Tracer arrival timing-insensitive technique for estimating flow in MR perfusion-weighted imaging using singular value decomposition with a block-circulant deconvolution matrix

Authors

  • Ona Wu,

    Corresponding author
    1. Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital/Massachusetts Institute of Technology/Harvard Medical School, Boston, Massachusetts
    • Mailcode CNY149-2301, MGH/MIT/HMS Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Boston, MA 02129
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  • Leif Østergaard,

    1. Department of Neuroradiology, Center for Functionally Integrative Neuroscience, Århus University Hospital, Århus, Denmark
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  • Robert M. Weisskoff,

    1. EPIX Medical, Cambridge, Massachusetts
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  • Thomas Benner,

    1. Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital/Massachusetts Institute of Technology/Harvard Medical School, Boston, Massachusetts
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  • Bruce R. Rosen,

    1. Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital/Massachusetts Institute of Technology/Harvard Medical School, Boston, Massachusetts
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  • A. Gregory Sorensen

    1. Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital/Massachusetts Institute of Technology/Harvard Medical School, Boston, Massachusetts
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Abstract

Relative cerebral blood flow (CBF) and tissue mean transit time (MTT) estimates from bolus-tracking MR perfusion-weighted imaging (PWI) have been shown to be sensitive to delay and dispersion when using singular value decomposition (SVD) with a single measured arterial input function. This study proposes a technique that is made time-shift insensitive by the use of a block-circulant matrix for deconvolution with (oSVD) and without (cSVD) minimization of oscillation of the derived residue function. The performances of these methods are compared with standard SVD (sSVD) in both numerical simulations and in clinically acquired data. An additional index of disturbed hemodynamics (oDelay) is proposed that represents the tracer arrival time difference between the AIF and tissue signal. Results show that PWI estimates from sSVD are weighted by tracer arrival time differences, while those from oSVD and cSVD are not. oSVD also provides estimates that are less sensitive to blood volume compared to cSVD. Using PWI data that can be routinely collected clinically, oSVD shows promise in providing tracer arrival timing-insensitive flow estimates and hence a more specific indicator of ischemic injury. Shift maps can continue to provide a sensitive reflection of disturbed hemodynamics. Magn Reson Med 50:164–174, 2003. © 2003 Wiley-Liss, Inc.

MR perfusion-weighted imaging (PWI) in acute stroke patients has been shown to be a sensitive indicator of tissue at risk of infarction (1, 2). A prevalent technique for calculating cerebral blood flow (CBF) and mean transit time (MTT) in PWI uses singular value decomposition (SVD) for deconvolving the tissue signal from the first pass of a bolus of contrast agent with an arterial input function (AIF). Although this method is robust to noise and is independent of underlying vascular structure and volume (3), it has been shown to be sensitive to both delay and dispersion, leading to grossly underestimated CBF in these situations (3–6). Furthermore, it has been demonstrated that in some cases of major vasculopathy wherein the tracer arrives earlier in some tissue than in the chosen AIF, CBF can be either over- or underestimated, depending on the extent of the tracer arrival difference and underlying hemodynamics of the measured tissue (7).

Although Fourier techniques are insensitive to delay in tracer arrival, they have been shown to provide poor estimates of high flow rates in the presence of noise (3, 8). Maximum likelihood expectation maximization techniques (9) or model-dependent approaches that fit for delay and dispersion (10) are also promising, because they are both less sensitive to noise than Fourier techniques and less sensitive to tracer arrival time differences. However, the performances of these techniques have yet to be evaluated in cases of ischemic disease or major vasculopathy, where the assumptions and models upon which they are based may no longer be valid. Region-of-interest (ROI)-based methods to compensate for delay by selecting one AIF and shifting it in time until it is synchronized with the tissue signal curve, using parametric curve-fitting (11) or numerically estimated bolus arrival times (12), may not translate well on a voxel-by-voxel basis due to a lower signal-to-noise ratio (SNR) (6, 8).

This work proposes to improve flow estimates in two ways: 1) by performing deconvolution with a block-circulant matrix to reduce sensitivity to tracer arrival differences between the AIF and tissue signal, and 2) by using local, rather than global, regularization. The performances of this method and the existing deconvolution approach using SVD (3) are evaluated in terms of flow estimation errors as a function of tracer arrival differences using numerical simulations where CBF values are known. In addition, techniques for estimating tracer arrival differences with respect to the AIF are evaluated. The performances of the methods are also compared qualitatively on clinically acquired human PW images. In particular, we examine two conditions that have been previously demonstrated to be problematic for SVD deconvolution: when internal carotid artery (ICA) stenosis (13) exists, and when the AIF regionally lags the tissue signal (7). We also examine two acute stroke cases in which initial PWI lesion volumes calculated using standard techniques are greater than lesion volumes detected on diffusion-weighted imaging (DWI).

