FULL PAPER Magnetic Resonance in Medicine 00:00–00 (2014)

Modeling and Correction of Bolus Dispersion Effects in Dynamic Susceptibility Contrast MRI Amit Mehndiratta,1* Fernando Calamante,2,3 Bradley J. MacIntosh,4 David E. Crane,4 Stephen J. Payne,1 and Michael A. Chappell1 dynamic susceptibility contrast MRI (DSC-MRI) requires the injection of a bolus of paramagnetic contrast agent followed by measurement of the MRI signal loss during its passage through the tissue (3–5). The quantification of DSC-MRI data is based on tracer kinetic theory (6–8), and requires the measurement of an arterial input function (AIF) for the voxel-wise deconvolution of the concentration time curve (CTC, C(t)) calculated from the MRI signal time series. On deconvolution of the CTC and the measured AIF, both tissue perfusion and the tissue residue function are estimated, with the residue function encapsulating information about capillary hemodynamics (9). There are multiple methods to perform deconvolution and nonparametric deconvolution techniques (3,10) have superseded the initially proposed model based analysis (8). Since then, singular value decomposition (SVD) has been commonly used for perfusion analysis (11). One of the typical assumptions in the DSC-MRI quantification process is the absence of bolus delay and dispersion between the site of AIF measurement and tissue voxel, i.e., the measured AIF is assumed to reflect the exact input function to the tissue voxel being analyzed. Partial volume effects in the AIF measurement arise due to the spatial resolution of a typical DSC-MRI acquisition, whereby both arterial and nonarterial components occupying AIF chosen voxels. To minimize partial volume effects, the AIF is typically measured at a major cerebral artery, such as the middle cerebral artery (MCA) (12–16). A global AIF is also sometimes calculated by selecting local AIFs automatically or semi automatically and merging these into a single global AIF (5,17). This single AIF is then used as the input function for the whole brain and this strategy can contribute to bolus delay and dispersion. Such errors might cause inaccuracies in perfusion quantification, particularly in patients with cerebral ischemia, but they may even be present to a lesser extent in healthy volunteers (18,19). In addition to perfusion the shape of the residue function is also of considerable interest. The residue function is related to hemodynamic variations in the capillary bed. It has been speculated that the residue function shape might undergo a considerable amount of change under condition of ischemic stress (20,21) which might be clinically useful for diagnostic and therapeutic management. The residue function might also hold potential information on flow heterogeneity (22,23) and oxygenation (24,25) of the tissue. It has been challenging to study the variations in the residue function in pathology with the commonly used SVD method (10) because of

Purpose: Bolus dispersion in DSC-MRI can lead to errors in cerebral blood flow (CBF) estimation by up to 70% when using singular value decomposition analysis. However, it might be possible to correct for dispersion using two alternative methods: the vascular model (VM) and control point interpolation (CPI). Additionally, these approaches potentially provide a means to quantify the microvascular residue function. Methods: VM and CPI were extended to correct for dispersion by means of a vascular transport function. Simulations were performed at multiple dispersion levels and an in vivo analysis was performed on a healthy subject and two patients with carotid atherosclerotic disease. Results: Simulations showed that methods that could not address dispersion tended to underestimate CBF (ratio in CBF estimation, CBFratio ¼ 0.57–0.77) in the presence of dispersion; whereas modified CPI showed the best performance at low-tomedium dispersion; CBFratio ¼ 0.99 and 0.81, respectively. The in vivo data showed trends in CBF estimation and residue function that were consistent with the predictions from simulations. Conclusion: In patients with atherosclerotic disease the estimated residue function showed considerable differences in the ipsilateral hemisphere. These differences could partly be attributed to dispersive effects arising from the stenosis when dispersion corrected CPI was used. It is thus beneficial to correct for dispersion in perfusion analysis using this method. Magn C 2014 Wiley Periodicals, Inc. Reson Med 000:000–000, 2014. V Key words: deconvolution; control point interpolation method; residue function; arterial input function; dispersion; Bayesian Analysis

