Experimental Validation of Perfusion Imaging with HOSVD Clutter Filters

Abstract

Novel pulsed-Doppler methods for perfusion imaging are validated using dialysis cartridges as perfusion phantoms. Techniques that were demonstrated qualitatively at 24 MHz, in vivo [18], are here examined quantitatively at 5 and 12.5 MHz using phantoms with blood-mimicking fluid flow within cellulose microfibers. One goal is to explore a variety of flow states to optimize measurement sensitivity and flow accuracy. The results show that 2–3 s echo acquisitions at roughly 10 frames/s yields the highest sensitivity to flows 1–4 mL/min. A second goal is to examine methods for setting the parameters of higher-order singular value decomposition (HOSVD) clutter filters. For stationary or moving clutter, the velocity of blood-mimicking fluid in the microfibers is consistently estimated within measurement uncertainty (mean coeff of variation = 0.26). Power Doppler signals were equivalent for stationary and moving clutter after clutter filtering, increasing approximately 3 dB per mL/min of blood-mimicking fluid flow for 0≤ q ≤4 mL/min. Comparisons between phantom and preclinical images show that peripheral perfusion imaging can be reliably achieved without contrast enhancement.

I. Introduction

Peripheral artery disease (PAD) caused by widespread atherosclerosis manifests prominently in the lower limbs [1]. It affects the lives of more than 12 million Americans, roughly 15% of the population over 55 years, as a loss of mobility and increased mortality [2]. The risk factors include smoking, diabetes mellitus, dyslipidemia, and hypertension. It often presents as patchy ischemia in foot and leg muscles creating numbness and exercise claudication, progressing to neuropathy, critical limb ischemia, and other cardiovascular deficits. The American Heart Association recommends PAD patients be monitored by assessing vascular flow and muscle perfusion throughout the lower extremities, coupled with evaluations of tissue oxygenation to assess wound healing potential [3].

Ankle-brachial index (ABI) measurements followed by CT or MR angiography are standard clinical assessments of PAD to detect and locate any large-vessel stenoses [3]. Microvascular occlusions resulting in regional ischemia are found applying molecular imaging probes and contrast media to optical [4], [5], nuclear [6], MR [7], and US [8], [9] modalities. Recently non-contrast-enhanced Doppler ultrasonic imaging of the microvasculature was successfully applied via plane-wave transmission and 2-D singular-value decomposition (SVD) clutter filtering [10], [11]. 2-D SVD clutter filters were shown to perform well at separating tissue from moving blood echoes when the clutter contribution to the singular spectrum is low rank [10], [12]–[16].

In previous reports, we describe a new SVD clutter filter applied to echo data acquired during long-duration temporal sampling in a manner that enhances perfusion echoes and clutter-blood separability [17]–[19]. Standard echo data acquisitions are arranged in a 3-D array with one spatial and two temporal axes. We showed that this approach enables visualization of the effects of angiogenic changes in an ischemic mouse hindlimb model and a melanoma tumor model without contrast enhancement [18]–[20].

So far, all of our reported measurements were made using a 24 MHz pulse frequency. This paper describes a set of phantom experiments at 5 and 12.5 MHz designed to validate the sensitivity and accuracy of our method for color-flow (CF) and power-Doppler (PD) imaging. At these lower pulse frequencies, we gain greater tissue penetration for future clinical exams at the cost of spatial resolution. With known phantom flow states, we explored and quantified the performance of the higher-order SVD (HOSVD) filter technique. Phantom and in vivo mouse data were compared to illustrate the strengths and weaknesses of fixed threshold versus statistical classifier approaches to setting clutter filter parameters.

II. Methods

A. Echo-data array structure and filtering for PD imaging

Data structure is important when implementing adaptive SVD clutter filtering techniques. As shown in Fig. 1 (left), the kth Doppler frame is composed of complex in-phase/quadrature (IQ) echo samples arranged in a 2-D array, $\tilde{X}\left(n,s\mid k\right)\in {ℂ}^{N\times S}$. N is the number of temporal samples (slow-time at a fixed spatial location) indexed along array columns and acquired at pulse-repetition rate 1/T₁, where 1 ≤ n ≤ N. S is the number of spatial samples recorded using a linear array for the nth Doppler pulse transmission. Spatial samples are indexed along the rows of array $\tilde{X}$ The S samples are composed of M axial samples acquired at rate 1/T and L lateral beams. Spatial samples are reordered by stacking echo lines end to end via $\tilde{X}\left(n,m,\mathcal{l}\mid k\right)\to \tilde{X}\left(n,s\mid k\right)$, where $s\triangleq m+\left(\mathcal{l}-1\right)M$, so that S = ML. Only one spatial axis of the data array is preserved for HOSVD processing because multiple spatial axes offer no more information than one axis when separating clutter, blood, and noise contributions. The third array dimension indexes the K Doppler frames acquired at frame-repetition rate 1/T₂ ≪ 1/T₁, as shown in Fig. 1 (top right), yielding $X\left(n,s,k\right)\in {ℂ}^{N\times S\times K}$.

