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Magn Reson Med. Author manuscript; available in PMC 2011 January 4.

Published in final edited form as:

PMCID: PMC3014609

NIHMSID: NIHMS257682

J.J. Lee,^{1} G.L. Bretthorst,^{2} C.P. Derdeyn,^{1,}^{3,}^{4} W.J. Powers,^{3,}^{5} T.O. Videen,^{1,}^{3} A.Z. Snyder,^{1,}^{3} J. Markham,^{1} and J.S. Shimony^{1}

Corresponding Author: Joshua S. Shimony, Mallinckrodt Inst. of Radiology, Washington University School of Medicine, Campus Box 8131, St. Louis, MO 63110. Email: ude.ltsuw@jynomihs

The publisher's final edited version of this article is available at Magn Reson Med

See other articles in PMC that cite the published article.

Compared to gold-standard measurements of cerebral perfusion with positron emission tomography (PET) using H_{2}[^{15}O] tracers, measurements with dynamic susceptibility contrast (DSC) MR are more accessible, less expensive and less invasive. However, existing methods for analyzing and interpreting data from DSC MR have characteristic disadvantages that include sensitivity to incorrectly modeled delay and dispersion in a single, global arterial input function (AIF). We describe a model of tissue microcirculation derived from tracer kinetics which estimates for each voxel a unique, localized AIF (LAIF). Parameters of the model were estimated using Bayesian probability theory and Markov-chain Monte Carlo, circumventing difficulties arising from numerical deconvolution. Applying the new method to imaging studies from a cohort of fourteen patients with chronic, atherosclerotic, occlusive disease showed strong correlations between perfusion measured by DSC MR with LAIF and perfusion measured by quantitative PET with H_{2}[^{15}O]. Regression to PET measurements enabled conversion of DSC MR to a physiological scale. Regression analysis for LAIF gave estimates of a scaling factor for quantitation which described perfusion accurately in patients with substantial variability in hemodynamic impairment.

Accurate measurement of cerebral hemodynamics advances our understanding of normal and abnormal brain physiology, but remains a difficult research and clinical challenge (1). Many common disorders of the brain, including stroke and neoplasm, manifest through hemodynamic abnormalities and could be better diagnosed and treated with more accurate measurements (2,3).

The gold standard for quantitative, cerebral hemodynamic measurement is positron emission tomography (PET) using H_{2}[^{15}O] for cerebral blood flow (CBF) and C[^{15}O] for cerebral blood volume (CBV). However, PET is not as widely available as magnetic resonance (MR) imaging and unlike MR entails significant radiation exposure to the patient (4,5). Following the pioneering work by Østergaard, *et al.* (6,7) cerebral hemodynamics measurements using dynamic susceptibility contrast (DSC) MR have become widely available on clinical scanners and have demonstrated early promise of clinical utility (2). Many variations of this method have been proposed (8–14) but few have been validated against a gold standard. Additionally, DSC MR methods yield relative measurements which require scaling to obtain quantitative physiologic values. This is typically done by assigning normal-appearing white matter a flow value of 20–22 mL/min/100 g of tissue (15).

Most clinical implementations of the DSC method require the selection of a large artery in the brain (usually a major branch of the middle cerebral artery) to define the AIF used in the analysis of the data. Traditionally, the magnitude of the native MR signal is logarithmically transformed into relative units of gadolinium concentration. One difficulty with this transformation is that at the peak of the bolus passage the signal may drop to the noise floor making accurate estimates of concentration difficult (16,17).

The use of a single artery or several averaged artery measurements to represent the AIF for the whole brain has its origin in the earliest work in tracer kinetic methods (18,19). For quantitative PET, activity is measured directly from a peripheral arterial source. The resulting time-activity curve is then decay-corrected and shifted in time to match the arrival of PET activity in the brain. One difference between the two methods relates to the use of a vascular tracer in DSC MR and a diffusible tracer in PET. Unlike DSC MR methods which use the complete time-varying AIF, PET uses the convolution of the AIF with the response function (a decaying mono-exponential). For these two reasons, flow measurements in DSC MR are significantly more sensitive than PET to errors in modeled delay and dispersion as well as to the exact shape of the AIF (20). Several investigators have explored the use of local AIFs (LAIFs) to improve hemodynamic measurements (9,21,22) including the use of voxel-specific AIFs using iterative blind-deconvolution (23,24).