INTRODUCTION Perfusion MRI is widely used in neurological diagnostic imaging for ischemia, stroke and brain tumor assessment (1,2). Cerebral blood flow (CBF) estimation using 1 Institute of Biomedical Engineering, University of Oxford, Oxford, United Kingdom. 2 Florey Institute of Neuroscience and Mental Health, Heidelberg, Victoria, Australia. 3 Department of Medicine, Austin Health and Northern Health, University of Melbourne, Melbourne, Victoria, Australia. 4 Medical Biophysics, Sunnybrook Research Institute, University of Toronto, Toronto, Canada

*Correspondence to: Amit Mehndiratta, M.B.B.S., M.M.S.T., D.Phil., Institute of Biomedical Engineering, Old Road Campus Research Building, University of Oxford, Headington, Oxford, OX3 7DQ, United Kingdom. E-mail: [email protected] Additional Supporting Information may be found in the online version of this article. Received 9 September 2013; revised 16 October 2013; accepted 4 November 2013 DOI 10.1002/mrm.25077 Published online 00 Month 2014 in Wiley Online Library (wileyonlinelibrary. com). C 2014 Wiley Periodicals, Inc. V

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the oscillatory nature of the resultant residue function. However, the recently introduced control point interpolation method has been able to demonstrate variations in the residue function in vivo (26–29). Bolus delay has long been recognized as a source of systematic error in CBF quantification and estimation of the shape of residue function and deconvolution strategies have been designed to be delay insensitive (5,10,26,30– 34). Dispersion causes a smearing of the AIF function in time, altering the bolus profile and affecting the features of AIF before its arrival at the tissue voxel (14,19,35). If the temporal spread is not considered in the perfusion analysis modeling, these distortions can lead to considerable errors in the quantification of the perfusion parameters (19,36), which can have serious implications for diagnosis and management of patients with cerebral ischemia. Accurate estimation of residue function is also likely to be effected by macrovascular dispersion and thus some form of correction or accommodation will be required if information on capillary hemodynamics from the estimation of the residue function could be used routinely. Of the various deconvolution algorithms that have been proposed in the past 15 years (5,10,26,30–32), none have fully addressed dispersion. In the current study, we thus examine the accuracy of two alternative proposed deconvolution methods (vascular model [VM] and control point interpolation [CPI] method) (26,31) in the presence of dispersion. Additionally, we extend both methods to explicitly correct for dispersion. We investigated the effects of dispersion in perfusion quantification on two levels: (i) in terms of CBF accuracy, and (ii) the ability to isolate dispersion effects from the residue function and thus to be able to derive quantitative measures of microvascular hemodynamics.

[2]

where Ca(t) is now the measured AIF. The residue function is defined as a monotonically decaying function with an initial value of one, R(0) 5 1. Once dispersion is included in the perfusion model, the sharp AIF profile is now “smeared out” over time by the VTF. Because convolution is associative, the VTF can instead be convolved with the residue function (keeping AIF unaltered) to examine the effects of dispersion on the residue function as would be estimated by perfect deconvolution. In such a scenario, the measured residue function may no longer start from one (i.e., R(0) 6¼1) (14,19). VTF Model The VTF depends on several factors, such as vascular topology, tissue type, site of AIF measurement, cerebral blood volume (CBV) and the pathological condition of the vessels (vessel lumen, elasticity, stenosis). Depending on these pathophysiological conditions, an assumed VTF kernel functional form could be used; for example, a gamma dispersion kernel (GDK) has been considered as a reasonable approximation to model the effects of dispersion (38): VTFðtÞ ¼

s1þsp  t sp  e-st Gð1 þ spÞ

[3]

where, s characterizes the “sharpness” of the kernel and p is the time-to-peak (ttp). When p ¼ 0 and as s ! 1 this kernel approximates a delta function resulting in zero dispersion; whereas with larger value of p and smaller value of s the kernel will be associated with a greater amount of dispersion (38). METHODS