THEORY

Background

Assuming a linear relationship between concentration of a high magnetic susceptibility contrast agent and change in transverse relaxation (ΔR2) in dynamic susceptibility contrast-weighted (DSC) MR images (14), the change in concentration over time during the passage of a bolus of contrast agent in a volume of tissue C(t) can be characterized as:

equation image(1)

where S0 is the baseline MR intensity, and S(t) is the signal intensity over time. By defining the residue function R(t) as the fraction of tracer remaining in the system at a given time t, the concentration of tracer within the voxel of tissue as a function of time can be modeled as (3):

equation image(2)

where Ca(t) is the AIF and Ft is the tissue blood flow. The AIF is typically measured directly from the MR images, which was shown in previous studies to correlate well with an AIF measured directly from arterial blood samples (15). R(t) can then be estimated using deconvolution with the measured ΔR2(t), which is often performed using SVD (3). In SVD, by expressing Eq. [2] in discretized format:

equation image(3)

where Δt is the sampling interval, and expanding Eq. [3] into matrix notation, the deconvolution problem can be formulated as an inverse matrix problem:

equation image(4)

Simplifying the above equation to c = A · b,one can solve for b, the elements of R(t) scaled by Ft. The measured AIF, Ca(t), is typically prefiltered to reduce noise contributions and to compensate for discretization errors (3), resulting in A with elements:

equation image(5)

By decomposing A = U · S · VT, where U and V are orthogonal matrices and S is a nonnegative square diagonal matrix, the inverse can be expressed as A–1= V · W · UT where W= 1/S along the diagonals, and zero elsewhere. Values of W corresponding to values where S is less than a preset tolerance threshold, PSVD (usually a percentage of the maximum value of S), are set to zero. The residue function scaled by Ft, b, can then be estimated by b =FtV · W · UT· c, and rCBF is estimated as b's maximum value.

Deconvolution Using a Block-Circulant Matrix

One of the assumptions built into Eq. [4] is causality, i.e., the voxel signal cannot arrive before the AIF. However, the AIF can lag C(t) by a time delay td in practice, since the measured AIF, Ca′(t), is not necessarily the true AIF for that voxel, Ca(t), thus resulting in Ca′(t) = Ca(t–td.). For example, this lag can occur when the chosen AIF comes from a highly diseased vessel. Therefore, the calculated R′(t) should be R(t + td) for C(t); but if causality is assumed, R′(t) cannot be properly estimated by inversion of Eq. [4]. However, by using circular deconvolution instead of linear deconvolution, R′(t) can be represented with R(t) circularly time shifted by td. Circular convolution has been shown to be equivalent to linear convolution with time aliasing (16), and is also mathematically equivalent to a standard inverse Fourier method. By zero-padding the N-point time series Ca(t) and C(t) to length L, where L2N, time aliasing can be avoided. Therefore, replacing matrix A with a block-circulant matrix, D, whose elements are di,j = aij for ji, and di,j = aL+i-j,0 otherwise, Eq. [4] can be reformulated as g = D · f, where g is the zero-padded c, and f is the residue function scaled by Ft. The inverse of D can be decomposed to D–1= Vc· Wc· Umath image. One can again make use of SVD techniques to solve for f by f= FtVc· Wc· Umath image · g. When using circular deconvolution, however, due to the discontinuities at t = 0 and t = L, leakage frequencies may be amplified, giving rise to spurious oscillations dominating the deconvolved signal (17). Increasing PSVD reduces the oscillations (18). Using a modified oscillation index from that described by Gobbel and Fike (19):

equation image(6)

where f is the scaled estimated residue function, fmax is the maximum amplitude of f, and L is the number of sample points, PSVD can be varied until the estimated residue function's oscillation index falls below a user-specified value (OI) and CBF set to fmax at that instance.

METHODS

Monte Carlo Simulations

An AIF was simulated using a gamma-variate function, which was shown by previous studies to correlate well with the shape of measured AIFs (3, 6, 9). The analytical expression for the AIF, Ca(t) was:

equation image(7)

with a = 3.0 and b = 1.5 s, representative of data from normal adult volunteers (3, 6, 9). Simulations used C0 = 1, t0 = 20 s over a time range of 200 s to avoid truncation for the longest MTT simulated (24 s). To evaluate the techniques' sensitivity to different underlying R(t), similarly to methods used in Refs. 3 and4, three different models were examined: exponential, box-shaped, and linear. In all models, the MTT was calculated as MTT = CBV/CBF from the central volume theorem (20). CBV was either 4% or 2%. These values were also used by other studies as representative for normal gray matter or white matter, respectively (3, 6). For CBV = 4%, flow values were varied between 10–70 ml/100 g/min in 10 ml/100 g/min increments. For CBV = 2%, flows were evaluated from 5 to 35 ml/100 g/min in 5 ml/100 g/min increments in order to maintain the same range of MTT values as for CBV = 4%. Analytical expressions for C(t) were derived by convolving Ca(t) with each R(t) (6).