Fig. 1.

(left) Diagram of K Doppler frames of echo data recorded and the associated acquisition timing. (right) The 3-D data array is parsed into stationary subregions and decomposed using HOSVD methods to give core tensor $\mathcal{G}$ and three eigenvector matrices U, V, W.

Echo-sample statistics are considered nonstationary because muscle perfusion varies regionally and slowly over time. So we decompose X into contiguous subregions that are scrolled along the spatial and frame-time axes, s, k, to give $\dot{X}\in {ℂ}^{N\times \dot{S}\times \dot{K}}$. See Fig. 1 (top right). Subregions are reduced in size until the echoes can be considered to be wide-sense stationary. In this way, the clutter filter is adaptive [17].

3-D decomposition of subregion $\dot{X}$ generates a core tensor $\mathcal{G}\in {ℂ}^{N\times \dot{S}\times \dot{K}}$ with elements g_n,s,k, and three eigenvector matrices U, V, and W [17], [21]. The nth column of N × N matrix U is the nth slow-time eigenvector u_n. Similarly for V and W. The data synthesis equation is

$$\dot{X}=\sum _{n=1}^N\sum _{s=1}^{\dot{S}}\sum _{k=1}^{\dot{K}}{g}_{n,s,k}{u}_n\times v_s\times w_k,$$ (1)

where u_n × v_s is the outer product of the two vectors [17].

Echo filtering requires that we identify for each subregion those core-tensor elements g_n,s,k that constitute the blood subspace ${\mathcal{G}}_B$; i.e., elements attributed to echoes from perfusing RBCs. The non-blood singular values are suppressed before echo data are resynthesized using

$${\dot{X}}_B=\sum _{\left(n,s,k\right)\in {\mathcal{G}}_B}{g}_{n,s,k}{u}_n\times v_s\times w_k.$$ (2)

These filtered echo data are further processed to generate power-Doppler (PD) or color-flow (CF) images via standard techniques [22], [23]. For example, the sth pixel of a PD images is found using [17],

$$PD\left(s\right)=\frac{1}{N\dot{K}}\sum _{n=1}^N\sum _{k=1}^{\dot{K}}{\left|{\dot{X}}_B\left(n,s,k\right)\right|}^2.$$ (3)

These pixels are reshaped into rectilinear coordinates for image presentation.

B. Phantom study

Similar to Li et al. [29], we adopted renal dialysis cartridges for experiments with perfusion-like fiber flows. The goal was to produce known flow conditions through parallel linear fibers that could be measured using color-flow and power-Doppler techniques. The phantom is illustrated in Fig. 2. The dialysis cartridge (Spectrum Laboratories Inc., Rancho Dominguez, CA) contains a bundle of 20 cm-long cellulose fibers with inner and outer diameters 0.63 mm and 0.87 mm, respectively. We cut 6 cm holes into opposing sides near the center of the thick plastic cartridge housing and covered them with a thin latex membrane. A thin-walled 8-mm-dia latex tube was placed adjacent to the cartridge to simulate large-vessel flow. The cartridge and latex elements were soaked for several hours in a weak concentration of soapy water to minimize air trapped in surfaces that would reduce acoustic penetration.

Fig. 2.

The perfusion phantom is a cellulose-fiber dialysis cartridge. A syringe pump infuses the fibers with tissue-mimicking blood. Clutter motion is introduced by the peristaltic pump introducing water pulse within the cartridge but outside the fibers. From [18].

A calibrated syringe pump (Harvard Apparatus, Holliston MA) determined the net flows. Fluid infused through the cellulose fibers was either degassed water (control) or blood-mimicking fluid (Doppler Refill Fluid, CAE Healthcare, Sarasota, FL).

Scattering from cellulose fibers ensured that stationary or moving clutter sources were always present. Clutter motion was introduced when a peristaltic pump injected a short-duration water pulse every 6.25 s into the cartridge housing but outside the flow fibers. The labels ‘stationary’ or ‘moving’ clutter in the results indicate the peristaltic pump is turned off or on, respectively. Phantom parameters are summarized in Table I.