Theory of Bolus Dispersion in Perfusion Imaging: Modeling and Analysis

Deconvolution Methods

According to indicator dilution theory, the CTC (C(t)) is the result of a convolution operation between an AIF and the tissue residue function (6–8): CðtÞ¼a  CBF  ðCaðtÞ  RðtÞÞ

CðtÞ¼a  CBF  ðCaðtÞ  VTFðtÞ  RðtÞÞ

[1]

where  represents convolution, Ca(t) the AIF, and R(t) the residue function, (describing the fraction of the bolus remaining in the tissue at time t, following the arrival of an ideal instantaneous bolus). The proportionality constant a is the measure of brain tissue density and difference in hematocrit between capillaries and large vessels (compensating for the fact that only plasma volume is accessible to contrast agent) (37). The residue function scaled with CBF is referred to as the tissue response function (TRF (t) 5 CBFR(t)). Bolus dispersion affects the AIF, by altering its shape and amplitude while in transit between the measured AIF location and the tissue. This can be reconciled by means of dispersion parameters in the DSC model: for example, through the inclusion of a vascular transport function (VTF) in convolution with the AIF (19). The VTF thus encapsulates the effects of dispersion in a simple mathematical function, giving a modified C(t):

The block-circulant SVD (oSVD) (10), VM (31), and CPI methods (26) were used in this study. Both the VM (31) and CPI (26) in their original implementation, were designed to model the residue function as per its fundamental definition with (R(0) ¼ 1). VM models the transit time distribution of the capillary hemodynamics with a gamma probability density function; thus calculating the residue function by estimating the shape parameter of the transit time distribution (31). The CPI method models the residue function with few control points, each having two degrees of freedom both in amplitude and in time (26). The complete shape of the residue function being calculated by performing a piecewise cubic spline interpolation between the control points (26). Both VM and CPI were implemented with a Bayesian optimization algorithm (26,31,39). To explicitly accommodate dispersion, both methods were modified, as described below. Dispersion Modification for VM A gamma dispersion kernel was implemented in conjunction with VM, referred to here as VMþVTF. The VM has three parameters, CBF, d and l, where d is the bolus arrival delay and l is the shape parameter of the gamma

Dispersion Correction with CPI in DSC-MRI

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FIG. 1. (a) The family of residue function shapes that could be achieved with VM; (b) corresponding family of dispersed residue function shapes which could be achieved with VMþVTF, GDK with p ¼ 2 and s¼0.7. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

probability density function to describes the transit times through the voxel (31). VMþVTF has five free parameters with the addition of s and p to describe the gamma dispersion kernel. Figure 1 shows the versatility available in VM with a range of l values (0.5–100) to represent various residue functions, and the corresponding versatility in dispersed residue function shapes for VMþVTF with a typical gamma dispersion kernel. Dispersion Modification for CPI In the original CPI, the residue function is estimated from a set of control points (CP) that form the basis of a smooth piecewise cubic spline interpolation (26). Each CP is allowed to vary in both amplitude and time, except for the first CP, which is set to 1 (CP1 ¼ 1). This CP1 restriction therefore does not permit assessment of dispersion (40). Two modifications were considered to accommodate dispersion in the CPI method. Figure 2 shows two implementations of CPI with dispersion correction: (i) CPI0: CP1 fixed at 0, (ii) CPIþVTF: CPI method implemented with the VTF. CPIþVTF attempts to characterize both the true underlying residue function (by means of the CPI residue function) and the dispersion effects (by means of the VTF), whereas, CPI0 variant retains the “model-free” nature of the original CPI because no assumptions are made for residue function shape or for dispersion kernel. The trade-off is that it does not allow to separate capillary hemodynamics from dispersion; thus neither the true residue function nor the dispersion kernel can be estimated separately. Implementation The delay insensitive variant of SVD, oSVD (blocked circulant SVD), was implemented as in Wu et al (10). The VM was implemented with priors supplied from the oSVD solution as described in (31), model-fitting being performed using a Bayesian nonlinear model fitting algorithm (39) which is broadly similar to that used by (31) (see dis-

