Advantages of frequency-domain modeling in ... - Wiley Online Library

Magnetic Resonance in Medicine 53:700 –707 (2005)

Advantages of Frequency-Domain Modeling in DynamicSusceptibility Contrast Magnetic Resonance Cerebral Blood Flow Quantification Jean J. Chen,1,2 Michael R. Smith,1,3* and Richard Frayne1– 4 In dynamic-susceptibility contrast magnetic resonance perfusion imaging, the cerebral blood flow (CBF) is estimated from the tissue residue function obtained through deconvolution of the contrast concentration functions. However, the reliability of CBF estimates obtained by deconvolution is sensitive to various distortions including high-frequency noise amplification. The frequency-domain Fourier transform-based and the time-domain singular-value decomposition-based (SVD) algorithms both have biases introduced into their CBF estimates when noise stability criteria are applied or when contrast recirculation is present. The recovery of the desired signal components from amid these distortions by modeling the residue function in the frequency domain is demonstrated. The basic advantages and applicability of the frequency-domain modeling concept are explored through a simple frequency-domain Lorentzian model (FDLM); with results compared to standard SVD-based approaches. The performance of the FDLM method is model dependent, well representing residue functions in the exponential family while less accurately representing other functions. Magn Reson Med 53:700 –707, 2005. © 2005 WileyLiss, Inc. Key words: perfusion-weighted imaging; dynamic susceptibility contrast imaging; cerebral blood flow (CBF); deconvolution; frequency-domain modeling; frequency-domain Lorentzian modeling

Accurate assessment of cerebral perfusion parameters, such as the cerebral blood flow (CBF) and mean transit time (MTT), is required for the timely prognosis and treatment of conditions such as ischemic stroke (1,2). Quantitative dynamic-susceptibility contrast (DSC) magnetic resonance (MR) imaging is a powerful tool for obtaining perfusion measurements (1–3) from arterial and tissue contrast concentration curves. The CBF estimate can be

1 Department of Electrical and Computer Engineering, University of Calgary, Calgary, Canada. 2 Seaman Family MR Research Centre, Foothills Medical Centre, Calgary Health Region, Calgary, Canada. 3 Department of Radiology, University of Calgary, Calgary, Canada. 4 Department of Clinical Neurosciences, University of Calgary, Calgary, Canada. Presented at the 21st International Symposium on Cerebral Blood Flow, Function and Metabolism, 1 July 2003, Calgary, Canada, and at the 11th Annual Scientific Meeting of ISMRM, 10 –16 July 2003, Toronto, Canada. Grant sponsor: Alberta Foundation for Health Research; Grant sponsor: Alberta Heritage Foundation for Medical Research; Grant sponsor: Canada Foundation for Innovation; Grant sponsor: Canadian Institutes of Health Research; Grant sponsor: Heart and Stroke Foundation of Canada; Grant sponsor: Natural Sciences and Engineering Research Council of Canada; Grant sponsor: University of Calgary. *Correspondence to: Michael Smith, Department of Electrical and Computer Engineering, University of Calgary, 2500 University Drive, NW, Calgary, Alberta T2N 1N4, Canada. E-mail: smithmr @ ucalgary.ca Received 15 March 2004; revised 7 October 2004; accepted 7 October 2004. DOI 10.1002/mrm.20382 Published online in Wiley InterScience (www.interscience.wiley.com).

© 2005 Wiley-Liss, Inc.

