Time-lagged Functional Ultrasound for Multi-parametric Cerebral Hemodynamic Imaging

Abstract

We introduce an ultrasound speckle decorrelation-based time-lagged functional ultrasound technique (tl-fUS) for the quantification of the relative changes in cerebral blood flow speed (rCBF_speed), cerebral blood volume (rCBV) and cerebral blood flow (rCBF) during functional stimulations. Numerical simulations, phantom validations, and in vivo mouse brain experiments were performed to test the capability of tl-fUS to parse out and quantify the ratio change of these hemodynamic parameters. The blood volume change was found to be more prominent in arterioles compared to venules and the peak blood flow changes were around 2.5 times the peak blood volume change during brain activation, agreeing with previous observations in the literature. The tl-fUS shows the ability of distinguishing the relative changes of rCBF_speed, rCBV, and rCBF, which can inform specific physiological interpretations of the fUS measurements.

I. INTRODUCTION

QUANTIFYING cerebral hemodynamic responses of cerebral blood flow speed (CBF_speed), cerebral blood volume (CBV) and cerebral blood flow (CBF) is important for studying blood flow dysregulation in pathologies and for quantitative understanding of neurovascular coupling during brain activation [1], [2]. Functional magnetic resonance imaging (fMRI) can be used to image hemodynamics in humans and animals but suffers from low spatiotemporal resolution and high cost [3]. In animals, blood flow and oxygenation can also be measured using a suite of optical methods including two photon microscopy (TPM) [4], optical intrinsic signal imaging (OISI) [5], optical computed tomography (OCT)[6] and laser speckle contrast imaging (LSCI) [7]. These methods, however, are limited to shallow penetration.

Over the last decade, the ultrafast ultrasound-based [8], [9] power Doppler functional ultrasound (PD-fUS) technique has been rapidly adopted for neuroscience research due to its unpreceded ability to image brain-wide hemodynamics in rodents with high spatiotemporal resolution and high sensitivity with relatively low cost [10]–[13]. The PD-fUS signal is the integration of the power of all the ultrasound signal fluctuations after clutter rejection. In addition to the influence from experimental setup alignment and acoustic pressure attenuation in tissue, multiple physiological parameters contribute to the PD-fUS signal, including local blood volume, blood flow, and blood flow speed [14]–[16]. PD-fUS is not able to differentiate these hemodynamic parameters, making it challenging to interpret the experimental results [17]. Conventional color Doppler measures either phase shift or Doppler frequency shift to provide information about blood flow speed but is limited to the axial flow speed component and unstable estimation due to the presence of noise [18]. In order to measure the in-plane total blood flow velocity independent of the Doppler angle, vector Doppler was introduced by retrieving the flow velocity from color Doppler measurements that obtained at different emission angles[19], [20]. The vector Doppler has been widely applied to measure the blood flow velocity in large vessels, and it was recently improved to measure the cerebral blood flow velocity of small animals, but still suffers from limited sensitivity to small vessels [21].

The ultrasound speckle decorrelation analysis-based velocimetry has been proposed to measure the blood flow velocity and show the ability to quantify both axial and lateral velocity components[22]–[24]. Recently, Tang et al. reported contrast-free and quantitative ultrasound velocimetry (vUS) for blood flow velocity measurements of the small vessels in the entire mouse brain [25]. Unlike the color Doppler which requires Doppler angle for the axial speed detection or the PD-fUS which simply sums the power spectrum of the ultrasound signal, vUS analyzes the ultrasound speckle decorrelation due to the movement of red blood cells (RBCs) in blood vessels to quantify both axial and transverse blood flow speeds. However, the vUS data processing procedure involves many variables requiring a complex and time-consuming non-linear estimation procedure, hindering its wider adoption for neuroscience studies.

In this work, we introduce an ultrasound speckle decorrelation-based time lagged functional ultrasound (tl-fUS) as a simpler and computationally efficient approach to differentiate CBF_speed, CBV and CBF responses during functional stimulations. We derive the theory of tl-fUS from the first order field autocorrelation function analysis ($g_1(\tau )$). By comparing the $g_1(\tau )$ decorrelation at different time lags, we obtain CBF_speed index and CBV index that enable us to quantify the relative change of rCBF_speed, rCBV and rCBF. For numerical simulation, to mimic the physiological situation where blood volume changes are mainly due to the dilation of arteries and arterioles, we simulated blood volume changes by increasing the vessel diameter instead of increasing the scatter density. For phantom validation, we tested our method to quantify the changes of rCBF_speed and rCBV at different preset flow speeds. For the in vivo experiments, we acquired data from awake head-restrained mice during whisker stimulation. Different activation maps of blood flow speed change (rCBF_speed) and blood volume change (rCBV) were observed.

II. Theory

Fig. 1A illustrates the theoretical model and Eq. (1) shows the derived first order field autocorrelation function $\left(g_1(\tau )=E\left[\frac{{\langle sI{Q}^{\ast }(t)sIQ(t+\tau )\rangle }_t}{{\langle sI{Q}^{\ast }(t)sIQ(t)\rangle }_t}\right]\right)$ of a three-dimensional ultrasound measurement voxel (for details please see APPENDIX).

$$g_1(\tau )=\frac{N_s{R}_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z}{N_s{R}_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z+R_e}{e}^{-\frac{{\left(v_x\tau \right)}^2}{4{\sigma }_x^2}-\frac{{\left(v_y\tau \right)}^2}{4{\sigma }_y^2}-\frac{{\left(v_z\tau \right)}^2}{4{\sigma }_z^2}}{e}^{i2k_0{v}_z\tau }$$ (1)

where, $N_s$ is the number of moving scatters within the voxel; $R_s$ is the reflection factor. We consider the reflectivity of the scatters (e.g., RBCs) are the same so that $R_s$ is a constant. ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$ are the Gaussian profile width at the $1/e$ value of the maximum intensity of the PSF in the x, y, and z directions, respectively; $k_0$ is the wavenumber of the central frequency of the transducer; and $R_e$ denotes the noise level of the imaging system.

Fig. 1.

