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Dynamic susceptibility contrast (DSC)-MRI is commonly used to measure cerebral perfusion in acute ischemic stroke. Quantification of perfusion parameters involves deconvolution of the tissue concentration-time curves with an arterial input function (AIF), typically with the use of singular value decomposition (SVD). To mitigate the effects of noise on the estimated cerebral blood flow (CBF), a regularization parameter or threshold is used. Often a single global threshold is applied to every voxel, and its value has a dramatic effect on the CBF values obtained. When a single global threshold was applied to simulated concentration-time curves produced using exponential, triangular, and boxcar residue functions, significant systematic errors were found in the measured perfusion parameters. We estimate the errors obtained for different sampling intervals and signal-to-noise ratios (SNRs), and discuss the source of the systematic error. We present a method that partially corrects for the systematic error in the presence of an exponential residue function by applying a linear fit, which removes underestimates of long mean transit time (MTT) and overestimates of short MTT. For example, the correction reduced the error at a temporal resolution of 2.5 s and an SNR of 30 from 29.1% to 11.7%. However, the error is largest in the presence of noise and at MTTs that are likely to be encountered in areas of hypoperfusion; furthermore, even though it is reduced, it cannot be corrected for exactly. Magn Reson Med, 2006. © 2006 Wiley-Liss, Inc.
Bolus contrast perfusion MRI, or dynamic susceptibility contrast (DSC)-MRI, is potentially important for assessing patients with stroke and other neurological conditions. Many methods for deriving relative perfusion values from the signal-time curves have been suggested and evaluated, such as the time to peak (TTP) (1), time to maximum (Tmax) (2), full width at half maximum (FWHM), first moment, and bolus arrival time (3, 4). However, these relative perfusion measures are prone to substantial fluctuations between examinations (5), which makes it difficult to make comparisons between patients, or in the same patient imaged on several occasions during the course of an illness (1).
Measurement of “absolute” cerebral perfusion using intravascular contrast agents may improve reliability both within and between examinations, and has been attempted in many studies using a framework developed by Meier and Zierler (6) and Zierler (7). In this approach an arterial input function (AIF) is used to characterize the delivery of tracer to the brain. The AIF is then used to deconvolve the tissue concentration-time curves in order to obtain an estimate of the residue function (R(t)) for an infinitely narrow (perfect) bolus. The residue function is related to the rate at which contrast leaves the tissue and is scaled by the cerebral blood flow (CBF). It is assumed that the peak height of CBF · R(t) is equal to the CBF. The cerebral blood volume (CBV) is calculated independently of the deconvolution process from the areas under the voxel signal-time curve and the AIF. The other quantity of interest, the mean transit time (MTT), is calculated by applying the central volume principal MTT=CBV/CBF. However, although the calculation of absolute perfusion values improves the reproducibility of the measurements both between and within subjects, it includes several assumptions that are not strictly applicable to DSC-MRI (8, 9). For example, it is assumed that the MRI perfusion data are cross-calibrated with values obtained from other modalities (10–12), that every voxel in the image is fed directly by the AIF (i.e., in parallel rather than sequentially), that the susceptibility is in direct proportion to contrast concentration, and that the constant of proportionality is independent of tissue type. These assumptions may mean that DSC-MRI with deconvolution will contain errors that vary in magnitude across the range of CBFs and MTTs that are likely to exist in low-flow areas, such as in ischemic stroke (5).