In this paper we describe a variation of the standard tissue perfusion model (7) that provides a local estimate of both the AIF and the residue curve for each voxel in the brain. The parameters appearing in this model are estimated using Bayesian probability theory and Markov-chain Monte Carlo methods (25). This approach has several advantages over existing methods. Specifically, it eliminates the difficulty in selecting a single AIF by estimating one for each voxel in the brain. It does not require numerical evaluation of the logarithm of the signal nor does it require numerical deconvolution. The new method estimates blood flow, blood volume and mean transit-time (MTT) as well as several additional model parameters, some of which may provide useful clinical information (25). Here we present the results of this method in comparison to PET measurements and to traditional DSC MR measurements performed on the same day in a population of patients with occlusive cerebrovascular disease (1,26).

The magnetization observed by echo-planar imaging (EPI) from a bolus of contrast agent may be represented as (27):

$$S(t)={S}_{0}exp[-k\xb7{C}_{T}(t)]$$

[1]

The detected signal, *S*(*t*), remains near a baseline value, *S*_{0}, until the arrival at time *t* 0 of the bolus, which is represented by the concentration time curve (CTC) in the tissue, *C _{T}*(

Conservation of mass and the fundamental central volume model of intravascular tracer kinetics (19) specify the CTC in the tissue as a convolution of a voxel-dependent residue function, *R*(*t*), with that voxel’s AIF, C_{A}(*t* ):

$${C}_{T}(t)=\text{CBF}\xb7\underset{0}{\overset{t}{\int}}R(t-\tau ){C}_{A}(\tau )d\tau .$$

[2]

The residue function depends on the details of the tissue microstructure. CBF is the local cerebral blood flow. A general expression for the residue function may be expressed as mono-exponential, as conventionally used in PET, multiplied by a power-series:

$$R(t)={e}^{-t/\text{MTT}}\left[1+\sum _{n=1}{c}_{n}{\left(\frac{t}{\mathit{MTT}}\right)}^{n}\right].$$

[3]

Model selection methods were used to investigate the need for the additional parameters, *c _{n}*. For the quality of our data, the model probabilities consistently favored the simplest model with

The bolus passage of the tracer was described as having three characteristic phases: (1) an initial rise in concentration with the main bolus arrival; (2) a subsequent rise in concentration from recirculation and (3) a steady state circulation phase. The concentrations of tracer in the main bolus as well as in the recirculation peak were each modeled with a normalized gamma function:

$$G(\alpha ,\beta ,{t}_{0},t)=\frac{1}{{\beta}^{\alpha +1}\mathrm{\Gamma}(\alpha +1)}{(t-{t}_{0})}^{\alpha}exp\left[-\frac{(t-{t}_{0})}{\beta}\right].$$

[4]

The arrival time of the bolus is *t _{0}*, and the phenomenological parameter

$${C}_{A}(t){c}_{0}(1-exp(-\gamma (t-{t}_{01})))+{c}_{1}G(\alpha ,\beta ,{t}_{01},t)+{c}_{2}G(\alpha ,\beta ,{t}_{02},t),$$

[5]

with *c*_{0}+*c*_{1}+*c*_{2}=1. In preliminary trials the posterior probabilities of models used independent values for parameters {*α*_{1},*β*_{1},*t*_{01}} and {*α*_{2},*β*_{2},*t*_{02}} for the initial bolus and recirculation, respectively. Bayesian model selection showed the data were more consistent with a simpler model using a single set of dispersion parameters for both boluses {*α*,*β*,*t*_{01},*t*_{02}}, but independent arrival times *t*_{01} and,*t*_{02}. Equations [3–5] were substituted into Eq. [2] and integrals were evaluated analytically. The resulting closed-form expressions were substituted into Eq. [1]. DSC MR provided time-dependent signals, *S*(t), for each voxel. Parameter estimation was performed using Bayesian probability theory (25).

Subjects with paired hemodynamic PET and DSC MR studies were retrospectively identified by review of laboratory records. These subjects were studied under two different research protocols. All subjects provided written, informed consent in compliance with the institutional review board of the Washington University School of Medicine. Both protocols involved hemodynamic PET studies and DSC MR studies obtained on the same day for the purposes of correlative analysis. One protocol was a sub-study of the Saint Louis Carotid Occlusion Study, a prospective, blinded, natural-history study of the role of hemodynamics in stroke risk for patients with atherosclerotic carotid artery occlusion (1,26). The other was a protocol specifically written for the purpose of comparing MR and PET methods of CBF measurement.