TABLE I.

Phantom Parameters

Parameter	Value
Control fluid	degassed water
Perfusion fluid	blood mimicking fluid
Flow rate range	0–4 mL/min
Clutter motion level 0	no motion
Clutter motion level 1	peristaltic rotation 0.16 Hz
Tapping motion clutter	0.5 Hz
Phantom fiber length	200 mm
Phantom passing time @ 2 mL/min	5′5″–5′30″ ($\overline{v}\prime$ =0.61–0.65 mm/s)
Phantom fiber bundle diameter	10–12 mm
Phantom fiber inner/outer diameters	0.63/0.87 mm
Doppler angle (θ)	75–83°

The phantom was scanned with a Vevo 2100 ultrasound imaging system transmitting 12.5 MHz Doppler pulses (FUJIFILM VisualSonics Inc. Toronto, Ontario, Canada) and a Sonix RP system transmitting 5 MHz Doppler pulses (Ul-trasonix Medical Corp., Richmond, BC, Canada). Blood-mimicking fluid flow was reversible; rates ranged from 0 to ±4 mL/min. Acquisition parameters are summarized in Table II.

TABLE II.

Acquisition Parameters

	Vevo 2100	Vevo 2100	Sonix RP
Probe	MS-200	MS-400	L14–5
Pulse center frequency (f0)	12.5 MHz	24 MHz	5 MHz
ST sample rate (1/T1)	1 kHz	1 kHz	1 kHz
ST ensemble size (N)	17 (0.017 s)	17 (0.017 s)	12 (0.012 s)
FT sample rate (1/T2)	15 Hz	8 Hz	20 Hz
FT ensemble size (K)	100 (6.7 s)	100 (12.5 s)	150 (7.5 s)
Fast-time sample rate (1/T)	12.5 Ms/s	12.5 Ms/s	20 Ms/s
Axial samples/frame (M)	248 (15.3 mm)	216 (10 mm)	448 (17.2 mm)
Lateral sample/frame (L)	53 (13.2 mm)	252 (15.1 mm)	78 (29.6 mm)
Scan-line density	4.02 lines/mm	16.67 lines/mm	2.63 lines/mm
Spatial samples (S = LM)	13144	54432	34944
Axial sub-block size ($\dot{M}$)	50 (3.1 mm)	17(0.5 mm)	80 (3.1 mm)
Lateral sub-block size ($\dot{L}$)	30 (7.5 mm)	16 (1 mm)	20 (7.6 mm)
Frame-time sub-block ($\dot{K}$)	30 (2.0 s)	16 (2.0 s)	40 (2.0 s)

ST is slow time and FT is frame time

C. In vivo mouse images

We compared in vivo results with phantom measurements by scanning the right hindlimb of a mouse at 24 MHz, but otherwise under similar conditions to the 12.5 MHz acquisition. The mouse was anesthetized (isofluorane 0.5 −1.5 l/min) and placed in the supine position on a 37°C heating pad. The hindlimb was scanned AP along a longitudinal cross section in a plane just lateral to the femur. Acquisition parameters at 24 MHz are summarized in Table II. All experiments were performed with the approval of the Institutional Animal Care and Use Committee of the University of Illinois at Urbana-Champaign following the principles outlined by the American Physiological Society on research animal use.

D. Flow speed estimation

Traditionally, peripheral perfusion is assessed using ultrasonic PD methods because the slow, multi-directional movement of RBCs in the microvasculature is best captured by the net echo power after clutter and noise filtering. Nevertheless, in this section we describe CF experiments aimed at estimating the speed of blood-mimicking fluid flow in the phantom. Slow directional flow in phantom fibers offers an opportunity to quantitatively validate our techniques, including the effectiveness of clutter filtering.

With a stopwatch and knowledge of fiber length, we measured the time of fluid flow through cartridge fibers to estimate the spatial mean velocity for blood-mimicking fluid, $\overline{v}\prime$ (Table I). Transit times were slow enough to clearly observe but flow speeds varied among the fibers. Repeated testing gave a coefficient of variation for $\overline{v}\prime$ less than 7%.