obtained through deconvolution of the tissue concentration function by the arterial concentration function (4 –10). Common deconvolution techniques are based on either the Fourier transform (FT) (2,5,6) or the singular-value decomposition (SVD) (2,5,6,11,12). Other less commonly used approaches include deconvolution by maximum likelihood (8) and Gaussian processes (13). Deconvolution is known to be a noise-sensitive operation (5,10 –15). Stability of the commonly used deconvolution algorithms is achieved by applying noise filters, either explicitly as in the FT algorithm (2,5,6,16) or implicitly as in the SVD algorithm (15,17,18). Application of these filters suppresses noise but also distorts the high-frequency signal components, leading to incorrectly estimated CBF values (5,15). One attempt to eliminate these signal distortions involves modeling the concentration signals in the time domain using a ␥-variate function (to remove noise, undersampling, and recirculation) (2–5,7,9,19,20). We describe a new approach to remove deconvolution artifacts through frequency-domain characterization of the tissue residue function parameters. Previous studies using parametric models for the residue function have shown that given a consistent vascular model across the brain, the CBF values obtained parametrically based on a single, wellmixed, vascular compartment (exponential model) agree best with results obtained using the nonparametric SVD method (5,11,12). This implies that the exponential model is not an unreasonable first model to use for an initial investigation of frequency-domain characterization of the residue function. The basic concepts of frequency-domain modeling are demonstrated through a parametric approach based on the equivalence between the standard time-domain exponential model (5) and a frequency-domain Lorentzian model (FDLM) (21,22). Using this simple, although not ideal, approach we attempt to provide a better understanding of the advantages and disadvantages of frequency-domain deconvolution techniques in the presence of signal distortions. THEORY The tissue residue function R(t) describes the fraction of contrast agent remaining in the tissue volume-of-interest (VOI) (2,4,5,7,8). The CBF can be estimated from the scaled residue function, R⬘(t), obtained by solving the convolution equation (2,4,5,7) c VOI共t兲 ⫽

700

␳ ␬H

冕

t

ca共␶兲关CBF 䡠 R共t ⫺ ␶兲兴d␶

0

⫽

冕

t

0

关ca共␶兲 䡠 R⬘共t ⫺ ␶兲兴d␶,

[1]

Frequency Domain Modeling in CBF Quantification

where ␳ is the brain tissue density, ␬H is the vascularcapillary hematocrit ratio, and ca(t) and cVOI(t) are the arterial input and tissue VOI concentration functions, respectively. If ␳/␬H is assumed to be 1.0 (4), then we can define R⬘(t) ⫽ CBF 䡠 R(t), so that CBF equals the peak of R⬘(t) (5,10). In practice, both ca(t) and cVOI(t) are obtained by tracking the bolus through the brain vasculature using a series of perfusion images acquired at time points separated by discrete intervals ⌬T. Therefore, the continuous ca(t) and cVOI(t) become discrete-time sequences ca[n⌬T] and cVOI[n⌬T], respectively, where ⌬T is most often equal to the pulse sequence repetition time (TR) (1). Current DSC perfusion imaging protocols have a sampling period ⌬T in the range of 1 to 2.25 s, corresponding to sampling rates of 1 Hz or less. In this paper, square brackets (i.e., ca[n⌬T]) will indicate the experimentally sampled function, while round brackets (i.e., ca(t)) will denote the true continuous function. To ensure algorithmic stability in the SVD method, noise filtering is implicitly imposed (18) through the use of a fixed or adaptive singular-value threshold (PSVD) (2,5– 7,11,12). This introduces distortion to the high-frequency signal components needed to properly characterize the peak of the residue function; introducing CBF biases that become more severe as the tissue MTT decreases, i.e., as the high-frequency residue function spectral components increase. Examination of the frequency components of the residue function prior to noise filtering permits the determination of the spectral components that are distorted by noise or other experimental artifacts such as those introduced when the rapid changes in the tissue and residue curves are not sampled at a rate that satisfies the Nyquist criterion (23). Inspection of the less distorted frequency-domain spectral data segments of the residue function allow the corrupted data points to be identified and then replaced by values calculated using this model (if it is suitable) to provide a better estimate of the true residue function. An advantage of frequency-domain modeling is that there is no necessity for assumptions regarding the shape of ca(t) or cVOI(t) in a clinical study. A further advantage is that frequency-domain modeling employs many points of the residue function in the CBF estimation process, unlike conventional deconvolution techniques in which the CBF is estimated from a single point of the residue function (i.e., max{R⬘(t)}), which may be unreliable because of noise or other distortions. After deconvolution, it is also possible to model the residue function as an exponential directly in the time domain. However, it is more difficult to separate signal from noise with this approach. In addition, the time-domain residue function would first be calculated from a signal already distorted by noise filtering in the deconvolution process, as is found in the SVD or FT algorithm, whereas frequency-domain modeling is performed before such distortions are introduced. METHODS Frequency-Domain Modeling Frequency-domain modeling can be performed using both parametric and nonparametric techniques (14). In this pa-

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per, the basic concepts of modeling the residue function in the frequency domain are demonstrated by applying a simple parametric model to the spectral components of the residue data. In continuous time, the exponential model is defined as