Fourteen subjects had complete PET and MR data suitable for analysis. Five and seven of these subjects have been previously reported in two correlative analyses of different MR methods (28,29). The primary inclusion criterion for the 14 subjects was angiographically confirmed atherosclerotic occlusion of one proximal internal carotid artery. Both asymptomatic and symptomatic subjects were enrolled. Symptoms included transient ischemic neurologic deficits, transient monocular blindness, and mild to moderate permanent ischemic neurologic deficits in the appropriate carotid artery territory. Symptomatic subjects were studied at least one month after their last event. Exclusion criteria were: inability to provide informed consent, inability to tolerate a supine position for PET and MR imaging, pregnancy, and contraindications to MR imaging, such as claustrophobia or MR-incompatible medical implants. PET and MR studies were obtained on the same day without randomization, with the PET done first and the MRI following within approximately an hour of completion of the PET study.

All PET scans were performed on an ECAT EXACT HR PET scanner (CTI PET Systems, Knoxville, TN) run in 2D-mode with extended intersection septa. All subjects received antecubital intravenous lines for receiving fluids and radioactive tracers. A radial-artery catheter was placed and connected to an automated continuous blood-sampler that recorded sampled activity in one-second bins. An attenuation scan was obtained. After venous injection of 50 to 75 mCi of H_{2}[^{15}O] and following initial arrival of activity to the head, a 40-second acquisition was recorded. PET scans were reconstructed with filtered back-projection using ramp filtering. The nominal spatial resolution of the detector array was 2.14 × 2.14 × 3.125 mm voxels. Back-projected PET activity counts were filtered with isotropic 2D-Gaussian kernels at 10.0 mm full-width at half-maximum (FWHM). CBF was measured using adaptations of the Kety autoradiographic method and permeability corrections (30,31).

CBF measurements with H_{2}[^{15}O] and quantitative PET underestimate true CBF at high flows because of the incomplete permeability of the blood-brain barrier to water. Our measurements were corrected for this finite permeability using permeability-surface area products for water and butanol as described in reference (31). To the authors’ best knowledge these corrections have previously been reported in only one prior correlative study of perfusion measurements by PET and DSC MR (29).

All MR scans were performed on a 1.5 T Magnetom Vision scanner (Siemens, Erlangen, Germany). All subjects received a large bore antecubital intravenous line appropriate for delivering normal saline and gadolinium contrast agent. Anatomic studies were performed with T1-weighted 3D fast low-angle shot gradient echo sequences (FLASH) with TR/TE = 18/8.4. Images were interpolated by the Vision console from 128 × 128 × 32 to 256 × 256 × 32 voxels; the field-of-view was 256 × 256 × 64 mm. Immediately following an anatomic study, a dynamic study was performed with single-shot gradient EPI with TR/TE = 800/54 and 90-degree flip angles. Images were interpolated from 128 × 128 × 8 to 256 × 256 × 8 voxels; the field of view was 240 × 240 × 48 mm. Axial EPI slices were contiguous and 6 mm thick. The image sequence was repeated 80 times for a total of 64 s. Approximately ten seconds following the start of the EPI scan, a bolus of gadodiamide (Omniscan, Nycomed, Zürich, Switzerland) was intravenously administered by power injection (Mallinckrodt, Inc., St. Louis, MO) at 5 mL/s to a final dose of 0.2 mmol/kg.

Posterior probabilities for all model parameters were computed voxel-by-voxel, using complete EPI time-series with Bayesian probability theory (32). There were ten parameters per voxel expressed in groups Ω, with voxel indices *i*:

$${\mathrm{\Omega}}_{i}={\{\mathit{CBF},\mathit{MTT},{t}_{01},{t}_{02},\alpha ,\beta ,{S}_{0},{c}_{0},{c}_{1},{c}_{2}\}}_{i}.$$

[7]

Two parameters were CBF and MTT. Two parameters expressed the primary and recirculation bolus arrival times, *t _{01}* and

Bayes’ theorem states that the posterior probability for the parameters, given data, *D _{i}*, and relevant background information,

$$P({\mathrm{\Omega}}_{i}{D}_{i},I)P({D}_{i}{\mathrm{\Omega}}_{i},I)P({\mathrm{\Omega}}_{i}I),$$