Acquiring and filtering echo data from the phantom gives ${\dot{X}}_B\left(n,s,k\right)$ via (2). We applied a conventional autocorrelation method [23] to calculate the mean frequency of the filtered echo data. At each spatial position s and pulse within the slow-time ensemble n, we computed the lag-one complex correlation function along the frame-time axis. At each spatial location s the mean correlation function is found by averaging over $\dot{K}-1$ pairwise frame-time frames and N slow-time ensembles. To calculate mean frequency $\overline{u}\left(s\right)$ we compute the phase (∠(.)) of the averaged correlation function,

$$\overline{u}\left(s\right)=\angle \frac{1}{2\pi N\left(\dot{K}-1\right)}\sum _{n=1}^N\sum _{k=2}^{\dot{K}}\left({\dot{X}}_B^{†}\left(k\mid n,s\right){\dot{X}}_B\left(k-1\mid n,s\right)\right),$$ (4)

where † denotes conjugate transpose.

The process is repeated at every spatial location to generate a spatial map of flow velocity,

$$\widehat{v}\left(s\right)=\frac{c}{2N{T}_2{f}_0\text{cos}\theta }\overline{u}\left(s\right).$$ (5)

The mean compressional wave speed is c, f₀ is the pulse carrier frequency, and θ is the Doppler angle.

To avoid aliasing, frame rate 1/T₂ ≥ (4f₀|υ_max| cos θ)/c, where |υ_max| is the highest velocity present in the echo signal. In situations where frame interval T₂ could not be reduced further, we adjusted Doppler angle θ. At higher flows, the extended autocorrelation method [24] was applied that combines amplitude and phase information to estimate mean velocities beyond the Nyquist limit.

Color-flow images are formed in a standard way by reshaping estimates into a 2-D image plane, $\widehat{v}\left(s\right)\to \widehat{v}\left(m,\mathcal{l}\right)$, and then superimposing them as a color overlay onto grayscale B-mode images. The spatial-mean velocity $\overline{v}={\sum }_{s=1}^{\dot{S}}\widehat{v}\left(s\right)/\dot{S}$ for a region of relatively homogeneous flow is also found.

E. Filter thresholds

The most challenging aspect of obtaining high-quality perfusion images is selecting clutter and noise filter thresholds. We previously described a statistical classifier for identifying core tensor elements g_n,s,k dominated by tissue clutter or acquisition noise sources [18]. Elements so classified are eliminated from inclusion in (2).

Classification is based on a five-feature vector, where features are computed from HOSVD output. Three features are normalized eigenvalues (singular values) computed from data along the three axes of the echo-data core tensor. Large eigenvalues are associated with clutter sources because the scattered amplitude from tissue is generally greater than that from blood and noise. A fourth feature measures correlations among spatial eigenvectors. The rationale is that highly correlated spatial eigenvectors are more likely to occur from tissue echoes than perfusing blood echoes. This measure is a component of the correlation matrix of spatial eigenvalues used by others to set 2-D SVD filter thresholds [16]. A fifth feature is the correlation of echo power; the goal is to sense coherent clutter movements among adjacent pixels from breathing and probe motion. Since each element of $\mathcal{G}$ is independently compared to statistical thresholds [18], the blood subspace within the core tensor is not contiguous as illustrated in Fig. 1.

The mean feature vector and its covariance matrix are estimated for tissue, blood and noise regions. These are combined using a Gaussian mixture (GM) model and trained to distinguish clutter from blood echoes as described in [18]. Acquisition noise was separated from blood signals using a minimum description length (MDL) method [30]. We call the combined clutter-noise filter a GM-MDL filter.

The statistical classifier appeared to function effectively in vivo [18]. Since echo data in this report are better characterized than in vivo measurements, we used data from flow phantoms and manually set filter thresholds to observe the filtering process quantitatively. Comparisons among the fixed threshold method described in this work and statistical classifier methods [16], [18] are presented. The remainder of Section II-E describes how the fixed filter thresholds listed in Table III were selected.

TABLE III.

Fixed HOSVD Clutter and Noise Filter Thresholds for phantom data. Last two columns give the eigen-indices set to define the blood subspace along the three data dimensions.