Signal curves were generated as S(t)=S0e−kC(t)TE, with baseline MR image intensity S0 = 100, and TE = 65 ms. For all simulations, a proportionality factor k was selected that resulted in a 40% peak signal drop at a flow rate of 60 ml/100 g/min and CBV = 4%, corresponding to values typically found in human gray matter (3, 6). The signal enhancement curve for the AIF, Sa(t), was similarly modeled as S(t), except that Ca(t) was substituted for C(t). The proportionality constant, k, in this case, was selected to generate a peak signal drop of 60%, which is a typical measured signal reduction for selected AIFs in our clinical PWI.

Using previously described techniques (21), noise was added to S(t) and Sa(t) to create signals with baseline SNRs of 20 and 100, respectively. To evaluate the sensitivity of flow estimates to differences in tracer arrival times between the AIF and tissue signal, S(t) was shifted up to ±5 s with respect to Sa(t) in increments of 1 s, resulting in a total of 11 shifts. To simulate shifts that are not multiples of the sampling period, signals were created with Δt = 100 ms, shifted after noise was added, and then resampled to TR = 1 or 1.5 s. A TR = 1 s was used primarily in our analysis to avoid confounds due to shifts that are not multiples of TR, and to be consistent with previous studies (3). Simulations were repeated at TR = 1.5 s to investigate the effects of sampling interval on the algorithms' performance.

Using approaches similar to those described in Refs. 3 and6, image data sets were created, resulting in a total of 1024 data points for each TR, SNR, shift, and flow. For sSVD and cSVD, PSVD was varied between 0 and 95%. For oSVD, OI was varied between 0 and 0.5. To determine absolute flow values, the calculated CBF values were rescaled by the k-factors used above for S(t) and Sa(t).

The following steps were repeated for each SNR and TR. For each PSVD and OI, the error at each iteration t (Et) was calculated as Et = 1/Nf Σ|F – F′|, where F is the true flow value, F′ is the calculated flow value, and Nf is the number of simulated flow values (Nf = 7). The optimal Psvd for sSVD and cSVD, and optimal OI for oSVD were determined as the values that minimized simultaneously over all assumed residue functions R(t) and Nt = 1024 iterations the average Et assuming zero time delays. The optimal PSVD and OI thresholds found in this step were then used to assess the performance of the techniques in terms of its mean error Et (Ē(D) = 1/Nt Σ Et) and standard deviation (σE(D)) over all Nt = 1024 iterations, as a function of tracer arrival time differences (D).

Tracer arrival timing differences between tissue and the AIF were estimated as the sample point, m, where the maximum R(t) occurs. For sSVD, the estimated shift D′ = m · TR. For oSVD, D′ = m · TR for m<L/2 and D′ = −(Lm) · TR for L/2 ≤ m < L, where L is the total number of points. The error in estimating tracer arrival time differences for each iteration t was calculated as EDt = 1/ND Σ|D – D′|, where D is the true time difference, D′ is the estimated difference, and ND is the number of simulated applied shifts (ND = 11). The PSVD and OI used for estimating flows at each SNR and TR were used to estimate timing shifts. The mean delay error EDt (ED(F) = 1/Nt ΣEDt) over all Nt = 1024 iterations, as well as the SD (σED(F)), were calculated at each flow rate, F.

Clinical MRI Acquisition

DSC MRI consisted of spin-echo, echo-planar images obtained during the first pass of 0.2 mmol/kg of a gadolinium-based contrast agent injected 10 s after start of imaging, at a rate of 5 ml/s, using an MRI-compatible power injector (Medrad, Pittsburgh, PA). Imaging studies were performed on 1.5 T GE Signa LX systems (GE Medical Systems, Milwaukee, WI). The parameters included TR/TE = 1500/65 ms, field of view (FOV) = 22 × 22 cm2 or 20 × 20 cm2, and acquisition matrix = 128 × 128. All studies consisted of 11 slices with a thickness of 6 mm and gap of 1 mm collected over 46 time points.

All data analysis was performed retrospectively, with approval from our institution's committee for human subject research. Four patients were retrospectively examined. Patient demographics are shown in Table 1. Based on the simulation results, our analysis was limited to sSVD and oSVD. Relative CBF (sCBF and oCBF) and Delay (sDelay and oDelay) maps were calculated using the same techniques as for the simulations. An AIF was selected from the ipsilateral hemisphere and used for analysis for both sSVD and oSVD. Selection of PSVD for sSVD and OI for oSVD were based on the optimal values found in the simulation section for SNR = 20 and TR = 1.5 s, which are typical for clinically acquired PW images at this institution. Relative cerebral blood volume (CBV) was calculated by numerically integrating the ΔR2(t) curves (14, 22). MTT values were calculated as sMTT = CBV/sCBF and oMTT = CBV/oCBF.