R共t兲 ⫽

再

0 共t ⫺ ATD兲 exp ⫺ MTT

冉

冊

t ⬍ ATD t ⱖ ATD,

[2]

where ATD is the arterial-tissue delay, the difference between the bolus arrival times at the arterial and tissue voxels (15–17). This time-domain representation of this single compartment model becomes a Lorentzian function in the frequency domain. This implies that the solution to Eq. [1] is the ideal frequency-domain representation of the scaled exponential residue function model, R⬘ideal关m⌬F兴 ⫽ CBF 䡠

␳ MTT 䡠 exp共 ⫺ j2␲共m⌬F兲 䡠 ATD兲 䡠 . ␬H 1 ⫹ j2␲共m⌬F兲 䡠 MTT

[3]

There are many approaches to determining the parameters of the Lorentzian model from the frequency spectrum of the computed residue function. A direct technique is to first square both sides of Eq. [3] in order to remove the frequency-domain phase modulation introduced by ATD. Then, by setting y[m] ⫽ 兩R⬘ [m⌬F]兩⫺2 and x[m] ⫽ m2, the residue function spectral components can be modeled as a linear expression of the form, y[m] ⫽ A 䡠 x[m] ⫹ B, 0 ⬉ m ⬍ N/2, where A and B are the slope and the intercept of the model, respectively. This rearrangement yields A ⫽ 4␲2␬H2/␳2CBF2 and B ⫽ ␬H2/␳2CBF2MTT2. To obtain a reliable model, only the first M data points should be used, such that 兩R⬘[m⌬F]兩 ⬎ v 䡠 兩R⬘[0]兩, 0 ⬉ m ⬍ M, where the parameter v is a percentage of the central peak of the residue spectrum, R⬘[0], chosen to best minimize the noise and other distortions. If the signal-to-noise ratio (SNR) is low, a large value for v should be used. The parameters A and B were obtained using a minimum mean-square error (MMSE) criterion calculated with a weighting window G[m],

冘

M⫺1

MMSE ⫽

兵关y关m兴 ⫺ A 䡠 x关m兴 ⫺ B兴2 䡠 G关m兴其,

[4a]

m⫽0

where G关m兴 ⫽

1 . 共m ⫹ 1兲 q

[4b]

This G[m] weighting was applied to reduce the impact of increasing high-frequency noise on the stability of the model. A preliminary empiric study was undertaken to find a suitable value for the weighting factor q for a variety of residue function characteristics. The MTT can theoretically be determined directly from the intercept of the model, i.e., B. However, in practice, although the slope of the linear fit was found to vary slightly with different

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FIG. 1. (a). The selection of undistorted raw data from the actual residue function spectrum (R⬘[m⌬FEXPT] or R⬘[m]; solid line) is required to obtain a reliable Lorentzian model (dotted line), especially in the presence of noise. In this representative example, the simulation parameters were MTT ⫽ 12 s, SNR was 60, and the FDLM noise threshold v ⫽ 0.2. The mean and standard deviations of 1000 FDLM runs at each frequency index were calculated and plotted (see text). R⬘[0] was 1.89 ⫾ 0.16 (mean ⫾ SD). (b) Linear fits, with x ⫽ m2 and y ⫽ 1/兩R⬘[m]兩2, were applied to each of the individual spectra using raw residue data that satisfied the condition R⬘[m] ⭌ v 1/兩R⬘[m]兩2, where v was 0.2 (equivalently, 1/兩R⬘[m]兩2 ⬍ 1/(v 䡠 R⬘[0])2). The mean of the ensemble of linear fits (solid line) was obtained by averaging the linear fits derived from each residue spectrum. Of the resulting models 95% fell within the interval delimited by the thick dashed lines. At frequency indices outside the window (corresponding to R⬘[m] ⬍ v 䡠 R⬘[0])), larger variations in R⬘[m] are observed; thus the data began to deviate from the 95% confidence limit and were therefore not used in the modeling process.

residue functions, the intercept proved to be more sensitive to model characteristics. The MTT was indirectly calculated using the central volume principle (4 –7,9,10). Computer Simulations Noiseless and noisy contrast concentration signals were synthesized at a sampling rate of 0.44 Hz, corresponding to a ⌬T of 2.25 s, comparable to our clinical protocols. The arterial input ca(t) was modeled as a ␥-variate function as per Ref. (4). The modeled temporal extent of the signal was 126 s. The analytic expression for the tissue curve is