	Pulse Frequency	Data Array Dimension	Stationary Clutter	Moving Clutter
Phantom	5 MHz	Slow-time	1–5	1–5
		Frame-time	3–10	7–14
		Spatial	5–20	7–25
Phantom	12.5 MHz	Slow-time	1–5	1–5
		Frame-time	3–20	7–25
		Spatial	5–30	8–30
Mouse Hindlimb In vivo	24 MHz	Slow-time	1–5	-
		Frame-time	3–10	-
		Spatial	5–20	-

1). Spatial thresholds:

Spatial eigenstates are very informative about the boundaries between the clutter, blood and noise subspaces. Although subspaces in $\mathcal{G}$ are defined in 3D, the boundaries set along the spatial axis are particularly critical for successful filtering. Fig. 3 gives examples of singular spectra along the spatial axis from phantom and in vivo mouse hindlimb data. In both situations, the blood subspace falls between the third and twentieth eigenvalues despite differences in experimental parameters. Stationary clutter occupies the first two eigenstates, while noise dominates the twenty-first state and above. The clutter peak broadens as the magnitude of moving clutter increases, which is automatically detected by the statistical classifier.

Fig. 3.

Largest 40 spatial eigenvalues for 5 MHz phantom data at 3 mL/min flow rate and (right) a 24 MHz perfused mouse hindlimb [17] at roughly the same average flow that we determined from the PD image intensity. The clutter is stationary in both measurements. Arrows indicate natural positions for setting the fixed filter thresholds. Since the clutter boundary occasionally extends to higher eigen-indices, we set the fixed lower spatial threshold to 5 as indicated in Table III.

Similarly, spatial eigenvectors (columns of V) reflect distinct features of the scattering sources. Each spatial eigenvector v_s can be reshaped into an M × L image ${\nu }_s\left[m,\mathcal{l}\right]$ that resembles structural features in the image associated with that data mode. Eigenvectors dominated by clutter resemble smooth B-mode image features as shown by the reshaped spatial eigenvector at index s = 2 in Fig. 4. Those dominated by acquisition noise, e.g., at s = 22 in Fig. 4, are featureless except for an uncorrelated noise pattern. Eigenvector s = 12 is considered dominated by moving blood signal power because it shares both features.

Fig. 4.

Three reshaped spatial eigenvectors (right) from the boxed region in the 5 MHz colorflow phantom image (left; 2 mL/min flow). Eigenimages are for the eigenindex value s indicated. Examples are from the clutter (s = 1, 2), blood (s = 3 – 20), and noise (s > 20) subspaces as identified by arrows in Fig. 3. Considering there are 1600 eigenvectors in V, the clutter and blood subspaces are sparse in the SVD representation.

Figs. 3 and 4 show that complementary information is shared by spatial eigenvalues and eigenvectors, although exact subspace boundaries are not always clear from visual inspection. For that reason, a statistical classifier determines filter thresholds for in vivo imaging. The fixed threshold values applied in this section are listed in Table III. Fixed values were applied to the stationary and moving clutter phantom measurements found in the Results section below.

2). Frame time:

The frame-time eigenvector matrix W provides some additional information for parsing subspaces, but is most valuable and essential for estimating the velocity of slow-moving targets. The most intuitive display of frame-time information involves eigenspectra. Eigenspectra are found by computing the magnitude of the discrete Fourier transform for each column of W, i.e., $\left|\mathcal{F}\left\{w_k\right\}\right|$ and displaying the results as a 2-D array as in Fig. 5. The temporal frequency axis was converted into velocity. Eigenspectra illustrate the relationship between the frame-time eigenindex (horizontal axis) and blood velocity (vertical axis). Doppler frequency spectra for perfusion are found by summing along rows of filtered eigenspectra as discussed below.

Fig. 5.

An example of an eigenspectrum, where the horizontal axis indicates the index of the first 40 frame-time eigenvectors and the vertical axis indicates velocity with zero at the center and negative velocities in the top half. Here we see the effects of flow toward the transducer (+υ) within stationary clutter as a diagonal linear pattern in the lower half. Eigenvectors at the first two indices are discarded to reject clutter and the last 15 eigenvectors are discarded to reject acquisition noise and aliased power. Data at frame-time indices 3–25 are entered into (4); others are set to zero. We further limited the velocity range included in the PD signal by setting the ±(υ_min, υ_max) bands shown and rejecting signal power outside these bands.

We may further limit the power mapped into a PD image to values within a specified velocity range, as illustrated by the horizontal bands in Fig. 5. Velocity limits are placed on both positive and negative velocities ranges.

Eigenspectra for the 12.5 MHz phantom data with stationary clutter are shown in the first two columns of Fig. 6 for flows toward (+υ) and away (−υ) from the transducer. The filtered Doppler spectra shown in the right two columns are found by summing transformed eigenvectors from the left two columns along rows in the range 3 ≤ k ≤ 20. These index bounds were selected to give consistent velocity estimates under the experimental conditions. With moving clutter, the filter thresholds change as listed in Table III. Eigenspectra are symmetric when there is no perfusion signal (first two rows of Fig. 6). Increasing the flow from 1–3 mL/min changes the slope of the flow spectrum such that faster flows begin to alias at progressively lower eigenindices, as indicated by the arrows in Fig. 6.