Principles of the time-lagged functional ultrasound (tl-fUS). (A) Illustration of the theoretical model. (A1) The flow speed effect on the Fourier spectrum (inset) and the $g_1(\tau )$ decorrelation. (A2) The flow volume effect (vessel size, d) on the Fourier spectrum (inset) and the $g_1(\tau )$ decorrelation. (B), (B1) and (B2) illustrate the speed and volume effects on Fourier spectrum (inset) and $g_1(\tau )$ decorrelation for horizontal flows. (C) and (D) Representative images of the CBF_speed index and the CBV index from a living mouse brain, respectively.

We notice that the decorrelation amplitude of $g_1(\tau )$ is affected by the number of scatters ($N_s$), i.e. the scatter volume, the speeds ($v_x$, $v_y$, and $v_z$), the system PSF (${\sigma }_x$, ${\sigma }_y$, ${\sigma }_z$), and the noise contribution $R_e$. The phase term of $g_1(\tau )$ is modulated by the axial velocity component ($v_z$). Assuming the system noise is fixed, the change of the scatter volume ($N_s$) will affect the signal-to-noise ratio (SNR), i.e. larger $N_s$ corresponds to higher SNR. According to the $g_1(\tau )$ decorrelation curves in Fig. 1A2&B2, we see that SNR term doesn’t affect the decorrelation time but only causes a drop of the $g_1(\tau )$ function at the first time lag. This is due to that the noise are uncorrelated and the $g_1(\tau )$ is indeed a statistical analysis, which is inherently resistive to noise.

Fig. 1A1 illustrates the effect of blood flow speed on the decorrelation of $g_1(\tau )$ under the same noise level for an angled flow scenario (i.e. with axial velocity component). We see that a higher flow speed (30 mm/s) leads to a larger Doppler shift and spectrum broadening in the frequency plot (inset), while the $g_1(\tau )$ decays faster compared to the lower speed (10 mm/s). Fig. 1A2 shows the effect of blood volume on the decorrelation of $g_1(\tau )$ for angled flow at constant speed. The blood volume ($N_s$) change was achieved by increasing diameter of the vessel. We see that a larger diameter, i.e. increased blood volume, leads to a higher amplitude power spectrum (inset) and a higher value of $\left|g_1(\tau )\right|$ at the first time lag. For the horizontal flows (Fig. 1B), there will be no Doppler frequency shift but only spectral broadening (inset of Fig. 1B1&B2). In contrast, the decorrelation curve of $g_1(\tau )$ will be similar for both angled and transverse flows, a result of the $g_1(\tau )$ analysis enabling measurement of the flow signal in all directions, as shown in Fig. 1B1&B2.

The observations in Fig. 1 (A1, A2, B1 and B2) indicate that the $g_1(\tau )$ decorrelation is not only related to flow speed but also the scatter volume, which lays the foundation to extract the hemodynamic information of CBF_speed and CBV with the tl-fUS analysis. Based on Eq. (1) and these observations, we derived two indices to distinguish the blood flow speed from blood volume. For the measurement of flow speed index (CBF_speed), the $g_1(\tau )$ value at two time lags are utilized to extract the total velocity $\nu$:

$$CB{F}_{\text{speed}}=\frac{1}{\sqrt{\left({\tau }_2^2-{\tau }_1^2\right)}}\sqrt{\left|\log \left(\frac{\left|g_1\left({\tau }_1\right)\right|}{\left|g_1\left({\tau }_2\right)\right|}\right)\right|}$$ (2)

$$=\sqrt{\frac{\cos {\theta }^2\cos {\beta }^2}{4{\sigma }_x^2}+\frac{\cos {\theta }^2\sin {\beta }^2}{4{\sigma }_y^2}+\frac{\sin {\theta }^2}{4{\sigma }_z^2}}\cdot v$$

where, $|\dots |$ denotes the absolute value; $\theta$ is the angle between the vessel and the horizontal plane (X-Y plane) and $\beta$ is the angle of the projected vessel in the X-Y plane to the X direction, as illustrated in Fig. 1A. Thus, $v_x=v\cdot \cos \theta \cos \beta$, $v_y=v.\cos \theta \sin \beta$, $v_z=v\cdot \sin \theta$. By dividing the $\left|g_1(\tau )\right|$ at the time lag ${\tau }_1$ with $\left|g_1(\tau )\right|$ at the time lag ${\tau }_2$, the term related to the blood volume ($N_s$) cancels out and therefore the CBF_speed index has a linear relationship with the total speed, as shown in Eq. (2). The selection of ${\tau }_1$ and ${\tau }_2$ will be discussed in the Numerical evaluation section.

Further, the exponential term of $\left|g_1(\tau )\right|$ at the first time lag is close to 1 when the sampling rate is high enough, i.e. inverse of frame rate much shorter than the decorrelation time, thus the decorrelation contributed by the speed terms in $\left|g_1\left({\tau }_1\right)\right|$ are negligible (APPENDIX Eq. (19)). With the assumption of a sufficiently high frame rate, we derived the blood volume index (CBV) as Eq. (3), which shows that the CBV-index is proportional to the number of scatters, $N_s$. Note that this proportionality depends on the overlap of the ultrasound imaging system’s PSF (${\sigma }_x$, ${\sigma }_y$, ${\sigma }_z$) with the blood flow.

$$CB{V}_{\text{index}}=\frac{\left|g_1\left({\tau }_1\right)\right|}{1-\left|g_1\left({\tau }_1\right)\right|}=\frac{R_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z}{R_e}\cdot N_s$$ (3)

Fig. 1C&D show representative results of the CBF_speed index and CBV index obtained with tl-fUS from a living mouse brain. We see that both indices can successfully detect the blood flow dynamics in the brain, but they are slightly different. Briefly, the CBF_speed index mainly detects the speed information from big vessels while the CBV index has a good detection of the blood perfusion in both big vessels and the parenchyma. We also notice that, compared to the CBV index, the CBF_speed index has a larger value ratio of big vessels-to-small vessels, and it is noisier in the region outside of the brain which is due to low signal-to-noise ratio in these regions after clutter rejection.

