each pixel has lateral dimension $1.04~\upmu \hbox {m}$, so we estimate $u_x \approx 0.104~\upmu \hbox {m}$. As mentioned above, for a frame rate of 29.6 Hz, $\Delta t = 33.8$ ms. Now suppose the root-mean-square single-frame displacement is $\Delta x = 2.158~\upmu \hbox {m}$ (this value is obtained empirically by performing particle tracking). Estimating the timing error $u_t$ for a two-photon microscope is subtle because images are produced not by making simultaneous measurements from an array of sensors but by making subsequent measurements as the single focal point rasters the field of view. If the focal point traces column-by-column, then measuring a particle moving to an adjacent column involves greater timing error than measuring a particle moving to an adjacent row. Considering image dimensions $512 \times 512$ pixels, the two timing errors would be roughly $\Delta t / 512$ and $\Delta t / 512^2$, respectively. To be conservative, we take $u_t \approx \Delta t / 512 = 66.0~\upmu \hbox {s}$. Using these values, we find $\epsilon _U = 3.08~\upmu \hbox {m/s}$.

In practice, we estimate the velocity with a higher-order method, convolving the measured position with a kernel that provides differentiation and smoothing. Explicitly, the numerical scheme used in our code makes the estimate

$$\begin{aligned} U_n = \frac{\alpha \Delta {\widetilde{x}}}{\Delta t}, \end{aligned}$$

(A2)

where $\alpha = 2 / (\pi ^{1/2} \textrm{erf}(3) - 6 e^{-9}) \approx 1.129$ and

$$\begin{aligned} \Delta {\widetilde{x}}= & {} -3e^{-9} x_{n-3} - 2e^{-4} x_{n-2} - e^{-1} x_{n-1} + e^1 x_{n+1} \nonumber \\{} & {} + 2e^{-4} x_{n+2} + 3e^{-9} x_{n+3}. \end{aligned}$$

(A3)

The velocity error is

$$\begin{aligned} \epsilon _U= & {} \left( \left( u_t \frac{\partial U_n}{\partial \Delta t} \right) ^2 + \left( u_{n-3} \frac{\partial U_n}{\partial x_{n-3}} \right) ^2 + \left( u_{n-2} \frac{\partial U_n}{\partial x_{n-2}} \right) ^2 \right. \ldots \nonumber \\{} & {} + \left( u_{n-1} \frac{\partial U_n}{\partial x_{n-1}} \right) ^2 + \left( u_{n+1} \frac{\partial U_n}{\partial x_{n+1}} \right) ^2 \left. + \left( u_{n+2} \frac{\partial U_n}{\partial x_{n+2}} \right) ^2 + \left( u_{n+3} \frac{\partial U_n}{\partial x_{n+3}} \right) ^2 \right) ^{1/2}, \end{aligned}$$

(A4)

where $u_{n-3}$ is the measurement error associated with location $x_{n-3}$, $u_{n-2}$ is the measurement error associated with location $x_{n-2}$, and so on. Assuming homogeneity implies that all those errors have the same value, which we again denote $u_x$. Then, the velocity error becomes

$$\begin{aligned} \epsilon _U = \alpha \left( \left( u_t \frac{\Delta {\widetilde{x}}}{\Delta t^2} \right) ^2 + \beta \left( u_x \frac{1}{\Delta t} \right) ^2 \right) ^{1/2}, \end{aligned}$$

(A5)

where $\beta = 18e^{-18} + 8e^{-8} + 2e^{-2} \approx 0.273.$ To estimate the value of $\Delta {\widetilde{x}}$, we consider the case in which a particle’s displacement between any two frames is the measured root-mean-square value $\Delta x$, implying $x_{n+1} - x_{n-1} = 2 \Delta x$, $x_{n+2} - x_{n-2} = 4 \Delta x$, and $x_{n+3} - x_{n-3} = 6 \Delta x$. Therefore, $\Delta {\widetilde{x}} = \gamma ^{1/2} \Delta x,$ where $\gamma = (2 e^{-1} + 8 e^{-4} + 18 e^{-9})^2 \approx 0.782$. Altogether, the velocity error is

$$\begin{aligned} \epsilon _U = \alpha \left( \gamma \left( u_t \frac{\Delta x}{\Delta t^2} \right) ^2 + \beta \left( u_x \frac{1}{\Delta t} \right) ^2 \right) ^{1/2}. \end{aligned}$$

(A6)

Comparing to Eq. (A1), we see that the error in velocity estimated with the higher-order numerical scheme differs from the error in the first-order velocity estimates only by factors of order unity. Again taking the same values for $u_t$, $u_x$, $\Delta t$, and $\Delta x$, the velocity error in the higher-order scheme is $\epsilon _U = 1.82~\upmu \hbox {m/s}$, about 40% lower than with the first-order estimate.