[8]

for which *P*(*D _{i}*|Ω

$$P({\mathrm{\Omega}}_{i}I)P(\mathit{CBF}I)P(\mathit{MTT}I)P({t}_{01}I)P({t}_{02}I)P(\alpha I)P(\beta I)\dots .$$

[9]

All prior probabilities were assigned Gaussian distributions bounded by appropriate physiologic ranges. The ranges selected for the parameters in our data set were: *CBF* (0–20.0) (dimensionless units that include the empirical constant *k*); *MTT* (0–20.0) s; *t _{01}* (0–40.0) s;

$$P({D}_{i}{\mathrm{\Omega}}_{i},I)=\int P({\sigma}_{i},{D}_{i}{\mathrm{\Omega}}_{i},I)d{\sigma}_{i}\int P({\sigma}_{i}I)P({D}_{i}{\sigma}_{i},{\mathrm{\Omega}}_{i},I)d{\sigma}_{i}.$$

[10]

Assigning Jeffreys’ prior probability to *P(σ _{i} |I* ) and assigning the Gaussian distribution to

$$P({D}_{i}{\mathrm{\Omega}}_{i},I)\frac{1}{2}\mathrm{\Gamma}\left(\frac{M}{2}\right){\left(\frac{{Q}_{i}}{2}\right)}^{-M/2}.$$

[11]

*M* is the number of data samples in time per voxel. *Ω _{i}* is the total squared residual:

$${Q}_{i}=\sum _{k=1}^{M}{({E}_{ik}-{S}_{ik})}^{2}$$

[12]

*E _{ik}* is the measured signal and

Computational implementations were custom written for the SGI Origin 2000 (Silicon Graphics, Inc., Sunnyvale, CA). Using 32 processor-cores the computation time per image slice, with extraparenchymal voxels masked out, was approximately 1.8 hours. Two of eight axial slices were analyzed for fourteen of the subjects. Only one of eight axial slices was analyzed in one of the subjects. The framework is currently distributed as Varian VNMRJ packages (http://bayesiananalysis.wustl.edu, http://www.varianinc.com/cgi-bin/nav?products/nmr/software/bayes). Recently we have formulated a non-linear algorithm that decreases the computation time by greater than ten-fold.

For comparison, perfusion was estimated using the method of maximum-likelihood expectation-maximization (MLEM) originally reported by Vonken, *et al.* (33). All computational details from the original implementation by Vonken, *et al.*, were implemented with two exceptions. One: all AIFs were expressed as absolute magnetizations in time then smoothed using nonlinear least-squares regression of gamma-variate forms (Eq. 4). Two: the matrix representation of the AIF was forced to have the circulant-Toeplitz form advocated by Wu, *et al.* (14). In a subset of the patient population described above, CBF maps from MLEM and OSVD had indistinguishable correlations with PET CBF maps. Limited tests of the sensitivity of our implementations of MLEM and OSVD revealed no observable dependence of AIFs on time-shifts forwards or backwards in time. On an IBM System ×3655 (IBM, Armonk, NY) running one processor-core the computation time per image slice was approximately 0.8 hours. AIFs were manually selected from perivascular voxels of M1-branches of the middle cerebral artery ipsilateral to patent carotid arteries. After identifying vessel-dominant voxels by scrolling through EPI time frames, 8–10 voxels were selected in the vicinity of major vessel branches, but exterior to areas of deep signal loss. Averaged AIFs were fitted with a gamma-variate using nonlinear least-squares (34).

For each DSC-MR series, EPI time-frames were co-registered to a single, early frame which best represented anatomical landmarks. 3D-FLASH series were aligned to the EPI spatial reference. H_{2}[^{15}O] studies from PET were co-registered to the aligned 3D-FLASH. In some cases, co-registration was improved by using native 3D-FLASH as an intermediate spatial reference, *e. g.*, PET was aligned to 3D-FLASH, then aligned PET was transformed to the EPI spatial reference. The final common processing for all series was transformation to the EPI reference. Linear, affine registration methods were used throughout. Minimization of mutual information was added in cases for which affine methods yielded inadequate alignment.