Fig. 6.

Phantom measurements at 12.5 MHz were acquired for stationary clutter and the flow rates indicated. The leftmost two columns are pre-filtered frame-time eigenspectra (see Fig. 5). Spectral power along positive velocities (first and third columns) specifies flow toward the transducer; that along negative velocities indicates flow away from the transducer (second and fourth columns). The rightmost two columns are post-filtered Doppler spectra [dB] found by summing linear pre-filtered eigenspectra on the left between the dotted white lines. Arrows indicate the wraparound points for aliasing that boost high frequencies in the Doppler spectra.

3). Slow time:

Eigenspectra formed from the slow-time eigenvector matrix U provided little flow information because at the 1 kHz pulse repetition frequency the echo spectrum did not have the frequency resolution needed to be sensitive to slow flow. Because the clutter power was also weak, including the first 5 slow-time eigenvectors resulted in passing blood signal power and rejecting noise.

III. Results

A. Color-flow images

Figure 7 displays examples of CF images of the phantom at 5 and 12.5 MHz for fixed filter thresholds. The directional organization of the fibers yields fairly uniform velocity maps of the 2 mL/min flows in both directions. Images with moving clutter were similar to those shown in Fig. 7.

Fig. 7.

CF images of the phantom for stationary clutter using fixed-threshold clutter filtering. Velocity measurements were spatially averaged to give results in Fig. 8.

B. Flow speed measurements

Figure 8 summarizes measurements where velocity estimates in a 7 × 15 mm region of the CF image are averaged. To mimic peripheral perfusion, Doppler frames immediately following the peristaltic pump pulse were removed leaving the gentle waving of fibers within the plastic housing to serve as clutter.

Fig. 8.

Spatially averaged velocities, $\overline{v}$ for N = 3 experiments, are estimated as a function of blood-mimicking fluid flow rate for stationary (top) and moving (bottom) clutter. Error bars indicate ±1 standard deviation. The narrow gray area near the prediction line indicates the range of stopwatch measurements of fluid velocity $\overline{v}\prime$. Uncertainty for both measurements increases with flow, reflecting the spatial variation in individual fiber flows. The center three measurements all indicate a no-flow state. The measurement labeled ‘water’ has stationary water in the fibers while the other two labeled ‘0 mL/min’ have stationary blood-mimicking fluid in the fibers. The points in black at ±2 and ±3 mL/min are found by reprocessing the 12.5 MHz data to estimate velocity using the extended autocorrelation method [24] to minimize aliasing. Fixed-threshold clutter filtering was applied.

The results indicate that spatially-averaged velocity $\overline{v}$ closely tracks the independent stopwatch measurements $\overline{v}\prime$. Deviations of $\overline{v}$ from $\overline{v}\prime$ at higher flow levels are attributed to limited frame rate adjustments causing aliasing along the frame-time axis most noticeably at 12.5 MHz. To minimize the influence of aliasing, velocities were recalculated using the extended autocorrelation method [24]. These estimates appear as points on the 12.5 MHz plots at ±2 and ±3 mL/min. Error bars reflect the spatial variability in velocity as seen in the examples of Fig. 7. Excluding the three control measurements in each plot of Fig. 8, the average coefficient of variation for other measurements at 5 MHz is 26%.

C. Power-Doppler images

Figure 9 illustrates additional properties of slow-flow power Doppler imaging. Because lateral resolution at 12.5 MHz for an f/2 probe is ≳ 0.25 mm, flow in the 0.87 mm fibers is resolvable. The top row of images are a control state with stationary water in the fibers. The second and third rows include flowing blood-mimicking fluid in the fibers. Appropriately, no flow is indicated in the control state for stationary clutter. However, flow is indicated in the control state for moving clutter from latex sheath refections (arrows) where echo strength and range of motion are both large. An oscillating latex sheath expands the clutter eigenbandwidth into the blood subspace in those regions. Although we adjusted clutter and noise filter parameters as shown in Table III, suppressing additional spatial and frame-time eigenvectors would inappropriately reduce the indication of true flow at 1 and 3 mL/min found in rows 2 and 3 of Fig. 9.

Fig. 9.