冉

c VOI共t兲 ⫽ 6 exp ⫺

冊

冉 冊

t⬘ t⬘ 䡠 ␭4 ⫺ exp ⫺ MTT ␭

⫻ 关␭t⬘3 ⫹ 3␭2 t⬘2 ⫹ 6␭3 t⬘ ⫹ 6␭4 兴, t ⬎ t0 ⫹ ATD,

[5]

where ␭ ⫽ (b⫺1 ⫺ MTT⫺1)⫺1, t⬘ ⫽ t ⫺ t0 ⫺ ATD, and t0 is the arterial bolus arrival time with cVOI(t) ⫽ 0 otherwise. Equation [5] can be rewritten as a sum of ␥-variates, showing that the common practice of fitting cVOI(t) to a single ␥-variate function (24) is not appropriate. Note that although the FDLM method is a parametric model, it still shares with the nonparametric SVD methods the characteristic that no assumptions are made on the shape of ca(t) or cVOI(t) in a clinical study, and therefore ␥-variate fitting is not necessary. To determine the sensitivity and robustness of the FDLM approach, we also tested our technique using frequency-domain data derived from box-shape residue functions (5), triangular residue functions (5), and dispersed exponential functions (4). For these three models, numerical convolution was performed. To ensure equivalent accuracy compared to analytic convolution, it is necessary that the simulated sampled signals be generated at a frequency much higher than TR; ca(t) and cVOI(t) were generated at intervals of 1/32 s (approximately TR/64) to avoid introducing convolution distortions due to Nyquist-related undersampling artifacts

prior to numerical convolution (c.f. Ref. 5) and then resampled (or decimated) to the required experimental sampling rates. Monte Carlo noise simulations (1000 iterations) were performed. MR signals were generated at signal-to-noise ratios (SNR) of 20 and 60, representing moderate and low noise conditions typical of DSC MR images, with SNR ⫽ S0/nRMS, where S0 is the precontrast MR signal intensity, and nRMSis the noise level. For comparison, we implemented the reformulated SVD algorithm (15), in which the arterial-tissue delay dependence artifact of the standard SVD algorithm (2,4 – 6) is removed. When solving the SVD deconvolution matrix equation, the [ca] matrix components were constructed directly from the AIF samples (15) and not from the prefiltered versions of the AIF samples (5,17). The metric for comparison of SVD and FDLM results was the mean and standard deviation (SD) of the measured-to-true CBF and MTT ratios. For the FDLM method, the threshold, v, was 0.2 both for the noiseless case and when SNR ⫽ 60, while v ⫽ 0.3 was required at SNR ⫽ 20. Since it is not standard practice to use a SVD noise threshold, PSVD, that is adaptive with varying MTT, average PSVD values that produce a similar thresholding effects to the equivalent frequency-domain noise threshold, v, were calculated (18), giving PSVD ⫽ 0.11 and 0.16 for v ⫽ 0.2, and 0.3, respectively. In Fig. 1a, we demonstrate the process of obtaining the characteristics of the Lorentzian model in the frequency domain using Eq. [3]. Clinical Study Clinical data were acquired on a 3-T MR scanner (Signa; GE Medical Systems, Milwaukee, WI). A volume of 20 mL of MR contrast agent (Magnevist; Berlex, Wayne, NJ) was delivered by a MR-compatible power-injector (Medrad, Pittsburgh, PA) at a rate of 5 mL s⫺1. Data from 10 acute stroke patients, judged by our stroke neurology team to

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FIG. 2. The performance of the FDLM and SVD methods in noise-free simulations is compared. The singular-value threshold PSVD is set to 0.11, while the FDLM threshold v ⫽ 0.2. When estimating (a) the CBF and (b) the MTT, the FDLM parametric modeling approach outperforms the SVD method for the exponential model, the triangle residue, and the exponential model with dispersion. Note that in the absence of noise, the FDLM method can equivalently model the triangle and exponential residue spectrums with an accuracy that increases with MTT. However, as expected, the box-car residue model is poorly fitted, since its characteristics substantially differ from those required by this parametric frequency-domain model.