Note that the goal of the proposed tl-fUS is not to quantify the absolute value of blood flow speed nor the blood volume of the brain, but to obtain the relative changes for dynamics and functional studies. Thus, with the CBF_speed index and CBV index, we finally reach the goal of differentiating and quantifying the relative changes of rCBF_speed and rCBV in response to functional stimulations, which can be obtained as:

$$rCB{F}_{speed}=\frac{v_{t1}}{v_{t2}}=\frac{CB{F}_{speed,\text{t}1}}{CB{F}_{speed,t2}};rCBV=\frac{N_{s,\text{t}1}}{N_{s,t2}}=\frac{CB{V}_{index,t1}}{CB{V}_{index,t2}}$$ (4)

where the $v_{t1}$ and $v_{t2}$ denote the speeds at time points of $t1$ and $t2$, typically an activation state versus the baseline state. Moreover, the relative cerebral blood flow (rCBF) is obtained by the definition of the product of rCBV and rCBF_speed.

$$rCBF=rCBV\times rCB{F}_{speed}$$ (5)

III. Materials and Methods

A. Numerical simulation

Ultrasound detection of flowing red blood cells in a vessel was mimicked through numerical simulation as illustrated in Fig. 1A. Point scatters were randomly distributed inside the vessel, moving along the vessel-oriented direction with a preset flow speed. The grid pixel size for simulation was 0.1 𝜇𝑚 and the particle density was set to be 1/20 𝜇𝑚² in 2D to approach the real red blood cell counts of 1/100 𝜇𝑚³ in 3D [26]. The ultrasound signal ($sIQ$) arising from the detected resolution volume was generated based on the APPENDIX Eq. (15) with a fixed system noise level. The white Gaussian noise was generated such that the first time lag of $g_1(\tau )$ function reaches to the typical value of 0.6 which was observed from in vivo measurements of mice brain vessels. Four seconds of the ultrasound signal collected at 5 kHz were used for the $g_1(\tau )$ measurement. Then the CBF_speed index and CBV index were calculated and compared with the preset speed and volume. We explored the impact of different velocities, vessel diameters, vessel orientations and PSF sizes on CBF_speed index and CBV index and statistically analyzed the results by repeating the simulation 5 times while the locations of point scatters and system noise were randomly generated each time.

B. Phantom experiments

A plastic microtube (inner diameter 580 𝜇𝑚, Intramedic Inc.) was buried in an agarose phantom with a horizontal angle of ~0° and 30° to mimic the transverse flow and the angled flow, respectively. The transducer array was placed along with the tube for in-plane detection, and orthogonal to the tube for through-plane detection. A volume of 1 ml blood solution which was obtained from an experimental rat and fixed in 10% formalin was pumped through the microtubes by a syringe pump (Harvard Apparatus) at 5, 10, 15, 20 mm/s for in-plane and through-plane flows. For phantom results, the first two highest singular values were removed to filter the background signals.

C. Animals

Animal experiments were approved by the Institutional Animal Care and Use Committee at Boston University and Southern University of Science and Technology and were conducted following the Guide for the Care and Use of Laboratory Animals.

In this study, four 12-to-16 weeks old C57BL/6 mice (22–28 g, either sex, Charles River Laboratories) were used. Animals were housed under diurnal lighting conditions with free access to food and water. Chronic polymethypentene (PMP) cranial windows were installed as previously described [27], [28]. Briefly, during the window installation surgery, mice were anesthetized with isoflurane. A custom-made aluminum head bar [27] was glued to the skull, bone extending −0.5 to −3.5 mm from bregma and laterally ±6 mm from midline was removed, a PMP film was inserted into the craniotomy and attached to skull edges with dental acrylic and glue. Animals were allowed to recover for 3 weeks after surgery. For 2 weeks before awake imaging sessions, mice were trained to be used to head fixation and sweetened milk was offered as reward every ~15 min during training and imaging sessions.

D. Ultrasound system and imaging protocol

We used a linear transducer probe (L22–14v, 18.5 MHz center frequency, 0.1 mm pitch, 128 elements, Verasonics Inc. Kirkland, WA, USA) connected to a commercial ultrasound imaging system (Vantage256, Verasonics Inc. Kirkland, WA, USA) for ultrafast ultrasound plane wave transmission and acquisition. The transducer was immersed in a customized water tank with water temperature maintained at 37 ± 0.5°C. We applied 2% agarose to fill the gap between the transparent film (Tegaderm, 3M) of the water tank and the mouse cranial window for acoustic coupling. The transducer probe was positioned ~4.5 mm above the cranial window.

We collected 5 titled angles of plane waves (−5°, −2.5°, 0°, 2.5°, 5°) to form one compound B-mode frame. Moreover, to ensure sufficient ensemble averaging for speckle fluctuations, we acquired 200 ms of ultrasound data at a compound image frame rate of 5 kHz, i.e., 1000 compounded frames were acquired to obtain a single tl-fUS result. Thus, the theoretical achievable frame rate for hemodynamic detection was 5 Hz. Limitations due to data transfer and saving reduced the effective frame rate to 1 Hz in this study.

E. Somatosensory stimulation

The stimulus consisted of air puff trains delivered to the whisker pad. An outlet from a pressure pulse generator (Picospritzer III, Parker Inc.) was connected to a plastic tube. The tube was placed at ~1 cm away from the mouse face and deflected the whiskers at a frequency of 3 Hz. Each imaging session consisted of 10 repeated trials and each trial consisted of a baseline period of 5 s followed by 5 s stimulation period (15 air puffs) and 20 s post-stimulus duration. We collected 8 datasets from 4 mice (2 datasets for each mouse) and each dataset is from one imaging session that contains 10 stimulation trials.

F. Clutter rejection

To remove tissue motion and extract blood blow signals, we applied a singular value decomposition (SVD)–based spatiotemporal clutter filter [29] to the raw data. The first 20 largest singular value components and the smallest 500 singular value components were removed for bulk motion rejection and high background noise suppression:

$$sIQ(x,z,t)=\sum _{i=21}^{N_{up}}{\lambda }_i{U}_i(x,z)V_i(t)$$ (6)

where $sIQ(x,z,t)$ is the clutter rejected ultrasound signal; $U_i$ and $V_i$ are the $i^{th}$ columns of the spatial singular matrix and temporal singular matrix, respectively; ${\lambda }_i$ is the descending-ordered singular values; $N_{up}=500$ is the upper range for SVD filtering.