The deconvolution required to produce the quantitative perfusion measurement can be performed in several different ways (13–18), but the one most frequently used in clinical research studies is singular value decomposition (SVD) (10, 11, 19–24). SVD uses a threshold parameter (sometimes referred to as the regularization parameter or PSVD) to limit the effects of noise on the calculated perfusion parameter estimates. Østergaard et al. (10) performed experiments with a signal model to identify optimal threshold parameters for standardized conditions thought to occur in the human brain. The standard conditions were MTTs of 2 and 4 s, and, assuming a brain density of 1.04 g/ml, CBFs of 19.2 and 38.5 ml/100 g/min (10). MTTs of 2 and 4 s are both within normal limits and well below the values found in an acute ischemic stroke (2, 25). However, a CBF of 19.2 ml/100 g/min is low, near the level where intracellular edema begins in an acute ischemic stroke and certainly below the level of cessation of normal neuronal electrical activity, which is thought to occur at about 35 ml/100 g/min (25). A CBF of 38.5 ml/100 g/min is therefore only just above the level of cessation of normal neuronal activity These values therefore do not represent the wide range of normal or abnormal values that might be encountered in normal and ischemic brain, but rather are a mixture of both and are not likely to be encountered biologically. From this signal model, a single optimum global threshold was then identified for use in all subsequent calculations, which was the average of the individual optimum local thresholds obtained from the four standard conditions described above. In subsequent patient studies, a constant global threshold parameter of 20% was used (11, 26).
We hypothesized that the chosen global threshold value could introduce errors into the measurement of the absolute perfusion parameters, and that the magnitude of these errors would vary with the temporal resolution, effective repetition time (TR), and signal-to-noise ratio (SNR), i.e., it would be related to low contrast concentration. Using simulated concentration-time curves produced with exponential, triangular, and boxcar residue functions, we investigated the effect of applying a single global threshold to simulated (but realistic) perfusion data covering a range of perfusion values, and measured the effect of different TRs and SNRs on the resulting systematic errors. We suggest a method to overcome the systematic error in the case of an exponential residue function, evaluate it, and discuss the source of the systematic error.
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- MATERIALS AND METHODS
In the following discussion we use perfusion and signal-processing terminology interchangeably, since CBF · R(t) represents the impulse response of some unknown system. Furthermore, we describe the SVD threshold and its effect on the estimates of CBF · R(t) by analogy with the frequency domain and filtering. This allows the effect of applying a threshold to be described in the terms of more easily visualized spectral behavior.
In order to understand the results produced by SVD-based deconvolution using a single threshold, it is most informative to first examine the behavior observed in the cases without noise. For a single fixed global threshold that is optimized for a range of CBF values, concentration-time curves with shorter MTT will be overfiltered, i.e., high-frequency information will be removed, leading to an impulse response that is overdamped. This overdamping leads to an underestimate of the CBF (peak height of CBF · R(t)), and hence via the central volume relationship an overestimate of the MTT, resulting in the migration of short MTT values to longer MTT values. It was noted that when the Fourier transforms of the data for MTT = 2 s at the simulated TR and the high-temporal-resolution data used for convolution were compared, the signal-time curves produced for the shortest MTT values appeared to be aliased. The concentration-time curves for the longer MTT values contain less high-frequency information, and in the absence of noise one could expect these to be relatively immune to the choice of threshold. However, this is not the case, and even for the SNR of 1000, the MTT estimates at these longer points are shortened. For longer MTT, the optimum local threshold values, i.e., the choice of threshold that minimizes the CBFerr for an individual concentration-time curve, are higher than the single optimum global threshold obtained for the population as a whole. At longer MTTs, the impulse response is underdamped and the maximum value of the residue function, CBF, is overestimated, leading to an underestimation of MTT. In other words, at longer (i.e., more abnormal) MTTs, where greater accuracy is needed to distinguish potentially viable from nonviable brain in acute ischemic stroke, the application of a single global threshold value leads to the least accuracy (compared to normal regions) and the estimates are most error prone.
Even in the absence of noise, an appropriate threshold must be applied before an estimate of CBF · R(t) is obtained. The MTT:CBF pairs with the shortest MTT have an optimum local threshold that is lower than the optimum global threshold; that is, if the simulations were run for a subset of the data comprised of the shortest MTT values alone, the threshold obtained (the local threshold) would be lower in value than that obtained for the entire range of MTT values (the global threshold). Since the high-frequency (short MTT) information is already overdamped due to the overfiltering by the optimum single global threshold, the addition of noise has minimal effect on these points except at the very lowest SNRs. A comparison of the positions of the MTT:CBF pairs in Figs. 1a and 2a show this skewed effect, with the estimated pairs with MTT = 2 s obtained at SNR = 30 being closer to the corresponding pairs at SNR = 1000 than the estimates at MTT = 20 s. In other words, areas of normal MTT are less affected by noise than areas of low flow and long MTT, where the greatest precision is actually needed.