Table 1. Patient Demographics, Diagnosis, and Imaging Times
PatientAge/sexDiagnosisInitial PWIFollow-up
167/FTransient ischemic attack due to left ICA stenosis2 weeksT2 same day as PWI
256/FRight MCA stroke4 hrs22-day FLAIR
362/MLeft MCA stroke7 hrs4 month FLAIR
452/FRight MCA stroke and complete right ICA occlusion11 hrs6 day FLAIR

RESULTS

Monte Carlo Simulations

For TR = 1 s and SNR = 100 and 20, the optimal PSVD's for sSVD were 4% and 20%, respectively; for cSVD, they were 5% and 10%; and for oSVD, the optimal OIs were .065 and .035. The performances of sSVD, oSVD, and cSVD in estimating flow assuming no phase shift between the AIF and C(t) were found to be comparable. As expected, greater underestimation of flow was found at low SNR than at high SNR. Figure 1 shows the results for TR = 1 s, and CBV = 4% and underlying monoexponential R(t) at SNR = 100 (a) and SNR = 20 (b). Similar results were found for different R(t) (graphs not shown), except in the case of a box-shaped R(t) at SNR = 100. In this condition, both sSVD and oSVD tended to overestimate high flows while cSVD did not. In addition, for CBV = 2% and SNR = 20 (graphs not shown), cSVD performed slightly worse at low flow rates with greater overestimated flow, e.g., at 5 ml/100 g/min, cSVD estimated flow as 6.9 ± 1.3, while sSVD overestimated flow as 6.1 ± 1.2 and oSVD estimated flow as 5.5 ± 1.1 ml/100 g/min.

Figure 1.

Comparison of performances of sSVD, oSVD, and cSVD in estimating flow for SNR = 100 (left column) and 20 (right column) with TR = 1 s, and CBV = 4% and underlying monoexponential R(t). a and b: Estimated flow (F′) as a function of true flow (F) assuming no phase shift between AIF and C(t) shows comparable performances among the three techniques, with greater underestimation of flow at (b) low SNR. c and d: Ē(D) and σE(D) as a function of tracer arrival timing shifts (D) show that sSVD's performance varies as a function of D, while oSVD's and cSVD's performances do not. For conspicuity, only the upper error bars (1 SD) are shown for sSVD and cSVD, and the lower error bars (1 SD) for oSVD.

For all simulation conditions, Ē(D) for sSVD varied with timing shifts (D), especially when the AIF lagged the tissue signal (D < 0), as shown in Fig. 1c–d. In contrast, both oSVD and cSVD were less sensitive to tracer arrival differences. At D = 0, oSVD was found to perform comparably or better than sSVD (Table 2). However, the minimum Ē(D) for sSVD (Ēmin) often did not occur at D = 0, but depended on simulation conditions. For SNR = 100, Ēmin occurred at D of +1, +1, +4, and +1 for monoexponential, linear, box-car R(t), and CBV = 2%, respectively. For SNR = 20, Ēmin occurred at D of –2, –1, 0, and –1.

Table 2. Ē(D) ± σE(D) for D = 0 and Ēmin for sSVD Compared to Average Ē(D) ± σE(D) of oSVD and cSVD for Different SNRs, Residue Functions and Blood Volume (CBV = 4% vs CBV = 2%)
Simulation conditionSNR = 100SNR = 20
sSVD (D = 0)sSVD (Ēmin)oSVDcSVDsSVD (D = 0)sSVD (Emin)oSVDcSVD
(ml/100 g/min)(ml/100 g/min)
Monoexponential R(t)8.42 ± .797.04 ± .607.03 ± .909.46 ± .4711.10 ± .774.98 ± .9610.56 ± 1.0911.64 ± .85
Linear R(t)4.19 ± .843.29 ± .533.64 ± .904.80 ± .539.03 ± .863.46 ± .958.15 ± 1.229.01 ± 1.04
Box-car R(t)9.57 ± 1.026.56 ± .779.30 ± 1.456.28 ± .654.34 ± .904.34 ± .907.60 ± 1.427.17 ± 1.09
CBV = 2%4.44 ± .613.43 ± .474.02 ± 0.514.71 ± .376.12 ± .703.06 ± .796.77 ± 0.736.73 ± .70

The extent by which sSVD over- or underestimated flow was found to depend on MTT, tracer timing shift, and R(t). On the other hand, the ratios of estimated flow (F′) to true flow (F) for both oSVD and cSVD did not vary with shifts but depended on MTT. This is illustrated in Fig. 2, which clearly demonstrates the variability of the magnitude of F′/F oscillations as functions of R(t) and MTT for sSVD. oSVD and cSVD produced results that were independent of tracer arrival differences for all R(t), i.e., producing straight lines such as that shown for the monoexponential R(t) (e and f). For oSVD, the ratios were also independent of CBV, while cSVD performed less accurately at lower CBV and longer MTT (Table 3).

Figure 2.