G. Directional filtering

The filtered blood signal $sIQ$ for each measurement pixel was further separated into the positive frequency signal component $sI{Q}_{pos}$ and the negative frequency signal component $sI{Q}_{neg}$ using directional filtering:

$$\mathcal{F}(sIQ)={\mathcal{F}}_{pos}(sIQ)+{\mathcal{F}}_{neg}(sIQ)$$ (7)

$$sI{Q}_{pos}={\mathcal{F}}^{-1}\left[{\mathcal{F}}_{pos}(sIQ)\right],sI{Q}_{neg}={\mathcal{F}}^{-1}\left[{\mathcal{F}}_{neg}(sIQ)\right]$$ (8)

where $\mathcal{F}$ and ${\mathcal{F}}^{-1}$ are the Fourier and inverse Fourier transform, respectively. $sI{Q}_{pos}$ captures descending blood flow (moving away from the detector) while $sI{Q}_{neg}$ represents ascending blood flow (moving towards the detector). In the following notation, $sIQ$ denotes either $sI{Q}_{pos}$ or $sI{Q}_{neg}$ for simplicity if not specified.

H. Improved color Doppler

The conventional color Doppler method which estimates the Doppler frequency shift using the whole spectrum is able to accurately estimate the Doppler frequency shift for a single-directional flow. However, this simple scenario of flow in a single-direction is not generally applicable for microvascular imaging in the rodent brain. The improved color Doppler ultrasound imaging method (iCD-fUS [25]) was achieved by applying directional filtering in combination with noise thresholding to estimate the Doppler frequency shifts.

$$v_{z,descend}=-\frac{c}{2f_0}\frac{{\int }_{F_{neg}}f\cdot \left|\mathcal{F}\right(sIQ){|}^2)df}{{\int }_{F_{neg}}\left|\mathcal{F}\right(sIQ){|}^2)df}$$ (9)

$$v_{z,ascend}=-\frac{c}{2f_0}\frac{{\int }_{F_{pos}}f\cdot |\mathcal{F}(sIQ){|}^2)df}{{\int }_{F_{\text{pos}}}|\mathcal{F}(sIQ){|}^2)df}$$ (10)

$$Threshold=1.5\cdot \max \left(\left|\mathcal{F}{(\text{sIQ})}_{\left|f\right|>2\text{kHz}}\right|\right)$$ (11)

where, $\text{c}$ is the sound speed in the medium and $\text{c}$ = 1540 m/s was used in this study; $f_0$ is the transducer center frequency; $F_{neg}$ is the signal frequency range after thresholding for the negative frequency component; $F_{pos}$ is the signal frequency range after thresholding for the positive frequency component; and $\mathcal{F}$ denotes the Fourier transform.

I. Power Doppler

PD-fUS measures the mean power intensity of the ultrasound Doppler signal [10]:

$$D(x,z,t)=\frac{1}{N}\sum _N^{i=1}sIQ{\left(x,z,t_i\right)}^2$$ (12)

Here, $N$ indicates the number of samples acquired per power Doppler image frame. $sIQ$ is the filtered blood signal after clutter rejection and directional filtering. Considering the noise term is not neglected in $sIQ$, it is obvious that the PD signal also includes the noise signal, which will degrade CBV sensitivity detected by PD-fUS.

J. Activation map

Activation maps were obtained by calculating Pearson’s correlation coefficient $r$ between experimentally obtained functional signal $s\left(t\right)$ and a predicted hemodynamic response $h\left(t\right)$.

$$r=\frac{{\sum }_{i=1}^{N_t}\left(s\left(t_i\right)-\overline{s}\right)\left(h\left(t_i\right)-\overline{h}\right)}{\sqrt{{\sum }_{i=1}^{N_t}{\left(s\left(t_i\right)-\overline{s}\right)}^2}\sqrt{{\sum }_{i=1}^{N_t}{\left(h\left(t_i\right)-\overline{h}\right)}^2}}$$ (13)

where, $N_t$ is the number of the image frames; $\overline{S}$ and $\overline{h}$ are the temporal average of $s\left(t\right)$ and $h\left(t\right)$, respectively. The functional signal $s\left(t\right)$ can be either CBF_speed index or CBV index depending on which hemodynamic parameter is observed. The hemodynamic response $h\left(t\right)$ was estimated by convolving the stimulation pattern with a predefined hemodynamic response function $(\text{HRF}):h(t)=Stim(t)\ast HRF(t)·HRF(t)$ was obtained by modeling two Gamma functions [25], [30] with the following parameters: delay of response 1.5 s, delay of undershoot 10 s, dispersion of response 0.5 s, dispersion of undershoot 1 s, ratio of response to undershoot 6 s, onset 0 s, length of kernel 16 s.

The correlation coefficient was further transformed into the Z-score through a Fisher’s transform, which is defined as:

$$z=\frac{\sqrt{N_t-3}}{2}\ln \frac{1+r}{1-r}$$ (14)

The level of significance was chosen to be z > 1.65 (𝑝 < 0.05, one tailed test) which corresponds to r > 0.3. Moreover, a connectivity filter [31] that kept only the activated regions containing more than 9 connected pixels was applied for removing false positive activated pixels.

IV. Results

A. Numerical simulation

We first evaluated the effect of the selections of ${\tau }_1$ and ${\tau }_2$ on the CBF_speed index measurement, as shown in Fig. 2A. In principle, any two time lags (${\tau }_1$ and ${\tau }_2$) that are shorter than the speckle decorrelation time $\left({\tau }_{\text{c}}\right)$ can be used for the CBF_speed index calculation. We tested the effect of different time lags on the speed index measurement using two simulated vessels with preset speeds of 10 mm/s and 30 mm/s. The time lag ${\tau }_1$ was set as the first time lag while the other time lag ${\tau }_2$ was changing from the 2nd to the 25th time lag. Fig. 2A shows the results for the CBF_speed index of the two simulated vessels (left panel) and the ratio (rCBF_speed index) of the 30 mm/s to the 10 mm/s. Fig. 2B shows the results for the CBV (left panel) and the ratio (rCBV, right panel) related to the selection of ${\tau }_1$ of two simulated vessels with diameters of d=30 𝜇𝑚 and d=60 𝜇𝑚 and under the same flow speed of 10 mm/s.