bear the clinical features consistent with minor to moderate stroke, were collected using T2*-weighted gradientecho echo-planar imaging sequences. The imaging parameters were TR/TE/flip ⫽ 2200 ms/25 ms/45°, with slice thickness of 5 mm separated by 2-mm gaps. Informed written consent was obtained from all patients. Postprocessing of the perfusion images, including identification of the arterial input function (ca(t)), was performed on a Unix-based workstation (R10000 Octane; Silicon Graphics, Milpitas, CA) using custom software (written in IDL; Research Systems, Boulder, CO). A single arterial input function was used for both the infarct and the normal regions and was measured in the middle cerebral artery (MCA) on the ipsilateral or contralateral side. A Blackman–Harris time-domain filter (24) was used to remove contrast recirculation. Since the average image SNR was measured to be 37.1, a FDLM noise threshold (v) of 0.3 was used, and correspondingly PSVD was set to 0.16. Deconvolution was performed on a pixel-by-pixel basis. Following Østergaard et al. (5), we applied a calibration factor to the measured CBF values so that average CBF in normal white matter was 22 mL min⫺1 (100 g)⫺1. It was anticipated from simulation studies that independently determined calibration factors would be required to scale results from the SVD and FDLM algorithms for each patient. The normal and infarct regions were defined by a stroke neurologist. Both areas contain a mixture of gray and white matter. A semi-automated tissue segmentation technique (25) was used to mask out the veins and arteries and facilitated calculation of average CBF in normal and abnormal gray and white matter individually. A paired t test was performed to evaluate the differences in the CBF values estimated using the FDLM and SVD approaches on a regional basis. RESULTS In Fig. 1a, a single residue spectrum is illustrated (MTT ⫽ 12 s). Low-frequency data, undistorted by noise, were obtained by excluding samples below the level (v 䡠 兩R⬘[0]兩).

The spectra were then individually fit to the FDLM based on Eqs. [4a, b]. A noise de-emphasis factor of q ⫽ 5/3 was empirically determined based on a compromise between the optimal results from all residue models and MTT considered. The mean and SD of the simulated residue spectra at each frequency index were plotted (Fig. 1b), with the SD found to increase as the frequency index increases. The values of parameters A and B obtained from the linear fits of all trials were used to determine the mean fit line and the interval within which 95% of the Lorentzian fits were found (within two standard deviations) for SNR ⫽ 60. It was observed that above the threshold corresponding to v ⫽ 0.2, the mean of the residue spectra began to fall outside this confidence range. Figure 2a compares the ratio of the theoretical (noiseless) measured-to-true CBF ratios. When tested on data generated using the exponential residue model, the SVD method systematically underestimated the CBF, more so with decreasing MTT as was previously observed by Østergaard et al. (5) and others. On the other hand, FDLM generally provided more accurate results over the range of evaluated MTT values, with the measured-to-true CBF ratio being nearly 1.0 regardless of the MTT of the simulated residue signal. With nonexponential residue models, the performance of both techniques was found to be dependent on the tissue model. For SVD, the CBF was found to be underestimated for all models except for the box-car model. The underestimation was highest with the dispersed exponential model and lowest with the triangle model and varied with MTT, as previously observed with the exponential model. The FDLM technique produced a 20% lower CBF underestimation than the SVD method for the dispersed exponential model. With increasing MTT, the FDLM accuracy improved. For the box-car, the FDLM severely overestimated the CBF (by ⬎100%). For a fixed CBV, CBF underestimation translates to a MTT overestimation (Fig. 2b). Correspondingly, the SVD MTT overestimation for most tissue models was most severe for short MTT, while MTT was underestimated for the box-car

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FIG. 3. CBF estimation using the FDLM and SVD approaches, at SNR of 60 (a, c, e) and 20 (b, d, f), are compared. (a) and (b) show measured-to-true CBF ratios for the data generated using the exponential residue model, while (c) and (d) correspond to results obtained using the exponential model with dispersion. The FDLM CBF was more accurate (ratio closer to 1.0) than the SVD CBF ratio in (a) to (d). Judging from noiseless simulation studies, noise leads to CBF overestimation for the exponential (a, b) and the triangle model (e, f). Note that while FDLM is more accurate for noiseless data, its stability varies with SNR. On the other hand, the SVD results are systematically more underestimated than those of FDLM, although more stable in noise (c.f. Fig. 2). As expected, the performance of FDLM depends on how much the signal characteristics deviate from the model.