Comparison of the three techniques' accuracy in estimating flow for different R(t) and MTTs. Shown are the ratios of F′/F as a function of shifts for SNR = 20, TR = 1 s, and CBV = 4% for sSVD and an R(t) that is (a) monoexponential, (b) linear, (c) box-shaped, or (d) monoexponential with CBV = 2%. Results for (e) oSVD and (f) cSVD assuming monoexponential R(t) are also shown. The upper error bar (1 SD) is shown for MTT = 24 s (F = 10 ml/100g/min for CBV = 4%; F = 5 ml/100 g/min for CBV = 2%), and lower error bars (1 SD) for MTT = 6.0 s (F = 40 ml/100 g/min for CBV = 4%; F = 20 ml/100 g/min for CBV = 2%) and MTT = 3.4s (F = 70 ml/100 g/min for CBV = 4%; F = 35 ml/100 g/min for CBV = 2%).

Table 3. Mean ± SD of F′/F for MTT Ranging From 3.4 to 24 s for oSVD and cSVD Techniques Assuming SNR = 20
Simulation conditionoSVD (ml/100 g/min)cSVD (ml/100 g/min)
Monoexponential R(t).64 ± .06 to 1.10 ± .16.60 ± .04 to 1.15 ± .17
Linear R(t).72 ± .06 to 1.14 ± .16.69 ± .05 to 1.25 ± .17
Box-car R(t).95 ± .09 to 1.51 ± .21.87 ± .05 to 1.60 ± .22
CBV = 2%.56 ± .07 to 1.11 ± .23.56 ± .06 to 1.39 ± .25

The effects of SNR and flow rate on the estimation of timing shifts are shown in Fig. 3a and b. Low flow rates produce the worst estimates. At high flow rates, oSVD and cSVD have lower ED(F) than sSVD. However, at low flow rates at SNR = 20, sSVD estimates have both lower ED(F) and σED(F) than oSVD and cSVD. Similar behavior was found for the other simulation conditions (graphs not shown). However, there is a systematic error in sSVD in estimating negative shifts at both high and low SNR, demonstrated in Fig. 3c and d for the case F = 60 ml/100 g/min. For positive delay values, sSVD performs comparably to oSVD and cSVD at high SNR. At SNR = 20, sSVD is more accurate in determining small shifts, but less accurate for large shifts compared to oSVD and cSVD. However, when the shift is negative, sSVD has a systematic bias since the technique inherently cannot distinguish negative time shifts from zero time shifts, whereas both cSVD and oSVD can. Similar findings were obtained under other simulation conditions (graphs not shown).

Figure 3.

Comparison of performances of sSVD, oSVD, and cSVD in estimating tracer arrival time differences for SNR = 100 (left column) and 20 (right column) with TR = 1 s, and CBV = 4% and underlying monoexponential R(t). a and b:ED(F) and σED(F) plotted as a function of F demonstrate that all techniques produce poor estimates at low flow. sSVD produces less ED (F) at low flow rates and SNR than oSVD and cSVD. c and d: Estimated shifts (D′) as a function of true shift (D) at F = 60 ml/100 g/min show that for D ≥ 0, sSVD performs comparably to cSVD and oSVD at SNR = 100, and overestimates large shifts to a greater extent at SNR = 20. For D < 0, sSVD has a systematic error, whereas oSVD and cSVD do not. For conspicuity, upper error bars (1 SD) are shown for sSVD and cSVD, and lower error bars (1 SD) are shown for oSVD.

Repeating the above simulations with TR = 1.5 s produced results similar to those obtained with TR = 1 s. For SNR = 100 and 20, the optimal PSVD's for sSVD were 4% and 10%, respectively; and for cSVD, they were 3% and 10%. For oSVD, the optimal OIs were .085 and .095. Greater underestimation of flow than at TR = 1 s was found for all techniques at high flow rates at both SNRs. The same sensitivity to time shifts for sSVD flow estimates were detected, while oSVD and cSVD were still shift-independent. Oscillations in Ē(D) were detected for shifts that were not multiples of TR for oSVD and cSVD; however, they were of smaller magnitude than oscillations due to noise, and oscillations produced by sSVD. The accuracy of time shift estimates continued to be poorer at low flow rates. In addition, time shift estimates were quantized to units of 1.5 s for all three techniques, since shifts that are not multiples of TR cannot be characterized in this present implementation.

Comparison of SVD and oSVD on Clinically Acquired Human MRI

From the results of the Monte Carlo simulations, we limited our clinical analysis to sSVD and oSVD. Based on the calculated optimal thresholds for SNR = 20 and TR = 1.5 s, PSVD = 10% was used for sSVD, and OI = .095 was used for oSVD perfusion analysis.