Fig. 2.

Numerical evaluation. (A) The effect of the ${\tau }_1$ and ${\tau }_2$ selection on the CBF_speed index estimation. (B) The effect of the ${\tau }_1$ selection on the CBV index. (C) CBF_speed index changes linearly with increasing blood flow speeds for different flow angles. (D) CBF_speed index doesn’t change with vessel diameter (blood volume) when fixing the flow speed. (E) CBV index changes linearly with increasing blood vessel diameter when the flow diameter is less than half of the system point spread function (PSF) and is nonlinear related to the vessel diameter for larger sizes; the inset illustrates the non-homogeneous detection when the vessel size is close or larger than the system PSF. (F) CBV index doesn’t change with flow speed when fixing the vessel diameter (blood volume).

We see that there is a large variation for the measurements of CBF_speed index and the rCBF_speed when ${\tau }_2$ is close to ${\tau }_1$. This is due to the limited data size for $g_1(\tau )$ calculation, which may result in non-smooth decorrelation, i.e. small fluctuations at adjacent time lags. When ${\tau }_2$ is longer than the decorrelation time ${\tau }_{\text{c}}$, the CBF_speed index is smaller than the preset value, since the noise and limited data size have a stronger effect on the $g_1(\tau )$ decorrelation beyond the decorrelation time. To obtain rCBF_speed, the ${\tau }_2$ shall be longer than 5 times of ${\tau }_1$ and shorter than the decorrelation time ${\tau }_{\text{c}}$ of the faster flow, as shown in the right panel of Fig. 2A. In this work, we used the 1st time lag (${\tau }_1$= 0.2 𝑚𝑠) and the 5th time lag (${\tau }_2$= 1 𝑚𝑠) for data processing, which satisfies the criteria that ${\tau }_2=1ms<{\tau }_{\text{c}}$, where ${\tau }_{\text{c}}$ ranges from 12 𝑚𝑠 to 1.2 𝑚𝑠 for velocities varying from 3 𝑚𝑚/𝑠 to 30 𝑚𝑚/𝑠 for our imaging parameters. The calculation of the CBV index is also affected by the selection of ${\tau }_1$. According to Fig. 2B, we see that the CBV index decreases nonlinearly with longer ${\tau }_1$ and the ratio of the two volumes (rCBV) is underestimated when increasing ${\tau }_1$. This under-estimation results from the non-negligible flow speed-caused decorrelation under large ${\tau }_1$.

We then evaluated the CBFspeed index for the measurement of different blood flow speeds. We performed a numerical simulation using a 50 𝜇𝑚 -wide vessel with different flow angles and speeds changing from 3 mm/s to 18 mm/s, as shown in Fig. 2C. Here, we fixed the ultrasound PSF at an experimentally measured value of (${\sigma }_x$, ${\sigma }_y$, ${\sigma }_z$) = (110, 420, 85) 𝜇𝑚. Fig. 2C shows that the measured CBFspeed index is proportional to the change of the preset flow speeds, and the proportionality depends on the angle of the flow. Note that this angle dependance cancels when calculating the relative change of the rCBF_speed index as the vessel angel doesn’t change over time. Fig. 2D shows the change of the CBFspeed index while varying the flow diameters (i.e. blood volume). We can see that it does not change with vessel size, suggesting that the CBFspeed index is not affected by blood volume.

We also investigated the accuracy of the CBV index for measuring blood volume changes. To mimic a physiological condition where blood volume increases due to vasodilation, we simulated changes in vessel diameter while the scatterer density in the vessel was fixed. Fig. 2E shows the change of the CBV index in response to increasing vessel diameters (i.e. $N_s$) at different PSFs. The lateral PSF size was determined by the numerical aperture during beamforming and was varied between 75 and 125 . We see that the CBV index is increasing with increased vessel diameters, and it is proportional to the blood volume change for vessel size when smaller than half of the PSF, but the changes become sub-linear for larger vessel size. In addition, we note that the linear-to-nonlinear turning point of the vessel diameter is increasing with larger system PSF. These results indicate that interpretation of the CBV change must consider this potential confounding factor and this phenomenon can be mitigated with a larger PSF. We also tested the CBV index change in response to total flow speed and noticed that the CBV index does not change with increasing speeds (as shown in Fig. 2F), confirming that our method is capable of differentiating CBV and CBF_speed.

B. Phantom validation

We further performed phantom validation experiments with blood samples flowing through a microplastic tube buried within a static agarose phantom, and the preset speeds were 5, 10, 15 and 20 mm/s. Fig. 3A shows the normalized CBF_speed index maps for through-plane flows (left column), in-plane angled flows (middle column), and in-plane transverse flows (right column). We can see that the CBF_speed index increases with the preset speeds for all flow directions. The relative rCBF_speed index further suggested that the CBF_speed index increases linearly with the preset speeds, as shown in Fig. 3B.

Fig. 3.

Phantom validation. (A) Normalized CBF_speed index maps obtained at different preset speeds for through-plane flows (left column), in-plane angled flows (middle column), and in-plane transverse flows (right column). (B) Relative changes of the CBF_speed index (rCBF_speed, left axis) and the relative CBV index and power Doppler value (right axis) with respect to the preset flow speeds; CBV index and power Doppler were obtained from an in-plane transverse flow. Error bars show the mean±s.e.m (standard error of the mean) value within the flow tube region.

We also calculated the CBV index from the phantom data sets and the results suggest that the obtained CBV index slowly decreases with higher flow speeds, as shown in Fig. 3B (right axis). This is due to the fact that the flow speed contribution to the $\left|g_1(\tau )\right|$ decorrelation at the first time lag is not negligible for fast flows. According to Eqs. (10) and (12), the CBV index was derived based on the assumption that the first time lag is short enough so that the decorrelation contributed by the flow speeds (exponential terms in Eq. (1)) can be neglected. However, in the experiment, the time lag is limited by data acquisition rate, which would result in increased values of $v\cdot {\tau }_1$ for higher speeds. In this case, according to Eq. (1), the resulting smaller value of $\left|g_1(\tau )\right|$ at the first time lag would lead to a smaller CBV index. Therefore, a high frame rate data acquisition is preferred to obtain accurate CBV index for fast flowing vessels. Nevertheless, compared to power Doppler imaging (PDI), the rCBV index measured with tl-fUS is much less affected by the flow speeds, as shown in Fig. 3B (right axis).