However, we can see from Fig. 2a that in the presence of noise, at longer MTTs an effect associated with the local CBF is also encountered. This is shown as the curvature in MTTest, with a reduction in CBFtrue causing shortening in MTTest, i.e.,MTTest is an underestimate of MTTtrue. Reference to Fig. 2a indicates that the presence of noise increases the positional error in the pairs with lowest CBF and longest MTT. In ischemic stroke, where long MTT and low CBF may be expected at low SNR, the results of our study suggest that quantifying MTT lesion area using the calculated quantitative maps may be problematic and underestimate MTT-based lesion volumes (1). For values of MTTtrue ≥ 14 s and CBFtrue ≤ 35 ml/100 g/min, an average MTT underestimation of 1.0 s was found. A 1.0-s error may not appear to be large, but it will lead to an underestimate of the “tissue at risk” (28) and provide erroneous information on the true perfusion values associated with salvageable vs. unsalvageable tissue, as well as for comparison with other modalities (1). The corrected data also restore the proper relationship between CBF and MTT at short and long MTT values, the distortion of which in the uncorrected data may have an effect on the interpretation on the area of abnormal MTT. Further work is required to characterize this effect and assess the magnitude of its impact on the interpretation of clinical images.
We can see from Table 1 that the optimum global threshold reduces with shorter TR. When we explore the relationship between the optimum local thresholds and the MTT, we see that the average gradient of the line is reduced as TR is reduced, i.e., the optimum global threshold is a better approximation of the individual local thresholds as the TR decreases. Therefore, in the absence of noise, as the TR is shortened the effect of the underfiltering is also reduced and hence the overall error is reduced. However, the error is still substantial, so increased temporal resolution alone is not a solution.
The systematic error demonstrated here can also be explained by considering the effect of the threshold as a filter in the frequency domain, and it also becomes clear what causes the oscillations seen in R(t). The basis functions formed by SVD for Caif(t) have some similarity to Fourier basis functions in that they have periods of oscillation that decrease with increasing row index, i.e., the first basis function approximates the DC component of R(t) while the last represents high-frequency components of R(t). Applying the threshold means that some of the basis functions are not included in the estimation of R(t). It now becomes clear that thresholding will have similar effects to applying a boxcar filter in the frequency domain, which is known to cause oscillations in the time domain due to the convolution property of the Fourier transform. For that reason we defined the optimum global threshold as that which minimized the error in CBFest, instead of attempting to minimize the error in R(t) as a whole. This difference is particularly important when high temporal resolutions cannot be achieved.
One limit to the general applicability of the current work is the assumption of an exponential residue function. The exponential model was chosen because it assumes the vasculature to be a single well-mixed compartment, and this would appear to be the most applicable in the presence of an intact blood–brain barrier. For simulations at the highest SNR using both square and triangular residue functions, the results showed features similar to those obtained for the exponential residue function, but the correction factors and optimum thresholds were different. Regardless of the underlying residue function, the reduction in estimated MTT for long-MTT, low-CBF pairs was still observed at lower SNR. At SNR = 1000, TR = 2.5 s, and MTT = 2 s, the MTTest's produced for the different residue functions were most similar. This is because there is the least information about the shape of R(t) available at the shortest MTTs, and therefore the CBFest obtained from the deconvolution differs the least. As mentioned above, the Cvoi(t) curves for an MTT = 2 s are also aliased, which increases the similarity of the estimates of CBFest obtained for different residue functions. As the MTT increases, so does the difference in the estimates of CBFest obtained for different residue functions. This is because at a given TR the shape of Cvoi(t) is more accurately recorded as the MTT is increased.