Three-dimensional-FLASH images were segmented into three to five clusters using fuzzy c-mean clustering (Matlab 7, The Mathworks, Inc., Natick, MA). Each cluster was inspected and categorized as a region of interest (ROI) for gray matter, white matter, cerebral-spinal fluid (CSF), extra-dural tissues or bone. From EPI, voxels rich in arteries, those with the greatest signal drop during bolus passage, were identified and assigned to a separate, arterial ROI. All ROIs were graphically superimposed on co-registered 3D-FLASH, EPI, and PET series and visually inspected to assure accuracy of placement. A board-certified neuroradiologist (JSS) confirmed the accuracy of all placements.

Correlative analyses were performed using averages over contiguous blocks of in-plane, native voxels in CBF maps. CBF maps from PET and DSC MR were matched for partial-volume effects by using blocks of voxels sized to 10 × 10 × 6 mm^{3}. Three-dimensional Gaussian filtering was applied to native PET counts per validated methods prior to block-averaging; filtering was never applied to MR CBF maps. High-resolution ROIs for arteries, CSF and viable parenchyma were each block-averaged and normalized for use with block-averaged CBF maps. Block-averaged ROIs were multiplied with block-averaged CBF maps. For block-averaged ROIs with mixed tissues an approximate weighting scheme was used. The averaged ROI voxel was assigned a real value from zero through unity, the relative weight of the composite high-resolution tissue ROI voxels, each of which was always set to zero or unity by the clustered segmentation scheme. Each block-average was analyzed with scatter plots and Bland-Altman plots.

All images were viewed and analyzed using Analyze AVW (The Mayo Clinic, Rochester, MN), FSL (http://www.fmrib.ox.ac.uk/fsl) (35), diplib (Quantitative Imaging Group, Delft University of Technology, http://www.diplib.org) and Matlab (The Mathworks, Inc., Natick, MA).

Representative plots from the parameter estimation phase of the Bayesian processing in a single voxel are shown in Fig. 1. During the parameter estimation phase the search space was explored using a Markov chain Monte-Carlo calculation and a population of parameter values were created by sampling this calculation, and were used to compute mean and standard deviation estimates for each parameter. Fig. 1a demonstrates the signal time curve of this voxel in red, with the model generated from the parameters that maximized the joint posterior probability in black, and the difference in green. Figs. 1b-1e show histograms of this sampling for four of the estimated parameters. The coefficients of variation (standard deviation divided by the mean) estimated from these plots is 1.4% for *t*_{01}, 12.3% for α, 8.1% for β and 13.2% for the CBF. Figs. 1f-1i illustrate sampling for pairs of parameters, which provide information on cross correlations between parameters in the model. The posterior probability density functions for *t*_{01} versus CBF and α versus CBF from Figs. 1g and 1i are consistent with no cross correlation between the parameters. Although not shown, there was also no cross correlation between CBF and β.

Plots describe the parameter estimation phase of Bayesian processing for a single voxel. Parameter estimates were produced by sampling the Markov chain for the data model. (a) The observed magnetization vs. time in the selected voxel is plotted in red. **...**

Details for a single patient from the study cohort are shown in Fig. 2. This 37 year-old female had a history of lacunar infarcts and infarction of the territory of the right middle-cerebral artery approximately one year prior to imaging. Cerebral angiography performed shortly before her MR and PET studies revealed the right internal carotid artery to be occluded. The T1-weighted 3D-FLASH image shows the sequela of the patient’s ischemic stroke, visible in the right hemisphere as encephalomalacia (radiologic display convention). The 3D-FLASH image was the reference for segmentation of ROIs and co-registration of PET images and is shown co-registered to EPI. Fig. 2 shows maps of hemodynamic parameters obtained by MR LAIF: CBF, CBV, MTT, α, β, *t*_{01}, and *c*_{1}. The right-frontal area of infarction and hemispheric asymmetries are discernable in all parameter maps except the map of α. This finding represents a uniform rate of in-flow of contrast to both hemispheres, as may be expected from collateral blood supply to the hemisphere ipsilateral to chronic carotid occlusion. However, the map of *β* is asymmetric between hemispheres, likely representing slower venous outflow in the affected hemisphere. The map of *t*_{0}, the delay in bolus arrival, is dramatically asymmetric between hemispheres showing delayed arrival in the hemisphere with complete carotid occlusion. The map of *c*_{1} demonstrates increases in the affected hemisphere that correspond to decreases in the recirculation of contrast agent; the map corresponds anatomically with regions of decreased *β*. The affected hemisphere also exhibits mildly decreased CBF, mildly decreased CBV and increased MTT,