In the case of patient 1 (Fig. 4), we found that sSVD overestimates the amount of tissue at risk of infarction (b and c), while oSVD maps (f and g) show much smaller regions of abnormalities, even though the sSVD and oSVD maps were created with the same AIF. Patient 1′s initial DWI (a) and CBV (e) show no abnormalities, while the regions of abnormalities on sCBF and sMTT (b and c) correspond with regions of the greatest phase shifts on the delay maps (d and h). The abnormalities on the oSVD maps correspond with chronic regions of hyperintensity on the patient's T2 fast spin-echo (FSE) (i) acquired at the same session. The patient was diagnosed with transient ischemic attack (TIA).

Figure 4.

Performances of sSVD and oSVD techniques in patient 1, an example case of ICA stenosis in which there are large sCBF and sMTT abnormalities (arrowhead) that do not infarct, and minimal oCBF and oMTT lesions (arrows). Imaging studies acquired 2 weeks after episodes of transient right-sided weakness: (a) normal-appearing DWI, (b) sCBF, (c) sMTT, (d) sDelay maps, (e) normal-appearing CBV, (f) oCBF, (g) oMTT, and (h) oDelay maps, and (i) T2 FSE with chronic regions of hyperintensities (arrow).

Figure 5 demonstrates that for patient 2, sSVD and oSVD can both acutely identify tissue that is ultimately infarcted on follow-up imaging (i) even though it presents with only minimal initial DWI (a) and CBV (e) abnormalities. Large perfusion abnormalities are evident in both sSVD (b and c) and oSVD (f and g) maps, and correspond with regions of large shifts on the delay maps for both techniques (d and h). This demonstrates that oSVD, although more specific as demonstrated by Fig. 4, can be still sensitive enough to detect tissue at risk of infarction that is initially normal appearing on CBV and DWI maps.

Figure 5.

Example case showing that both sSVD and oSVD can detect regions of tissue at risk of infarction that is initially normal in the DWI. Acute imaging studies for patient 2: (a) DWI, (b) sCBF, (c) sMTT, (d) sDelay maps, (e) CBV, (f) oCBF, (g) oMTT, and (h) oDelay maps. i: A 22-day follow-up FLAIR shows that the infarct lesion matches with abnormalities on both sSVD and oSVD perfusion maps (arrows).

Figure 6 demonstrates that oSVD can provide a more specific indicator of tissue at risk of infarction than sSVD by showing an example in which there is mismatch in DWI and PWI lesion volumes using sSVD, and no mismatch when using oSVD, as in the case of patient 3. The sSVD perfusion maps (b and c) show a much larger volume of tissue at risk of infarction than the amount of tissue shown to have infarcted on the 4-month follow-up T2 fluid-attenuated inversion recovery (FLAIR) (i). In contrast, the oSVD maps (f and g) do not demonstrate a mismatch in lesion volumes with the DWI (a) and CBV (e), and correspond well with the follow-up lesion volume (i).

Figure 6.

Example case of a mismatch in DWI and PWI lesion volumes when using sSVD, and no mismatch when using oSVD. Acute imaging studies for patient 3: (a) DWI, (b) sCBF, (c) sMTT, (d) sDelay maps, (e) CBV, (f) oCBF, (g) oMTT, and (h) oDelay maps. i: A 4-month follow-up FLAIR shows an infarct that is well matched with the patient's acute DWI, CBV, oCBF, and oMTT studies (arrows).

Figure 7 compares the sensitivity of both techniques to AIF selection in the case of patient 4. An AIF selected from the ipsilateral hemisphere along the expected course of the MCA (AIF 1) results in sSVD perfusion maps (b and c) that appear hypoperfused in regions that are inconsistent with the patient's initial DWI (a), initial CBV (b), and follow-up FLAIR study (c). From the delay maps, we see that regions identified as hypoperfused by sCBF correspond to negative time differences on the oDelay maps (i). In regions that appear hyperperfused (d, arrows), AIF 1 slightly lags the tissue signal, as indicated by the ΔR2(t) curves for AIF 1 and ROI 2 (j). The results of oSVD (g and h), on the other hand, show regions of hypoperfusion that correspond well with the patient's DWI (a) and CBV (b) lesions, although they underestimate the amount of tissue that is identified as infarcted on follow-up imaging (c). The oDelay map (i), however, shows regions with large positive delays that correspond well with the patient's follow-up lesion (c). Choosing instead an AIF from the patient's right posterior cerebral artery (AIF 2) results in sSVD maps (k–m) that are quite different from their counterparts using AIF 1 (d–f), and that correspond better with the patient's follow-up infarct volume (c). oSVD maps, on the other hand, are comparable to those created with AIF 1, with the exception that the oDelay map in this case is positive. The relative sCBF ratio of ROI 2/ROI 1 was 401% for AIF 1 and 35% for AIF 2. For oSVD, the oCBF ratio of ROI 2/ROI 1 was 84% for AIF 1, and 88% for AIF 2.

Figure 7.