cussion in Chappell et al for more details) (39). The CPI was implemented using the same Bayesian nonlinear model fitting algorithm (39) using priors as elaborated in Mehndiratta et al (26). The six deconvolution methods (oSVD, VM, CPI, VMþVTF, CPI0, and CPIþVTF) were implemented in MATLAB using software written in-house. Note that four of the six methods were in principle able to model a dispersed residue function oSVD, VMþVTF, CPI0, and CPIþVTF. The Bayesian nonlinear model fitting algorithm used for optimization also required prior information for the VTF. Priors for the VTF were provided in the form of a Gaussian distribution with high probability for low-tomedium level of dispersion (based on empirical preliminary analysis, data not shown); priors used on p and s were log(2) 6 2 (mean 6 SD) and the same values were used for initialization of the optimization routine. Parameters p and s were estimated on a logarithmic scale, because this makes the model more linear with respect to these parameters, aiding the convergence of the fitting algorithm, which depends upon a local approximation to the function using a Taylor expansion (39). Simulations. An initial set of simulations was performed with no dispersion. Next, a range of values of bolus dispersion and delay were introduced. In all cases, accuracy was calculated as the ability to estimate CBF and to separate the effects of dispersion from the simulated tissue hemodynamics (residue function). The AIF was simulated using a gamma-variate function (3,10,26,31). Simulations were performed with CBV ¼ 4 mL/100 g, representative of normal gray matter (41), and CBF in the range 10–60 mL/100 g/min, in increments of 10 mL/100 g/min. Thus, mean transit time (MTT) varied from 4 s to 24 s. A well-known model of the vasculature bed as a single well-mixed compartment was assumed to model tissue hemodynamics (3): the residue function was

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FIG. 2. (left) CPI methods as elaborated in (26); (right) top: dispersion variants of CPI method, CP1 fixed to zero (CP1 ¼ 0), bottom: CPIþVTF variant of CPI accounting for dispersion with GDK as a function of time-to-peak (p), sharpness (s), and time (t) as VTF. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

hence modeled as an exponential decay function. The Gamma VTF (GDK) was used to model dispersion (38,40); a range of low to high dispersion values were evaluated using the parameters shown in Table 1. Figure 3 shows the effect of low-high dispersion (GDK) on MR signal for a typical gray matter voxel (CBF ¼ 60 mL/100 g/min). The increase in the level of dispersion is associated with an increase in the time-to-peak, decrease in the signal drop, and broadening of the tissue signal; the area under the corresponding CTC remains the same, as was also illustrated by Calamante (40). To evaluate sensitivity of the methods with variation in microvascular residue function, in addition to exponential residue function, linear (26) and Lorentzian (42) residue function were also simulated at low level of dispersion with GDK. To evaluate the sensitivity of the methods with variation in macrovascular dispersion, in addition to GDK further simulations were performed with two more dispersion kernels: exponential dispersion kernel (EDK) (19) and lognormal dispersion kernel (LNDK) (16). t 1 EDKðtÞ ¼ e u u

[4]

where u represents a time constant for the exponential dispersion kernel, such that larger the value of u, the Table 1 Gamma Dispersion Kernel (GDK) Coefficients Used in Simulation for Low–High Dispersion in the MR Signal Level of dispersion

p

s

Low (L) Medium (M) High (H)

1 3 5

2 1 0.5

larger the dispersion and in the limit u ! 0, VTF(t) tends toward the Dirac delta function with no dispersion (19). Value of 1 s, 2 s, and 4 s were used for u to model low, medium, and high level of dispersion. ðln ðtÞ  mÞ2 1 2s2 LNDKðtÞ ¼ pffiffiffiffiffiffi e ts 2p