T1-weighted MR study and maps of hemodynamic parameters from the LAIF method for a patient with a history of lacunar infarcts and cerebral vascular accident involving the right middle-cerebral artery. CBF, CBV and MTT were estimated with LAIF. Additional **...**

Representative blood-flow maps for three subjects from the study cohort are shown in Fig. 3. The variability of CBF maps between PET, MR LAIF and MR MLEM, with and without block-averaging of native voxels is illustrated across columns. The variability of perfusion maps between human subjects is illustrated by the rows. Statistical analyses required comparison of perfusion maps from MR and PET with matched spatial resolutions. The results of block-averaging are illustrated across the columns of Fig. 3. Not shown are the block-averaged ROIs used to exclude artery-rich voxels, CSF-rich voxels, voxels from extra-dural tissues, and background extracranial voxels.

CBF maps from MR LAIF and PET for three representative patients. Column (a) shows anatomical references. Columns (b—d) show CBF maps with native spatial resolutions for MR and PET. Columns (e—g) show results of block averaging to 10 × **...**

Fig. 4 illustrates graphically for a single patient the association of quantitative PET CBF to (a) MR LAIF CBF and (b) MR MLEM CBF. Each plotted point represents the average over a contiguous block of voxels; MR and PET voxels were each sized to 10 × 10 × 6 mm. Arbitrary units for LAIF and MLEM indicate that the value of scaling coefficient *k* is unknown. Both artery-rich voxels, plotted in red, and CSF voxels plotted in green confound the analysis of CBF as measured by either MR method. For the single, representative patient, the squared Pearson correlation coefficient of PET with LAIF is 0.491. For PET with MLEM it is 0.446. The graphical analysis of Bland and Altman (36) for comparing alternative measurement methods is shown for (c) the deviation of standardized first moments of LAIF CBF from PET CBF and (d) the deviation of standardized first moments of MLEM CBF from PET CBF. Standardized moments show scale-invariant trends. Similar results were obtained for plots of percentage of the mean (37).

Analysis of correlations between PET CBF and MR CBF for a representative patient (patient 1). Plotted points are block-averages of parenchyma voxels only. Blocks containing a majority of CSF or artery-rich voxels are excluded. (a) Scatter plot of PET **...**

Fig. 5 illustrates graphically for all 14 subjects the pooled association of PET CBF to (a) MR LAIF CBF and (b) MR MLEM CBF. Each plotted point represents a single block-averaged voxel. ROI masks were employed to exclude artery-rich voxels and CSF-rich voxels. The squared Pearson correlation coefficient for PET with LAIF is 0.314. For PET with MLEM, it is 0.246. The Bland and Altman analysis is shown for the pooled patient data for (c) the deviation of LAIF CBF from PET CBF and (d) the deviation of MLEM CBF from PET CBF. The scaling coefficient *k* is *a priori* not universal and constant for each patient, as illustrated previously by Takasawa (37) as well as Figs 5a and 5b. For any single patient, correlations are strong between LAIF and PET, and between MLEM and PET, as detailed in Table 1. The mean value of *k* over 14 patients was calculated to be 0.054 ± 0.0089 for LAIF, and 0.0068 ± 0.0013 for MLEM. Relative standard errors in *k* were 16% for LAIF and 19% for MLEM, suggesting that an empirically determined, constant *k* may be useful in practice. Variation of *k* corresponds predictably with decreased correlation between MR and PET, for which case regressed slopes tend toward zero.

The LAIF method has several advantages over current techniques. The selection of a single, global AIF, which can be challenging in a DSC MR study, is obviated in LAIF. During contrast-bolus injection the signal in large vessels will often saturate to the noise floor making accurate estimation of the AIF difficult (16). It should be noted that this effect is unique to MR and is not shared by AIFs used in perfusion imaging by PET and CT. Many investigators avoid voxels which are suspected to lie within or very near large vessels because of confounding effects from signal saturation. These criteria have been codified in automated AIF selection methods, which employ statistical clustering algorithms to segregate tissue concentration curves into groups with similar temporal shapes (38). A cluster is sought which best satisfies criteria such as: large area under the curve of the CTC, early arrival, rapidly and smoothly rising peak concentrations and quick wash-out. Clusters are averaged, filtered and often fitted with a gamma-variate curve for use as the global AIF. Global AIF selection methods run the risk of incorporating some tissue voxels into the AIF (39). The LAIF method models AIF saturation, using the assumption of a gamma-variate shape, by directly using the MR signal Eq. [1] without resorting to a logarithm transformation to concentrations.