model. Because of the poor performance seen for the boxcar, it was not included in the subsequent simulations. At SNR ⫽ 60, for the exponential residue model (Fig. 3a), SVD considerably underestimated the CBF, while FDLM slightly overestimated it. However, the average accuracy of the FDLM CBF estimates was higher than that of the SVD method (mean CBF ratio across the range of MTT: 1.02 ⫾ 0.08 versus. 0.67 ⫾ 0.07). CBF was also underestimated by the SVD method for both the triangle and the dispersed exponential residue models at SNR ⫽ 60 (Fig. 3c and e), with the degree of underestimation decreasing as MTT increased. FDLM overestimated CBF for the triangle residue model (Fig. 3c), while underestimating it for the dispersed exponential model. However, the FDLM method outperformed SVD for the dispersed exponential (mean CBF ratio 0.82 ⫾ 0.06 versus 0.63 ⫾ 0.05) model. At SNR ⫽ 20 (Fig. 3b, d, and f), variations in the FDLM results were higher than in the SVD results for all models studied. These variations also decreased as MTT increased. As previously observed, the FDLM CBF estimates were higher at SNR ⫽ 20 than at SNR ⫽ 60 and more accurate than the SVD estimates for the exponential model and dispersed exponential models (mean CBF ratio 1.09 ⫾ 0.24 versus 0.73 ⫾ 0.14 and 0.91 ⫾ 0.15 versus 0.67 ⫾ 0.11, respec-

tively). A larger FDLM average overestimation was observed for triangle model than at SNR ⫽ 60 (mean CBF ratio 1.21 ⫾ 0.10 versus 1.55 ⫾ 0.38). The SVD estimates also fluctuated more with increased noise, but remained less accurate but more stable than FDLM estimates for the first two models, especially at short MTT. In Table 1, semi-quantitative (uncalibrated) and quantitative (calibrated) CBF values obtained using both the FDLM and the SVD methods are summarized for the 10 patients studied. For a representative patient, calibrated quantitative CBF maps are shown in Fig. 4. The uncalibrated CBF estimates obtained using the FDLM method were nearly twice as that high as obtained by SVD, with the difference between estimates higher for normal tissue, as would be expected from simulation studies. The calibrated FDLM/SVD GM/WM CBF ratios were close to 1.00 for normal tissue. The CBF in normal GM was 40.0 mL min⫺1 (100 g)⫺1 and 39.7 mL min⫺1 (100 g)⫺1 for FDLM and SVD, respectively. In the infarct, the FDLM estimates were lower than SVD, with the average FDLM/SVD being 0.72 and 0.70 for GM and WM, respectively. The average GM/WM CBF ratio was 1.82 and 1.80 in normal tissue and 1.75 and 1.43 for infracted tissue for FDLM and SVD, respectively.

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Table 1 Uncalibrated and Calibrated CBF Estimates Were Obtained for 10 Patients with Minor to Moderate Stroke

Uncalibrated (arbitrary units) Normal GM WM GM/WM Infarct GM WM GM/WM Calibrated (mL min⫺1 (100 g)⫺1) Normal GM WM GM/WM Infarct GM WM GM/WM