[5]

where, s and m are the shape and location parameter of the log-normal function, respectively. As suggested by Calamante (16), the low level of dispersion was simulated with s ¼ 1 s and m ¼ 1, medium level with s ¼ 0.75 s, m ¼ 0.15 and high level with s ¼ 0.78 s and m ¼ 0.59. CTCs were generated as described in (3,10). A linear relationship was assumed between tissue relaxivity (R2*) and C(t) (4). The CTC was converted to signal curves S(t) with S(t0) at arbitrary set to 100 and TR/TE ¼ 1 s/65 ms. The linear proportionality constant was chosen such that 40% peak signal drop was achieved at a flow of 60 mL/100 g/min with CBV ¼ 4 mL/100 g (10). The AIF signal curve was generated similarly, but with a constant that produced a 60% peak signal drop. Zero mean Gaussian noise was added to the signal curves to produce a baseline SNR of 50. Values of 0–5 s were used to evaluate the models for sensitivity with respect to bolus arrival delay (d) and its effect when combined with dispersion. The simulations were thus performed under three conditions: I) no dispersion and no delay; II) dispersion with no delay: low, medium, and high dispersion; III) dispersion with delay: low, medium, and high dispersion and delay (d ¼ 0–5 s) For each combination of CBF, dispersion kernel, dispersion coefficients and delay a total of 100 CTC were generated. Thus a total of 43,200 (6  3  4  6  100) concentration curves were analyzed during each simulation process with six deconvolution methods: (a) delay

Dispersion Correction with CPI in DSC-MRI

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FIG. 3. The effect of low, medium, and high level of dispersion on MR signal (a) and CTC (b) from typical gray matter (CBV ¼ 4 mL/100 g, CBF¼60 mL/100 g/min) using GDK. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

insensitive SVD (oSVD) (10); (b) vascular model (VM) (31); (c) VMþVTF; (d) CPI (26); (e) CPI0; and f) CPIþVTF. In Vivo Clinical Data For empirical evaluation, the various methods were retrospectively applied to DSC data acquired on a Siemens 3T Trio using single-shot gradient-echo EPI: TR/TE ¼ 1.5 s/30 ms, 78 volumes, 128  128  22 matrix, 1.7  1.7  5 mm3 voxels, flip angle ¼ 70 , generalized autocalibrating partially parallel acquisition (GRAPPA) with acceleration factor 2. An intravenous bolus injection of 0.1 R was performed at injection rate of mmol/kg MagnevistV 10 mL/s followed by a 20 mL saline flush. DSC perfusion data were acquired from one healthy participant (age ¼ 40 years, male) and two patients with a history of atherosclerotic disease. Both of the patients had bilateral internal carotid artery (ICA) stenosis: Patient I: 65 years female, left ICA ¼ 90% occlusion, right ICA ¼ 40% occlusion and Patient II: 76 years male, left ICA ¼ 75% occlusion, right ICA ¼ 20% occlusion. Both the patients underwent left carotid endarterectomy (CEA) and DSCMRI data were acquired both pre- and post-CEA. Images were acquired under an Institutional Review Board approved protocol for DSC-MRI study as part of a larger study (43,44). Diffusion-weighted imaging (DWI) (TR/ TE ¼ 4.4 s/93 ms, b-values ¼ 0, 1000 s/mm2, 27slices, 1.6  1.6  3 mm3 voxels) was also acquired along with the DSC data. The AIF for the in vivo data was selected using an automated algorithm written in-house, from a 30-mm slab centered at the Circle of Willis. CTC features including the first moment (FM), peak height (Pmax), time-topeak (ttp), and bolus arrival time (Td) were extracted and, using a K-means clustering algorithm, voxels were grouped into five classes (grey matter, white matter, vessel, CSF, and background). The AIF was defined in the vessel subclass with highest Pmax and lowest FM, and a venous output function (VOF) in the same class but with