PD images of the phantom at 12.5 MHz with stationary (left) and moving (right) clutter for three flow states. The view is magnified 2x compared with CF images in Fig. 7. Arrows indicate the sheath membrane encompassing the fibers. Color bars indicate signal power in dB. Fixed-threshold clutter filtering was applied.

Figure 10 shows that power increases exponentially with flow, which appears as a linear increase for power displayed in dB. The slope of power versus flow is roughly 3 dB per mL/min for flows 0 ≤ q ≤ 4 mL/min at both 5 and 12.5 MHz. When blood mimicking fluid is not flowing, the clutter filter appropriately removes its contribution to signal power as it is indistinguishable from stationary clutter. So we see a sudden drop in power for blood mimicking fluid at q = 0 mL/min compared with q = 1 mL/min in stationary clutter. Signal power in states without flowing blood-mimicking fluid is 5–10 dB larger for moving clutter compared with stationary clutter, showing clutter signals are not completely eliminated by filtering.

Fig. 10.

Spatially averaged post-filtered echo power is plotted versus input flow for stationary and moving clutter. Error bars indicate standard errors for N = 3 experiments. There is no flow in the fibers for the first two points plotted. The 0 mL/min values have stationary blood-mimicking fluid present, which generates more signal power than the stationary water state. Fixed-threshold clutter filtering was applied. Linear fits are for the moving clutter data when blood-mimicking fluid is present (excluding the water points).

D. Comparison of three SVD clutter filter methods

Figs. 11 and 12 display results from which three methods for setting SVD clutter filter parameters can be qualitatively compared. Using the same 5 MHz phantom data, the fixed threshold method described in this report with parameters from Table III may be compared with results using the adaptive spatiotemporal method of Baranger et al. [16] and our adaptive GM-MDL statistical classifier [18] in Fig. 11. Two of the methods may be compared for perfusion in the mouse hindlimb in Fig. 12. Generally, the fixed threshold approach eliminates more low-level echo power than the other two. Both statistical approaches to SVD filtering give similar results as seen comparing Fig. 11b with c and e with f.

Fig. 11.

PD images of the phantom at 5 MHz for a 3 mL/min blood-mimicking fluid flow rate. The first row includes stationary clutter, while the second row includes moving clutter. Images (a) and (d) apply the fixed SVD filter thresholds listed in Table III. (b) and (e) are processed with the adaptive spatiotemporal auto-thresholding method of [16]. (c) and (f) are processed using the GM statistical classifier clutter filter with MDL noise filtering [18].

Fig. 12.

PD images of a mouse hindlimb processed with (a) the fixed SVD clutter filter thresholds listed in Table III and (b) the GM classifier clutter filter with MDL noise filtering.

Fig. 13 displays the effects on signal power in the blood subspace when we vary the first eigen-index threshold from 1 to 9 while maintaining the second threshold at 20. The first threshold is the boundary between clutter and blood. The second threshold, fixed at 20, sets the boundary between blood and noise. Data show that thresholds set at 5–20 result in some loss of blood-mimicking fluid flow power, which is necessary to eliminate clutter.

Fig. 13.

Spatially averaged post-filtered echo power is plotted for 5 MHz phantom data with stationary clutter as a function of the spatial threshold at the clutter-blood interface. As the threshold increases from 1 to 9, blood power is lost above a setting of 3. Error bars indicate standard deviations. For all data, the frame-time and slow-time thresholds are fixed at 3–10 and 1–5, respectively. The three sets of threshold values define the blood subspace within the core tensor as diagrammed in Fig. 1.

E. Dependence of PD image sensitivity on acquisition time

We extended acquisition time τ₂ in an effort to obtain high Doppler-frequency resolution. High frequency resolution increases the number of independent samples applied to power or velocity estimates, which improves slow-flow sensitivity. Fig. 14a is the result of an experiment to observe how post-filtered power depends on τ₂. In principle, measured echo power should depend on flow but not τ₂. That is essentially the case for τ₂ ≥ 3 s, but at lower acquisition times filtered power diminishes as the number of samples included in the estimate declines.

Fig. 14.

(a) Dependence of power Doppler image magnitude on acquisition time τ₂ for stationary clutter (see timing diagram in Fig. 1). At 5 MHz, 20 Doppler frames are recorded each second. However, the SVD filter along the frame-time axis passes only 25% of the available 20 eigenvectors/s, as seen in (b) between the two vertical dashed lines in eigenspectra for τ₂ = 1, 3, 5s. Since eigenspectra scale with τ₂, longer acquisitions increase the number of samples from which velocity is estimated.