Comparison of sensitivity of sSVD and oSVD to AIF selection for patient 4. Lesions on initial (a) DWI and (b) CBV studies (small arrows) extend into larger infarct volume shown on (c) 6-day follow-up FLAIR. Shown are (d) sCBF, (e) sMTT (f) sDelay, (g) oCBF, (h) oMTT, and (i) oDelay maps using AIF1. Regions darker than background on the (i) oDelay map represent negative values (small arrow). j: ΔR2(t) for both AIFs and for two 5 × 5 ROIs. k: sCBF. l: sMTT. m: sDelay. n: oCBF. o: oMTT. p: oDelay maps using AIF2. Regions of hyperperfusion (arrows) or hypoperfusion (arrowheads) depend on AIF selection for sSVD (d and k) but not for oSVD (g and n).

DISCUSSION

Monte Carlo Simulations

We have presented model-independent techniques (oSVD and cSVD) that are insensitive to tracer arrival time differences. Importantly, they also perform comparably to the standard SVD (sSVD) technique when there are no differences between the tracer arrival time of the AIF and the tissue signal (Fig. 1). In contrast, sSVD flow estimates have a high dependence on tracer arrival time differences (D), MTT, and the underlying R(t) (Fig. 2). This is especially true for negative shifts and at high SNR. We have extended previous studies that showed that sSVD was independent of underlying R(t) (3) by examining sSVD's performance as a function of varying R(t) and delayed tracer arrival. These techniques all depend on MTT because of insufficient temporal sampling with which to properly characterize high flow rates.

Interestingly, for sSVD at low SNR, the effects of D on flow estimation have less variation than at high SNR due to the use of a higher PSVD (20% vs. 4%), demonstrating the importance of PSVD selection. Low PSVD (< 5%) has been shown to lead to greater overestimation of flow even when D = 0 (18). Because the extent of overestimation increases for D < 0 (Fig. 1), the use of a low PSVD resulted in even greater overestimation of flow. Because PSVD was determined assuming D = 0, a low PSVD led to more accurate flow estimates (Fig. 1) when D = 0, but led to larger Ē(D) for D < 0 at high SNR. Ēmin for SNR = 100 was noted to occur at D > 0, whereas for SNR = 20, it occurred for D ≤ 0. Furthermore, Ēmin for SNR = 20 was sometimes lower than Ēmin for SNR = 100 (Fig. 2), since some flow values typically underestimated for D > 0 (e.g., MTT = 3.4 s) were accurately determined at D > 0 at SNR = 20. In contrast, for SNR = 100, flow values are overestimated for D < 0. For D > 0, Ē(D) for sSVD reaches steady-state values that fluctuate slightly about Ē(D) for oSVD and cSVD, as shown in Fig. 1c.

Our findings suggest that even in the absence of tracer arrival time differences, sSVD CBF values are underestimated. This makes absolute quantification of CBF difficult (Figs. 1 and 2), consistent with results from previous simulation studies (3). However, our results show that if MTT is preserved, the accuracy of CBF estimates in gray and white matter would be comparable, e.g., estimates at CBV = 4% and 2% (Fig. 2a and d), producing ratios of gray to white matter CBF values that correlate well with those from other quantitative imaging modalities. This is consistent with findings from studies performed on normal human volunteers in which MR CBF was compared with positron emission tomography (PET) (5) and single photon emission-computed tomography (SPECT) CBF (23) values.

We also investigated the effects of changing the sampling rate, and the implications of these effects for flow determination. As the TR increases, the performance of all of the algorithms degrades for estimating high flow rates. This is almost surely due to insufficient temporal sampling to properly characterize the hemodynamic properties of the measured tissue. Furthermore, the optimal thresholds for sSVD, oSVD, and cSVD techniques also change, becoming less stringent, with more data points in W and Wc kept. In the presence of arrival time differences that are not multiples of TR, oSVD and cSVD also exhibit slight oscillations. Although this is the most likely situation in clinically acquired data, the fluctuations that result are smaller in magnitude than those due to noise and those measured in sSVD. Furthermore, by interpolating the signal to higher sampling rates, one can potentially compensate for flow errors due to these non-sample-unit shifts.

Of the techniques investigated in this study that use block-circulant matrices, the oSVD technique (which varied the SVD threshold on an individual voxel basis) showed improved estimation over cSVD (which used a global SVD threshold). These findings are consistent with the results of Liu et al. (18), who showed that an adaptive technique with sSVD based on each voxel's concentration characteristic provided improved performance over sSVD using a global threshold. We did not separately compare improvements that may be due to voxel-based adaptive threshold techniques with those resulting from the use of a block-circular matrix (cSVD), since sSVD did not include cSVD's robustness to AIF lags to the tissue signal. In addition to the proposed technique (Eq. [6]), there may be other methods to adaptively minimize oscillations based on each individual voxel's tissue signal characteristics, which may further improve oSVD flow estimates (17, 18, 24).