highest Pmax and greatest FM. The AIF was then scaled such that the mean value after the first passage of bolus (the tail of AIF) was equal to the tail of the VOF using the tail scaling technique (11). The time course of the “tail” was made up of the last 39 volumes of the total 78 volumes (volumes 40 to 78) including the recirculation peak. Analysis of Simulated Data The ratio of CBF (CBFratio) as estimated CBF / true CBF was calculated to quantify the absolute error in CBF estimation. rCBFratio (relative CBFratio) was also calculated by normalizing the estimated CBF in presence of dispersion against the estimated CBF with no dispersion. Calculation of rCBFratio simulates a clinical scenario where relative CBF is often measured by normalizing the whole brain with the CBF value in a contralateral region of interest (ROI) (where no or minimal dispersion might be expected). The accuracy of residue function estimation for each method was evaluated by calculating the sum of squared errors (SSE) between the estimated and true residue function shapes. Statistical significance at p < 0.05 was calculated using the Student paired t-test for comparison among methods. Analysis of In Vivo Data In addition to CBF, MTT, and CBV parametric maps, the residue functions obtained from deconvolution methods were also analyzed for spatial variations in shape both pre- and post-CEA. For visualization purposes, residue functions from two ROIs (2  2  1 voxels), one in infarcted and another in normal tissue, will be shown. Residue function is a monotonic decreasing function; thus, the time taken for the residue function to drop to 10% of its maximum value was calculated and referred to as R10 maps. Larger values of R10 would represent more time taken by the residue function to decrease, which might be an indication of tissue under hemodynamic stress (27,45). To see the spatial variations in

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dispersion, p and S (S ¼ 1/s) parametric maps were also analyzed; larger values of p and S are likely to be associated with a higher amount of dispersion. Time to peak from the residue function from CPI0 was also recorded and treated in the same way as the parameter, p in CPIþVTF as a measure of dispersion. RESULTS Simulations Figure 4a shows the CBFratio (Estimated/True CBF) for the six deconvolution methods for no, low, medium, and high levels of simulated dispersion. Figure 4b shows the rCBFratio (Estimated CBF in Dispersion/No Dispersion) for low, medium, and high level of simulated dispersion. The results in Figure 4 represent the mean (6s) CBFratio across simulated CBF values. A nonlinear relationship was observed between estimated and true CBF values particularly at high values (not shown), as has been observed previously (10,26,31). No Dispersion and No Delay CPI (CBFratio ¼ 1.00 6 0.12) was the most accurate in case of no dispersion, followed by VM (CBFratio ¼ 0.95 6 0.11),

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which significantly underestimated CBF (p < 0.05). CPIþVTF (CBFratio ¼ 1.1 6 0.14) significantly overestimated CBF in case of no dispersion; whereas CPI0 and VMþVTF overestimated CBF by approximately 1.6–2 times. Table 2 shows the SSE in estimation of residue function; the methods having the lowest SSE are marked in bold where (b) indicates statistical significance (p < 0.05) and c indicates methods with statistically significant lower SSE within the groups but which could not be differentiated among themselves. In the case of no dispersion and no delay CPI had the lowest error, but the error with the CPI, VM, and CPIþVTF methods showed no significant difference.

Dispersion with No Delay As expected the value of both CBFratio and rCBFratio decreased when dispersion levels went from low to high (Fig. 4). All the methods underestimated CBF in the presence of dispersion with poorer performance at the highest level of dispersion (Fig. 4). CPIþVTF was most accurate for absolute CBF estimation with a small standard deviation at low (0.99 6 0.13) and medium (0.81 6 0.13) dispersion values. CPI0 tended to overestimate CBF at low and medium levels of dispersion

FIG. 4. CBFratio (estimate/true CBF) (a) and rCBFratio (estimated CBF with dispersion/no dispersion) (b) with six deconvolution methods using GDK for simulations at no delay; black line representing line of unity. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Table 2 Sum of Squared Error (SSE) for Simulated and Estimated Residue Function with oSVD, VM, VMþVTF, CPI, CPIþVTF, and CPI0 at No, Low, Medium, and High Dispersion with and without Delaya Dispersion

oSVD

VM

No Low Medium High

0.9160.28 1.9160.35 4.2860.51 7.2561.00

0.1760.16c 0.2460.20 0.8360.42 2.6960.72

Low Medium High

4.7760.49 6.8160.82 9.1861.18

0.2660.24 0.8560.42 2.7160.07

VM þ VTF No delay 1.1360.90 0.8360.89 0.6360.47 2.3360.74b With delay 0.3260.46 0.5860.42c 2.3660.75b