FDLM CBF

SVD CBF

P value

FDLM CBF/SVD CBF

0.40 ⫾ 0.32 0.21 ⫾ 0.16 1.82 ⫾ 0.18

0.16 ⫾ 0.08 0.09 ⫾ 0.04 1.80 ⫾ 0.18

0.040 0.035 0.288

2.51 ⫾ 1.91 2.19 ⫾ 1.77

0.09 ⫾ 0.05 0.07 ⫾ 0.04 1.75 ⫾ 1.68

0.07 ⫾ 0.04 0.05 ⫾ 0.03 1.43 ⫾ 0.63

0.011 0.009 0.406

1.53 ⫾ 0.61 1.41 ⫾ 0.34

40.0 ⫾ 3.9 22.0 1.82 ⫾ 0.18

39.7 ⫾ 3.9 22.0 1.80 ⫾ 0.18

0.288 N/A 0.288

1.01 ⫾ 0.03 N/A

11.2 ⫾ 3.3 9.3 ⫾ 5.0 1.75 ⫾ 1.68

15.7 ⫾ 5.2 11.8 ⫾ 3.6 1.43 ⫾ 0.63

0.012 0.013 0.406

0.74 ⫾ 0.27 0.73 ⫾ 0.29

Note. The uncalibrated FDLM estimates were higher than SVD estimates, for both GM and WM. Also, the difference between FDLM and SVD estimates was higher for normal tissue, corresponding to shorter MTTs, than for infarct tissue, as predicted from the simulation study. The average GM/WM ratio was higher for normal tissue than for infarct tissue for FDLM and SVD. After cross-calibration the FDLM and SVD estimates are identical for WM (by definition) and very close for normal GM (P ⫽ 0.288). Greater differences between the two methods are now found for infracted tissue, with the FDLM estimates lower than SVD (P ⫽ 0.012 and 0.013 for GM and WM, respectively). The GM/WM CBF ratios are unchanged with calibration.

DISCUSSION AND CONCLUSIONS Although parametric, the FDLM method may be used in clinical situations without requiring ␥-variate fitting or other assumptions on the shapes for ca(t) or cVOI(t). Nonetheless, this simple frequency-domain modeling approach is model-dependent and cannot be expected to perform equally well for all residue models, in particular on models with significant high-frequency content, such as the box-car and triangle functions. In the frequency domain, these two functions transform to sinc and sinc2, respectively, which contain many large high-frequency signal components. Compared to the exponential and dispersed exponential functions, these characteristics can theoreti-

cally be expected to require a more complex frequencydomain model (14) than a simple parametric Lorentzian model. Examination of the spectral characteristics of the residue function also raises the question of whether the residue models typically used in current simulation studies are physiologically realistic, especially those with infinitely sharp edges (exponential, box, and triangle). Use of standard, but nonphysiologically realistic, models in simulation studies may be unrealistically degrading the apparent performance of FDLM, SVD, and other methods. An observation from noise-free and noisy simulations was that the accuracy and precision of the FDLM estimates increased with increasing MTT. This implies that FDLM

FIG. 4. Calibrated (quantitative) CBF maps for a patient of moderate stroke calculated using the FDLM (a) and SVD (b) approaches appear comparable. The absolute CBF in both images have been calibrated based on an average CBF of 22 (mL min⫺1(100g)⫺1) in normal white matter (5,6) and the maximum displayed CBF set to 100 (mL min⫺1(100g)⫺1). The infarct (left parietal white matter) can be discerned in both maps. In the FDLM map, the high-flow regions appear in a less smooth pattern compared to in the SVD map. In normal gray matter, the average FDLM CBF estimate is 41.5 (mL min⫺1(100g)⫺1), compared to 37.1 (mL min⫺1(100g)⫺1) given by SVD. In the infarct, the average FDLM CBF estimates were 13.4 (GM) and 7.5 (WM) (mL min⫺1(100g)⫺1), compared to 27.3 (GM) and 13.1 (WM) (mL min⫺1(100g)⫺1) for SVD. The normal GM/WM CBF ratio is 1.89 for FDLM and 1.68 for SVD.