C. In vivo comparison

To evaluate the performance of tl-fUS for in vivo imaging, we compared the measured CBF_speed index and the CBV index to the results of the commonly used color Doppler (CD-fUS) and power Doppler (PD-fUS) techniques, respectively. We first calculated the results using the full frequency range of the same data set, as shown in the top row of Fig. 4A. For speed measurements, we see that the CBF_speed index obtained with tl-fUS can detect more vessels, while the $v_z$ obtained with regular color Doppler suffers from the mutual frequency cancellation of positive and negative frequency components, which results in underestimated measurement of the blood flow speed, as we discussed in our previous publication [25].

Fig. 4.

In vivo comparison. (A) CBF_speed index, color Doppler measurements, CBV index, and power Doppler measurements obtained with full spectrum (top row), negative frequency components (middle row, descending flows), and positive frequency components (bottom row, ascending flows). (B) Point-to-point scatter plots comparing the CBF_speed index with color Doppler measurements (top panel), and CBV index with power Doppler measurements (bottom panel).

We then performed a directional filtering [10] to obtain the CBF_speed index, CBV index, and power Doppler results for negative frequency components (descending flow) and positive frequency components (ascending flow), and the $v_z$ was obtained with an improved color Doppler method [25]. With directional filtering, we see that the improved color Doppler can obtain more speed measurements compared to the full spectrum-based regular color Doppler calculation. Comparing the speed measurements, the directional filtering-based CBF_speed index results are overall quite close to that obtained with the improved color Doppler method except for three notable differences. Firstly, we see that the CBF_speed index can detect more vessels, specifically transverse flowing vessels, than the improved color Doppler method, as indicated by the cyan colored arrows in the middle row. Secondly, the CBF_speed index may underestimate the blood flow speed for fast flowing vessels as indicated by the white arrows in the bottom row. This is due to the fact that the limited temporal frame rate (5 KHz for this study) is not fast enough to resolve the decorrelation curve for high-speed flows, i.e. the autocorrelation function $g_1(\tau )$ may fully decorrelated in a very short time period and the ${\tau }_2$ = 1 𝑚𝑠 used in this study may be too long as shown in Fig. 2A. Also, it is worth noting that, the high $v_z$ value obtained with the improved color Doppler may be incorrect as the low SNR and high frequency noise may bias the estimation of the Doppler phase shift [25]. Therefore, to improve the detection accuracy of CBF_speed index, a higher data acquisition frame rate is preferred. Lastly, we noticed the improved color Doppler is not able to detect some flows, as indicated by the magenta-colored arrow in the bottom row. This is probably due to the threshold used in the improved color Doppler for the removal of high frequency noise [25]. For the blood volume measurement, we don’t see any notable difference between the CBV index and the power Doppler images in Fig. 4A.

Fig. 4B further compared the normalized value between the CBF_speed index and the improved color Doppler measurement (top panel), and between the CBV index and the power Doppler measurement (bottom panel). The point-to-point scatter plot shown in the top panel suggests that, in the lower speed range, the CBF_speed index has a higher relative value compared to the $v_z$ obtained with the improved color Doppler. This is due to the fact that the $g_1(\tau )$ function is sensitive to flows in all directions while the color Doppler is only sensitive to the axial flow component, therefore, the CBF_speed index represents total flows while the color Doppler only detects the axial flow. In the higher speed range, especially for ascending flows, we see that the improved color Doppler has a larger value than the CBF_speed index. This is due to the fact that the speed is too fast for tl-fUS to resolve the dynamic change in a sufficiently short time period as we discussed before. For the comparison between the CBV index and the power Doppler shown in the bottom panel of Fig. 4B, a high linearity was observed between the two measurements (R=0.958) but with a slope of 0.808 indicating that the power Doppler measured value is larger than the CBV index.

D. Functional imaging of evoked response

To evaluate the functional imaging ability of the tl-fUS, we acquired functional data from awake mice during whisker stimulation (see Materials and Methods). Fig. 5 shows the correlation coefficient maps of CBF_speed index, Color Doppler measured $v_z$, CBV index, and power Doppler measurements in response to whisker stimulation. We note that both CBF_speed and CBV measured with tl-fUS exhibited strong activation in the primary somatosensory barrel field (S1BF) in the contralateral side, and they show similar activation maps to color Doppler and power Doppler measurements, respectively.

Fig. 5.

Correlation coefficient maps obtained with CBFspeed index, color Doppler method, CBV index, and power Doppler method. Primary somatosensory barrel cortex (S1BF, orange) and ventral posterior medial nucleus (VPM, green) are delineated from Allen Mouse Brain Atlas[32]. The control region is the S1BF on the ipsilateral side. Scale bars: 1 mm. The vascular image in the background is obtained from power Doppler imaging anatomy.

We also noticed differences among those measurements. Comparing the correlation coefficient maps obtained with CBF_speed index and color Doppler $v_z$, we see that the CBF_speed index shows a larger activation region, especially for the results obtained with all frequency components (top row). This is due to the fact that the standard color Doppler measurement suffers from mutual frequency cancellation when both negative frequency (descending flows) and positive frequency (ascending flows) exist within a measurement voxel. For the improved color Doppler processed with directional filtering (middle and bottom rows), the CBF_speed index results also show a larger activation region which is potentially due to the CBF_speed index being sensitive to both axial and transverse flows, while the color Doppler is only sensitive to axial flow component.

The CBV index also shows a larger activation region and stronger correlation compared to the power Doppler measurements (with the same z score thresholding of z>2.5). This may due to that the Power Doppler signal is affected by both the blood volume and blood flow speed while the CBV index is only sensitive to blood volume. As Fig. 3B shows, we note that the Power Doppler signal is decreasing when the flow speed increases, which indicates that the blood volume signal increase may be canceled out due to the opposite effect of speed increasing, especially in the big vessels. In contrast, the speed change has little effect on the CBV index (Fig. 3B). It’s know that the speed increase in big vessels would result in a larger blood volume, especially in the diluted arterials. Comparing the the CBV index results with the results obtained with both speed methods (CBF_speed Index and color Doppler), we note that the activation maps agree well with each other in location, which suggests that the CBV index may be more accurate in detecting cerebral blood volume change compared to Power Doppler during functional activation.