CPI

CPI þ VTF

CPI0

0.1460.18c 0.2260.34 1.1961.26 6.1963.83

0.1660.16c 0.1660.17b 0.4760.62b 3.1862.39

2.0460.12 2.0860.14 2.2460.18 2.7560.23

0.5660.66 1.9561.38 3.3262.38

0.1660.21b 0.5860.79c 3.0662.36

2.0960.14 2.360.17 2.8260.22

a

Lowest SSE are marked in boldface type. The method with statistically significant lowest SSE (p < 0.05). c Methods with statistically significant lowest SSE (p < 0.05) but are no better than each other. b

but was most accurate compared with other methods at the highest level (1.04 6 0.06) of dispersion. VMþVTF was found to overestimate CBF at a low dispersion level by 1.6 times and to underestimate at medium and high dispersion levels. VMþVTF also had the highest level of uncertainty in CBF estimation among all other methods (VMþVTF: Low ¼ 1.57 6 0.87, Med ¼ 0.94 6 0.50, High ¼ 0.60 6 0.27). rCBFratio calculations (Fig. 4b) showed that at low level of dispersion all the methods were very similar in rCBF estimation, highest rCBF observed with VMþVTF (0.94 6 0.71), which also had the highest level of uncertainty (large standard deviation). oSVD, CPI0 and CPIþVTF had the next best rCBFratio (with reasonable level of uncertainty) and were very similar in rCBF estimation at low level of dispersion, rCBFratio ¼ 0.91 6 0.11, 0.91 6 0.11 and 0.91 6 0.17, respectively. At medium and high levels of dispersion, oSVD had the highest accuracy with rCBFratio ¼ 0.82 6 0.11 and 0.67 6 0.09 respectively, closely followed by CPI0 at rCBFratio ¼ 0.80 6 0.08 and 0.63 6 0.07, VM at rCBFratio ¼ 0.76 6 0.12 and 0.57 6 0.08 and CPIþVTF at rCBFratio ¼ 0.75 6 0.15 and 0.49 6 0.11, respectively.

Figure 5 shows the residue functions for typical gray matter (CBF ¼ 60 mL/100 g/min), together with the mean of the estimated residue functions over the 100 realizations from oSVD, VM, CPI, and with dispersion corrected methods. Figure 5a shows that in the presence of dispersion VM and CPI tend to estimate a residue function that is broader and that decays more slowly than the true residue function. CPIþVTF could estimate the residue function more accurately as illustrated in Figure 5b. Table 2 shows the sum of squared errors in the estimation of residue function shape with a low, medium, and high degree of simulated dispersion. At low and medium dispersion levels, CPIþVTF had the lowest error in residue function shape estimation, significantly lower than other methods. The accuracy in estimation of residue function decreases with an increase in the level of dispersion for all methods. At the highest level of dispersion, all the methods were found to perform poorly, among which VMþVTF showed the lowest error.

FIG. 5. Residue function simulated for typical gray matter (CBF ¼ 60 mL/100 g/min) with GDK (p ¼ 1, s ¼ 2) and delay ¼0 s and mean fit achieved with oSVD, VM, CPI (a); VMþVTF, CPI0, CPIþVTF (b). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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FIG. 6. Error in estimation of delay, p (one of the GDK parameters), and sum of the two for CPIþVTF (a) and VMþVTF (b) method, with simulated delay of 1–5 s. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Dispersion with Delay The methods that attempt to account for dispersion were found to perform better in the case of dispersion combined with delay (shown in Supp. Table S1, which is available online). Introduction of delay together with dispersion led to an impaired performance by both VM and CPI, which was found to exacerbate as dispersion and/or delay was increased. CPIþVTF was found to be the most accurate in CBF estimation at low (CBFratio ¼ 0.99 6 0.13) and medium (CBFratio ¼ 0.80 6 0.14) levels of dispersion; whereas CPI0 was more accurate at the highest (CBFratio ¼ 1.03 6 0.06) level of dispersion simulated. oSVD always underestimated the CBF (CBFratio ¼ 0.78 6 0.08 to