Examining the eigenspectra in Fig. 14b, we see that the number of samples passed by the clutter filter increases from 5 to 15 to 25 as τ₂ increases from 1 s to 3 s to 5 s; it is 25% of the available samples for these parameter setting. Short duration echo acquisitions include fewer quantized samples in power Doppler measurements. For τ₂ ≲ 2 – 3 s, echo power sensitivity is reduced. Power levels peak at 3 s for all nonzero flows. However, when 3 s values are compared to those at 2 s, we find t-statistic probabilities of 0.20, far above the 0.05 level expected to reject the null hypothesis.

IV. Discussion

Our objective is to report the results of validation studies for the perfusion imaging methods introduced in [17], [18]. These methods were previously demonstrated on preclinical animal models qualitatively since the flow states were unknown. Here, we use phantoms with measurable flow geometries and speeds. In addition, all tests were conducted at 5 and 12.5 MHz pulse frequencies where the depths of penetration are consistent with the requirements of clinical exams.

The results of Figs. 8 and 10 summarize the results of velocity and power measurements as a function of phantom perfusion rate with and without clutter motion. The clutter filter is effective for all blood-mimicking fluid flow measurements where q > 0. At q = 0 some clutter power is passed through the signal filter influencing power but not velocity measurements. When the sudden transient from the peristaltic pump is included in the echo signals processed (results not shown), we find more clutter power mixes with blood power, further reducing the effectiveness of the HOSVD clutter filter. We decided to eliminate the sudden clutter transient to more closely mimic clutter conditions found during peripheral perfusion imaging in vivo.

The two main determinants of perfusion measurement accuracy are (a) the presence of a spatiotemporal signature in the echo signal that characterizes blood flow as distinct from clutter, and (b), given that the subspaces are separable, choosing SVD filter thresholds that correctly define the blood subspace.

Preclinical studies showed that the clutter subspace for in vivo peripheral perfusion imaging is low rank [18]. A low-rank clutter component in the echo signal is separable from the blood component. In contrast, a full-rank clutter component cannot be separated from flowing blood-mimicking fluid. The predominant clutter motion in clinical peripheral vascular imaging is from patient and probe motions that generate spatially-coherent echo signatures forming a low rank clutter subspace.

When the clutter and blood subspaces are separable, SVD filter thresholds are key to successfully isolating blood echoes. Results, including Figs. 11 and 12, show that the adaptive thresholds provided by statistical classifiers offer clutter and noise filtering options that are more sensitive to the motion of perfusion than fixed thresholds. Fixed thresholds are suboptimal but quickly implemented and perform reasonably well.

Arranging the echo-data array into spatiotemporal dimensions provides physical meaning to the eigenvectors emerging from the HOSVD process. Appropriate sampling along the frame-time dimension (8–20 Hz) ensures sensitivity to perfusing blood. The higher slow-time sampling (1 kHz) yields little sensitivity to slow flows, e.g., < 4 mL/min at speeds < 1.2 mm/s, although averaging signals along slow-time axis reduces noise in power estimates. The spatial and frame-time dimensions provided all of the information necessary for rejecting low-rank clutter echoes. A strength of any SVD filtering approach is its ability to adapt to characteristics of each data set.

A distinguishing feature of 3-D SVD clutter filters over 2-D filters is preservation of the frame-time data axis as opposed to frame averaging. The duration of the frame-time axis determines the frequency resolution and hence the number of independent samples in the post-filtered echo signal from which power is estimated. As seen in Fig. 14, τ₂ = 2 – 3 s maximizes blood-echo power under these experimental conditions. Shorter frame-time acquisitions reduce the number of samples in the quantized post-filtered echo spectrum, which reduces sensitivity of the echo signal to slow-moving blood echoes. Longer acquisitions increase the chance of encountering time-varying echoes that generate nonstationary echo-signal statistics. Finally, the SVD filter thresholds set for phantoms apply equally to in vivo perfusion imaging provided the clutter and blood are separable.

V. Conclusions

The sensitivity of blood perfusion imaging with commercial instruments to slow-moving blood echoes is enhanced by processing frame-time echo data. Power Doppler imaging is conveniently implemented by forming a 3-D echo-data array with one spatial and two temporal dimensions that is filtered using HOSVD methods. We found that velocity and power were accurately measured for flows 0 ≤ q ≤ 4 mL/min. Echo power increased 3 dB per mL/min over the same range. These results hold for situations in which the clutter component to the echo signal is low-rank.