In addition, comparing the ‘speed’ measurements (CBF_speed index and color Doppler) with the ‘blood volume’ measurements (CBV index and power Doppler), we note that the ‘speed’ activation regions are mainly within big vessels and also shows a larger activation region compared to the Power Doppler measurements. This is due to the fact that the blood flow speed change is more significant and detectable in big vessels. During neural activation, more blood is supplied to the activated region, and the surrounding arterials and venules were also involved to supply or drain the increased blood flow, leading to increased blood flow speed in these vessels. These blood vessels are not only within the activation region but also in the outer regions. Thus, the activation map obtained by the speed measurements of CBF_speed index and Color Doppler would have activated voxels in big vessels and also outside of the “neuronal activation region”

E. Different responses of arterioles and venules in the cortical region

Based on the fact that the arterioles are flowing downward while the venules are flowing upward in the cortex, we can differentiate the arterioles from venules in cortex with the directional filtering (see Materials and Methods). Fig. 6A show the obtained correlation coefficient maps of rCBF_speed, rCBV, and rCBF (blood flow) of arterioles (downward flows, reddish) and venules (upward flows, bluish) in response to functional stimulation. Fig. 6B shows the averaged time traces of all 8 datasets for arterioles and venules with each dataset consisting of 10 repeated stimulation trials.

Fig. 6.

Comparison of functional responses of arterioles and venules within the cortex. (A) Representative directional activation maps in the cortex for rCBF_speed, rCBV and rCBF. The upward flow (i.e. venules in the cortex, bluish) is overlayed on the downward flow (i.e. arterioles in the cortex, reddish). Scale bar: 1 mm; Dn: downward. (B) Time courses of arterioles and venules measured with rCBF_speed, rCBV and rCBF, respectively. Results are averaged from 8 datasets from 4 mice with colored shades representing the standard error of the mean. Each dataset was obtained by averaging the 10 repeated stimulation trials. The horizontal solid black line indicates when the stimulus is on.

We noted that arterioles show a significant higher peak change of rCBV than venules. This is consistent with fMRI studies and it implies that arterioles’ diameter changes play a dominant role in the total CBV increase during neuronal activation [33]–[35]. Microscopic studies[36]–[38] also suggested that vasodilation was found in arterioles but not venules, consistent with a higher total hemoglobin change in arterioles [39]. We also observed that ascending venules have a larger change of CBF_speed than the descending arterioles. Considering that total flow from arterioles to the venules generally needs to be conserved, the increased CBV in arterioles thus would cause a higher speed change in the venules. The small discrepancy of the rCBF measurement in arterioles and venules shown in Fig. 6B is possibly due to that more flow is preserved in the capillary bed during this period [37].

F. Blood flow-to-volume relationship

The coupling between blood flow and blood volume change is an important factor in interpreting the fMRI BOLD signal and physiological models of blood flow regulation. Here, we investigated the ratio changes of CBF and CBV measured by tl-fUS during brain activation to validate the ability of tl-fUS for quantifying blood flow and volume changes. We averaged functional responses of rCBFspeed, rCBV and rCBF of all 8 datasets from 4 mice (Fig. 7A) with each dataset averaged from 10 trials and compared the peak relative changes of the three hemodynamic parameters as shown in Fig. 7B. The total functional change was calculated as the sum of arterioles and venules functional changes weighted by the number of pixels identified as arterioles and veins in the activation region. Note that for each dataset all functional signals were averaged on the same ROI to ensure flow and volume comparison with the same vascular compartments. The ratio of peak flow and peak volume change was around 2.5, agreeing with literature reports of 2 to 4[40]–[42], which indicates that tl-fUS enables accurate measurement of physiological parameters and is good for analyzing the flow-volume coupling.

Fig. 7.

Blood flow-to-volume relationship. (A) Time courses of total functional changes of rCBF_speed, rCBV and rCBF for all datasets (gray, n=8), and the average time courses (black). Each dataset was obtained by averaging the 10 repeated stimulation trials. The horizontal solid black line indicates when the stimulus is on. (B) Bar plot of the mean of the peak changes of rCBF_speed, rCBV and rCBF. Error bar represents the standard deviation.

V. Conclusion

In this work, we introduced the tl-fUS technique to differentiate the three hemodynamic parameters of CBF_speed, CBV and CBF that are often mixed in existing fUS techniques. We first derived the theoretical model, conducted numerical simulations for evaluation, and validated this method using phantom experiments. We then compared the proposed method with the widely used color Doppler and power Doppler measurements and performed whisker stimulation experiments on head-fixed awake mice. The results suggested that the tl-fUS technique can measure large-scale hemodynamic signal changes and can differentiate rCBF_speed, rCBV and rCBF.

We showed that the relative speed index obtained with tl-fUS linearly represents the preset flow speeds. Compared to color Doppler, the CBF_speed index obtained with tl-fUS is sensitive to both axial and transverse flows and is thus capable of imaging the total speed value in more vessels. For the blood volume measurement, the CBV index obtained with tl-fUS has a linear relationship to the power Doppler results but with a lower value. From the functional results, we saw that a larger responding region and stronger responses were observed for both speed and volume measurement using tl-fUS. Note that the multiangle emission-based vector Doppler [21] has the ability to obtain the total blood flow velocity, which may be of interest to applicatios where quantification of blood flow velocity is needed. Unlike color Doppler or vector Doppler, the tl-fUS is not intended to obtain the absolute value of the blood flow speed, but to obtain the relative change and differentiate it from blood volume change during functional stilumations with a more time and compuational effective approach.