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parametric modeling is better for long mean transit times, corresponding to a narrow-band residue spectrum, than for short mean-transit times, which have a wide-band residue spectrum. A possible explanation for this behavior is that at large mean transit times (generally associated with slow flow), R⬘[m⌬F] is generally narrower, which means that more of the required data points lie at the lower frequencies and are relatively undistorted by noise. At lower signal-to-noise ratios, the selection of valid data points is expected to become more difficult, since a higher fraction of the necessary points would be distorted by noise. This may explain the dependence of the stability of the FDLM method on the SNR. The patient GM/WM CBF ratio was 1.80 and 1.82 for rSVD and FDLM, respectively. They are lower compared to values reported by Østergaard et al. (6) using the standard SVD algorithm, now known to have a higher ATD sensitivity that varies with MTT (15,17). However, our results compare favorably with ratios presented in a more recent DSC MR study (26), as well as results from CT perfusion (27), a technique that has been rigorously evaluated in animal studies (28). No explanation is offered as to why the GM/WM ratios reported in this study and in Refs. (26) and (27) are low by comparison to the wide range of PET values summarized in Ref. (7). We note that in the CBF maps, the CBF values have been scaled so that normal WM CBF is 22 mL min⫺1 (100 g)⫺1. However, as seen from the noise-free simulation (Fig. 2), it is actually inappropriate to apply a single calibration factor to either the SVD or the FDLM method in order to convert the measured CBF estimates to values with absolute units [mL min⫺1 (100 g)⫺1], as the degree of calibration required (29) is dependent on the tissue MTT (18,21,22). This known dependence has been confirmed in our patient studies by a better agreement between the calibrated FDLM and SVD CBF estimates for normal GM and WM (which are distorted in a similar manner because of their similar MTT values) than for infarct tissue (Table 1). That is, the absolute CBF calibration is theoretically dependent on MTT, and since a single calibration factor for both algorithms was calculated based on normal tissue values (short MTT), the CBF estimates in the normal regions approached each other, while the differences in the CBF for infarcted regions (long MTT) were emphasized. Furthermore, since FDLM and SVD have different MTT sensitivities (Fig. 2), and since each of the selected regions of interest contain a mixture of different tissues, the chosen calibration factors may not be entirely appropriate, further contributing to the difference between the FDLM and SVD estimates. The simulation studies hint that both SVD and FDLM deconvolution approaches can be expected to introduce biases into the CBF estimates from patient studies in a manner that changes with MTT. As expected, the simple FDLM approach has the tendency to bias (overestimate) CBF for nonexponential residue models, giving the appearance of hyperperfused tissue, with the most significant CBF overestimation found at short MTTs. In most other techniques, including the most commonly used SVD algorithm, CBF underestimation is most severe at short MTTs, increasing the chances for normal tissue to appear hypoperfused. However, CBF overestimation (hyperperfu-

Chen et al.

sion) was also reported for the oSVD method (17) for infarcted tissue. The results indicate that after cross-calibration has, by definition, forced a common WM CBF value of 22 mL min⫺1(100g)⫺1 upon the results of the two algorithms, the SVD algorithms will now consistently produce hyperperfused values and the FDLM algorithm will produce hypoperfused values for tissues with MTT ⬎ 4.8 s. This is a complete inversion of the relative behavior prior to cross-calibration Thus, MTT-dependent CBF biases exist in different forms in different applications of deconvolution algorithms, and clinicians should be aware of these caveats and their consequences. The potential benefits of extending frequency modeling to nonparametric techniques (14) can be visualized through the preliminary success of the parametric FDLM concept. Techniques to improve the stability of modeling in the presence of noise are needed. Furthermore, in theory, since the contrast recirculation peak varies much more slowly than the initial peak, the frequency components from the recirculated bolus would distort only a few low-frequency points in the residue spectrum, contributing to lower sensitivity of the FDLM method to recirculation. This potential advantage of frequency-domain modeling needs to be further studied. In summary, current standard perfusion quantification algorithms lead to errors in the CBF measurement. Specifically, the application of noise filtering to ensure algorithmic stability causes substantial CBF biases (15,17,21,22). In response, we investigated some of the advantageous properties of the frequency-domain modeling concept. First, it is possible to visualize noise components to identify and use only the relatively undistorted signal components in the frequency domain. Second, instead of truncating noisy signal components they can be replaced by values calculated using this model to potentially correct the CBF biases caused by noise filtering. These concepts were demonstrated through the simple FDLM. Although the FDLM approach is a parametric approach and, as any model-dependent method, has practical limitations, no fitting of either the arterial or tissue concentration signals is needed. Furthermore, FDLM has demonstrated the potential of frequency-domain modeling in improving CBF estimation accuracy. ACKNOWLEDGMENTS The involvement and support in both patient recruitment and MR imaging of the Calgary Stroke Program is greatly appreciated. In particular, we acknowledge the assistance of M.S. Bristow, B.Sc., and J.E. Simon, M.B., Ch.B., in brain tissue segmentation. R.F. is an AHFMR Medical Scholar, a HSFC Research Scholar, and a Canada Research Chair. J.J.C. is a NSERC Canada Graduate Scholarship awardee. REFERENCES 1. Baird AE, Warach S. Magnetic resonance imaging of acute stroke. J Cereb Blood Flow Metab 1999;18:583– 609. 2. Wirestam R, Andersson L, Østergaard L, Bolling M, Aunola J, Lindgren A, Geijer B, Holtas S, Stahlberg F. Assessment of regional cerebral blood flow by dynamic susceptibility contrast MR using different deconvolution techniques. Magn Reson Med 2000;13:691–700.

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