To overcome this difficulty, Liu et al. (19) introduced a measure of the local SNR, SNRc, or contrast-to-noise ratio (CNR). They published thresholds as a function of SNRc, allowing for a local estimate of the optimum threshold. However, their simulations were performed only at an MTT of 3 s. Unfortunately, the relationship between the local threshold and SNRc is likely to be different at different MTT values. This is because one can produce the same value of SNRc for two voxels with different MTTs by changing the CBF; however, the voxels will have different spectrums and hence optimal thresholds.
Murase et al. (22) examined the relationship among the optimum threshold, CBF estimate, and SNR. They showed that the value of the threshold has a large effect on the measured CBF values, and that the optimum threshold is a function of SNR. The results of their study are broadly similar to those presented here. However, their simulations covered only a limited combination of CBF, CBV, and MTT values, and consequently the systematic error introduced by SVD-based deconvolution was not demonstrated clearly and was not quantified. However, they did report that for a constant threshold the error in the estimate of CBF is dependent on the MTT. In two studies Sourbron et al. (23, 24) investigated different methods of choosing and applying the threshold. One of these methods, termed “generalized cross validation with standard form Tikhonov regularization,” appears to be a robust method for choosing a local threshold, and the results of the simulations and its application to clinical data are encouraging. As part of these studies they examined the systematic error in the estimates of CBF and R(t) produced using a fixed threshold. Their results are broadly similar to the present ones; specifically they also demonstrate the underestimation of MTT for long MTT values at low CBF. However, the effect of this underestimation on the relationship between the estimates of CBF and MTT produced for different true MTT and CBF values is not clearly demonstrated.
As developments in magnet and gradient technology provide improved TR and SNR, methods for regularizing individual voxels in an image will become applicable. For example, in simulations at shorter TRs (1.0–1.5 s), Gaussian process deconvolution was shown to produce superior results to SVD-based deconvolution with a single global threshold (18).
In addition to the assumption of an exponential residue function, our correction method has other limitations. For example, the use of a subject-averaged AIF is problematic, since the averaging process has a low-pass filtering effect on the resultant average AIF. Consequently, although the SNR is increased, this is at the expense of distortion in the spectral properties of the average AIF. An alternative approach would be to calculate the perfusion parameters for every voxel using the individual-subject AIFs defined by a region of interest (ROI) drawn over an artery and then calculate the mean parameter values from the results obtained for the individual AIFs. This approach was not taken here due to the lengthy nature of the simulations required for a single AIF.
Several papers (9, 20, 21) have shown that arrival-time delay between the AIF and the tissue concentration-time curve can also have a major impact on the MTT estimates produced by SVD deconvolution. In the current work the delay was set to zero. However, this in itself may cause difficulties. To investigate this problem, experiments are now being performed with a discretization scheme suggested by Wu et al. (20, 21) and nonzero delay times. Another possible limitation to the current simulations is the precise scaling factor used to match the peak signal drop to the model. This will have an important effect on the achievable SNRc, since the scaling is affected by factors such as the AIF, tissue susceptibility, contrast concentration, and pulse sequence. However, regardless of the scaling factor the systematic error will still be observed. The effect of a reduction in signal drop would be to reduce the SNRc, causing the curvature observed at long MTT and low CBF to start at shorter MTT values. It is also interesting to note that since the scaling of the signal is produced to match a standard condition, it is likely that the corrections applied to the data close to the MTT for the standard condition of 4 s (10) will be smallest and hence the technique will be most accurate. Although the effect of changing the standard condition on the results was not investigated, certainly Figs. 1b and 2b suggest that this may be the case. The effect of cross calibration with values obtained for white matter by positron emission tomography (PET) was investigated by Chen et al. (12). They concluded that if cross calibration was applied, it was more important that the deconvolution algorithm applied to the data would produce more accurate estimates of MTT than CBF.
In conclusion, this work empirically demonstrates the existence of a significant systematic error within the MTT estimates produced by SVD deconvolution when a single global threshold is applied. Further work is currently under way to produce a more mechanistic analysis to increase our understanding of these empirical observations.