For the analysis of arteriole and venule signals measured by tl-fUS, a higher volume change was observed in arterioles while a higher blood flow speed change was found in venules. This verifies for the first time with ultrasound measurements that arteriole vessel compartments contribute more to sensory-evoked increase in blood volume than the venule compartments. Our results support previous findings that venules manifest flow speed increases but no obvious dilation for a short stimulus of 4 seconds [39] and that upstream arterioles dilation was observed [38], indicating that arterioles play the primary role in CBV increase through vessel dilation and that blood volume and flow speed are inherently related and constrained through the conservation of blood flow within the whole volume. The estimated ratio of peak flow and peak volume measured with tl-fUS was around 2.5 times which is consistent with observations in fMRI studies [35], [43] and physiological modeling [41], [42]. This provides a further validation of our method for quantifying neural evoked hemodynamic signal changes.

Ultrasound speckle evolution contains invaluable underlying physiological information such as blood flow dynamics. In this work, we proposed a novel tl-fUS technique as an efficient approach to analyze ultrasound speckle dynamics and differentiate relative changes of rCBF_speed, rCBV and rCBF quantitively. This will make it a powerful tool to study neurovascular coupling, understanding the vascular origins of BOLD signals, and evaluating changes in cerebral hemodynamics arising from disease.

APPENDIX

Derivation of the normalized field autocorrelation function for a 3D ultrasound voxel

We consider that the time-varying ultrasound signal detected from a measurement voxel is composed of the integration of backscattered echoes from all moving scatters within the voxel and a noise term. The ultrasound complex-valued field signal at a time point $t$ for a given pixel can thus be written as:

$$\begin{gathered} sIQ\left(x_0,y_0,z_0,t\right)= \\ \sum _j^{N_s}{R}_s\cdot e^{-\frac{{\left(x_j(t)-x_0\right)}^2}{2{\sigma }_x^2}-\frac{{\left(y_j(t)-y_0\right)}^2}{2{\sigma }_y^2}-\frac{{\left(z_j(t)-z_0\right)}^2}{2{\sigma }_z^2}}{e}^{i2k_0\left(z_j(t)-z_0\right)}+R_{e}w(t) \end{gathered}$$ (15)

where, $N_s$ is the number of moving scatters within the voxel; $R_s$ is the reflection factor. We assume the reflectivity of the scatters (e.g., RBCs) are the same so that $R_s$ is considered a constant. ($x_j$, $y_j$, $z_j$) is the position of the $j^{th}$ scatter; ($x_0$, $y_0$, $z_0$) is the central position of the measurement voxel;${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$ are the Gaussian profile width at the $1/e$ value of the maximum intensity of the PSF in the x, y, and z directions, respectively; $k_0$ is the wavenumber of the central frequency of the transducer; $w\left(t\right)$ is the complex-valued white Gaussian noise with variance of 1 and $R_e$ denotes the noise level of the imaging system. The movement of the scatters induces amplitude and phase fluctuations of the ultrasound field detected at the center of the measurement voxel. The correlation of the temporal signal variation can be described by the field autocorrelation function $g$:

$$g_1(\tau )=E\left[\frac{{\langle sI{Q}^{\ast }(t)sIQ(t+\tau )\rangle }_t}{{\langle sI{Q}^{\ast }(t)sIQ(t)\rangle }_t}\right]$$ (16)

where, $E[\dots ]$ means the average over random initial positions of the particles; ${\langle \dots \rangle }_t$ represents an ensemble temporal average; $\tau$ is the time lag and * indicates the complex conjugate. In this study, we calculated $g_1(\tau )$ for $0<\tau <20ms$ with a time lag step of 0.2 𝑚𝑠.

Considering that all particles move from the same direction with the same speed, the position of the $j^{th}$ particle at time $\left(t+\tau \right)$ becomes: $x_j(t+\tau )=x_j(t)+v_x\tau$, $y_j(t+\tau )=y_j(t)+v_y\tau$, $z_j(t+\tau )=z_j(t)+v_z\tau$, where $v_x$, $v_y$ and $v_z$ are the velocities along the x, y and z direction, respectively. In order to relate $g_1(\tau )$ to the number of scatters $N_s$, we assumed that all scatters have the same probability distribution in space. Putting (15) into (16), we get:

$$g_1(\tau )=\frac{N_S{R}_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z{e}^{-\frac{{\left(v_x\tau \right)}^2}{4{\sigma }_x^2}-\frac{{\left(v_y\tau \right)}^2}{4{\sigma }_y^2}-\frac{{\left(v_z\tau \right)}^2}{4{\sigma }_z^2}}{e}^{i2k_0{v}_z\tau }+R_e\delta (\tau )}{N_s{R}_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z+R_e}$$ (17)

Here, $\delta (\tau )$ is the delta function which describes the autocorrelation of the noise with a time lag $\tau$. At zero time lag, $\delta (\tau =0)=1$ so that the numerator is equal to the denominator and $g_1(\tau =0)=1$. For a non-zero time lag $(\tau >0)$, Eq. (17) reduces to:

$$g_1(\tau )=\frac{N_s{R}_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z}{N_s{R}_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z+R_e}{e}^{-\frac{{\left(v_x\tau \right)}^2}{4{\sigma }_x^2}\frac{{\left(v_y\tau \right)}^2}{4{\sigma }_y^2}\frac{{\left(v_z\tau \right)}^2}{4{\sigma }_z^2}}{e}^{i2k_0{v}_z\tau }$$ (18)

Further, if the frame rate is fast enough the magnitude decorrelation of $\left|g_1(\tau )\right|$ at the first time lag can be simplified to Eq. (19) since the exponential term is close to 1 for this very short time lag and therefore the effect of the velocity on $\left|g_1\left({\tau }_1\right)\right|$ is negligible:

$$\begin{gathered} \left|g_1\left({\tau }_1\right)\right|=\frac{N_s{R}_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z}{N_s{R}_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z+R_e}{e}^{-\frac{{\left(v_x{\tau }_1\right)}^2}{4{\sigma }_x^2}-\frac{{\left(v_y{\tau }_1\right)}^2}{4{\sigma }_y^2}-\frac{{\left(v_z{\tau }_1\right)}^2}{4{\sigma }_z^2}} \\ \approx \frac{N_s{R}_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z}{N_s{R}_s\pi \sqrt{\pi }{\sigma }_x{\sigma }_y{\sigma }_z+R_e} \end{gathered}$$ (19)

The data processing code is available through shared file.