Selection of a local AIF has the advantage of better representing the arterial supply to the brain in subjects with acute or chronic hemodynamic impairment. Using a single AIF provides accurate results in perfusion imaging with quantitative PET since PET uses the convolution integral of the AIF rather than its timecourse. Additionally, PET uses a diffusible tracer for which the residue curve is an exponential that requires the estimation of a single transit-time parameter. Compared to the use of AIFs in PET, the use of AIFs for DSC MR is significantly more sensitive to the effects of delay and dispersion. Underestimations as well as overestimations of regional CBF have been observed and attributed to the delay, dispersion and mixing with tissue voxels that result from the construction of a global AIF (14,39,40). Delay and dispersion in the AIF are hallmarks of hemodynamic impairment; thus DSC MR with deconvolution may compound inaccuracies in this clinical scenario. The significant variations from normal cerebral hemodynamics in our patient cohort bring out these overestimations and underestimations in Fig. 5. Delay, but not dispersion, effects may be substantially mitigated by ensuring that numerical deconvolutions are performed with circulant-Toeplitz kernels, thereby preserving temporal causality (14). There is great variability in perfusion measurements with variations in the choice of AIFs, as has been shown in numerous studies (39,41,42). The LAIF method avoids the complications of delay and dispersion from a distal AIF.

An additional benefit of the LAIF model is the estimation of new parameters that may provide additional valuable clinical information. These parameters include the arrival time for the main bolus and the recirculation peak (*t _{01}* and

Within the limitation of our assumption of certain shapes for the AIF and residue curves, the LAIF method avoids the need to perform numerical deconvolution, which is inherently unstable, especially with non-Gaussian noise. Various methods have been developed to overcome this problem: truncating the rank of the deconvolution kernel by SVD (14), Tikhonov regularization (8), and also regularization by parameterized reduction of oscillations in the residue function (40,43). The LAIF model makes the assumption of analytical curve shapes for the tissue concentration curve and the residue curve and then, using these analytical curves, performs an analytical convolution. The parameter estimation then relies on Bayesian probability theory and Markov-chain Monte Carlo and does not require computation of deconvolutions, thus avoiding these issues altogether.

It should be noted that when substituting Eq. [2] into [1] the CBF is multiplied by the unknown constant *k*, and without a way to determine *k* only relative perfusion can be evaluated. In this study we show that PET perfusion data may be used to calibrate this constant for a population of patients with chronic, cerebral hemodynamic impairment. Although theoretically *k* varies across different tissue types and details of pulse sequences, in practice, a single value may be adequate for the MR CBF ranges of clinical significance. The determination of this constant may provide a scaling factor to better quantify flow measurements with DSC MR.

We present a tissue perfusion model that provides a local estimate of both the AIF and the residue function for each voxel in brain perfusion image. The parameters of the model are estimated using Bayesian probability theory and Markov-chain Monte Carlo. This technique has several advantages over existing methods. It obviates the difficulties of determining a single, global AIF. It does not require numerical deconvolution. In addition to estimating CBF, CBV and MTT, several novel model parameters are estimated, which may provide useful clinical information.

We thank the ASNR Neuroradiology Education and Research Foundation, Boston Scientific-Target Fellowship in Cerebrovascular Disease Research (JSS) and the Mallinckrodt Institute of Radiology – Bayer Healthcare Clinical Research Fellowship (JJL) for their generous support during this project. PET and MR data acquisition was supported by National Institutes of Health research grants R01 28497 (WJP) and K08 NS02029 (CPD)

- AIF
- Arterial input function
- CBF
- Cerebral blood flow
- CBV
- Cerebral blood volume
- CTC
- Concentration time curve
- DSC
- Dynamic susceptibility contrast
- EPI
- Echo-planar imaging
- FLASH
- Fast, low-angle shot
- FWHM
- Full-width at half-maximum
- LAIF
- Localized arterial input function
- MLEM
- Maximum-likelihood expectation-maximization
- MTT
- Mean transit time
- MR
- Magnetic resonance
- OSVD
- Oscillation-minimized singular-value decomposition
- PET
- Positron emission tomography
- ROI
- Region of interest
- STLCOS
- Saint Louis Carotid Occlusion Study
- SVD,SSVD
- Standard singular-value decomposition

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