Finally, in this study we also examined techniques for assessing the time difference between tracer arrival in tissue and the AIF by using the time of the peak of the deconvolved R(t). At high flow rates, all techniques performed comparably (Fig. 3). At low flow rates, all techniques had larger errors. However, sSVD cannot estimate negative shifts (Fig. 3c and d), and thus oDelay maps are preferable.

Comparison of SVD and oSVD on Clinically Acquired Human MR Images

We have presented examples of how the errors demonstrated by Monte Carlo simulations may translate in clinically acquired PW images, and how the proposed oSVD technique may compensate for them. Furthermore, our findings suggest that sCBF and sMTT maps are weighted by delayed tracer arrival. In three of four cases (Figs. 4–6), the regions identified as hypoperfused with sSVD (b and c) correspond with regions of greatest delay (d and h). In regions of mismatch between sSVD and oSVD in tissue that did not infarct (Figs. 4 and 6), the ratio of the flow with respect to the contralateral hemisphere was reduced by as much as 50% for phase shifts > 2 s. This is consistent with our simulation results (Fig. 2), which showed that for time delays in this range, CBF in normal gray matter (F = 60 ml/100 g/min) can be underestimated by as much as 50%, whereas tissue without tracer delay is estimated more accurately. This is most evident assuming a box-car residue function, which has been reported to correlate well with acquired human PWI (3). Furthermore, a 50% underestimation of CBF values has also been measured experimentally in previous studies of the effects of delays on acquired human PWI, in which single AIFs were artificially shifted in time (5, 7). As a result, when taking relative flow ratios in sSVD, one may erroneously detect an overall 50% reduction that could be due to tracer arrival differences. Although oSVD can also underestimate flow by close to 50% (Fig. 2, Table 3), it does so for tissue with and without tracer arrival differences. Therefore, when relative CBF analysis is performed, assuming equivalent flow rates, the ratio of tissue with delayed to nondelayed tracer arrival will be close to unity, resulting in less relative differences between the hemispheres in the oSVD-generated maps. Furthermore, oSVD's use of an adaptive local threshold may provide additional accuracy over sSVD for different tissue types (18).

Our findings also demonstrate that delay-insensitive CBF estimates using oSVD may provide results that are robust not only to differences in tracer arrival times, but also to AIF selection (Fig. 7). This suggests that oSVD maps will be less sensitive to user variability in AIF selection, which will perhaps make automated AIF selection algorithms (25) more feasible. To determine whether oSVD provides more accurate quantitative relative flow estimates, correlation with PET flow values in humans and experimental animal stroke models should be performed in future studies.

With less contamination of flow estimates by tracer arrival time differences between the AIF and tissue signals, it may be possible to obtain improved identification of salvageable tissue. Due to tracer arrival delay, the existing sSVD technique may produce estimates of flow that indicate ischemia in regions that are actually oligemic or normally perfused. This in turn may contribute to the large sensitivity and lack of specificity in traditional sCBF and sMTT maps (2). However, one can argue that, clinically, what is desired is a sensitive indicator of tissue with disturbed hemodynamics, and that therefore delay contamination in sCBF and sMTT maps is a clinically useful feature. The shift maps may represent tissue that is downstream from an occlusion or stenosis, but which may still receive sufficient flow at the time of imaging. The oCBF maps, on the other hand, represent the instantaneous flow at the acquisition time. Since CBF is a dynamic process, a single snapshot may not be a sensitive predictor of future infarction. Therefore, we speculate that the oDelay maps presented here can continue to provide sensitivity in identifying tissue with disturbed hemodynamics, while the oCBF maps provide a more specific snapshot of the severity of ischemia in tissue at risk of infarction. The clinical utility of decoupling flow from delayed tracer arrival still remains to be proven in a larger study.

The clinical findings presented here are clearly preliminary, and were meant to demonstrate the techniques in practice rather than prove their accuracy. Additional studies involving a larger cohort of patients are necessary to better determine the predictive power of oSVD perfusion maps. Retrospective analysis of perfusion studies with follow-up MRIs should be performed in which lesion volumes identified by oCBF, oDelay, CBV, sCBF, and sMTT are correlated with lesion volumes on follow-up imaging studies. Alternatively, one may utilize predictive modeling algorithms that comprise the different sets of perfusion parameters, and evaluate which generated risk maps are more predictive of tissue infarction in terms of sensitivity and specificity (26).

CONCLUSIONS

We have extended previous studies that examined the sensitivity of the SVD technique to delay and dispersion (3–6, 10) by investigating conditions in which the AIF lags the tissue signal, assuming different residue functions and CBV. We have proposed a technique that is less sensitive to tracer arrival timing differences than the current SVD technique, without sacrificing performance for different underlying residue functions and CBV. We have also demonstrated that the new technique may provide improved accuracy in the identification of tissue at risk of infarction in clinically acquired PWI.

Acknowledgements

The authors express their gratitude to Joanie O'Donnell, R.N., for chart review, and the Departments of Neuroradiology and Neurology, Massachusetts General Hospital, for their assistance in clinical data acquisition.

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