0.58 6 0.5) and VMþVTF overestimated CBF at low dispersion (CBFratio ¼ 1.30 6 0.69) and tended to underestimate at medium and high (CBFratio ¼ 0.85 6 0.35 and 0.56 6 0.05, respectively) dispersion levels. At the medium dispersion levels, the mean CBFratio for VMþVTF was found to be close to CPIþVTF, but VMþVTF had three times higher variability than CPIþVTF (Supp. Table S1). Table 2 shows the sum of squared errors from the estimation of residue function at low, medium, and high degrees of dispersion along with delay. At low dispersion, CPIþVTF had the significantly lowest SSE (p < 0.05) but at the medium level of dispersion CPIþVTF and VMþVTF both performed similar in residue function shape estimation.

FIG. 7. MTT maps: CPI (a) and CPIþVTF (b), showing ROI selected in healthy and in DWI positive ischemic tissue. (c) CPI and CPIþVTF shows similar residue function in contralateral ROI with no dispersion. (d) Infarcted ROI shows broader and slower decay in residue function with CPI compared with CPIþVTF. CPIþVTF also showed slower decay than ROI in healthy ROI as in c. Larger amount of dispersion was observed in ischemic ROI than normal ROI as seen with much broader residue function with CPI0. Data from a 65 years old female with ICA stenosis (left ¼ 90%, right ¼ 40%) scheduled for left carotid endarterectomy. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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FIG. 8. DWI (a,e), R10 (b,c,f,g), S (dispersion) (d,h) maps for CPI (b,f) and CPIþVTF (c,d,g,h) showing area of infarction (from DWI) with higher values of R10 with both CPI and CPIþVTF. (b) R10 with CPI also showed higher values in left MCA territory which improved post-CEA (f). (c) R10 with CPIþVTF showed much reduced values in left MCA territory where high amount of dispersion was found (marked with arrow in d). Dispersion was found to disappear post-CEA (h). Data from a 65 years old female with ICA stenosis (left¼ 90%, right ¼ 40%). Data shown both pre- and post-left carotid endarterectomy.

Figure 6 shows the error in the estimation of delay and the time-to-peak of the gamma vascular transport kernel (p) with CPIþVTF and VMþVTF. With CPIþVTF, delay was underestimated for low dispersion and overestimated with medium and high dispersion. For VMþVTF delay was overestimated and p underestimated. However, for both deconvolution methods, the sum of delay and p (i.e., delayþp) had consistently smaller error at all dispersion values. Supplementary Table S2 shows the sum of squared error in estimation of linear and Lorentzian residue functions. For both linear and Lorentzian function CPI and CPIþVTF, both methods had the lowest error but no better than each other. Supplementary Table S3 shows sum of squared error in estimation of the true residue function for simulated EDK and LNDK at low, medium, and high dispersion values. The CPIþVTF consistently showed significantly lowest error compared with alternative methods. In Vivo Data The oSVD estimates of CBF were the lowest among all methods; in agreement with the simulation study, VMþVTF consistently estimated the highest CBF within any given voxel (data not shown). MTT values estimated by oSVD were observed to be higher than other methods; however, a larger variation in MTT was observed with other methods. Figure 7 shows the MTT estimates from a representative patient calculated with CPI and CPIþVTF; higher values of MTT being observed in the left parietal white matter region. The DWI confirmed a region of ischemic tissue in parietal white matter (shown in Fig. 8). CPI and CPIþVTF also showed higher values of MTT in parts of the left MCA territory surrounding DWI positive ischemic tissue. Figure 7 also shows the residue functions from within normal and infarcted tissue from one patient. The residue function with CPI (Fig. 7d) was broader and decayed more slowly in infarcted ROI compared with healthy ROI (Fig. 7c) (as was also observed during simulations for dispersion in Fig. 5). The residue

function from CPIþVTF showed a faster decay compared with CPI in the infarcted ROI (Fig. 7d), but was still broader in infarcted ROI (Fig. 7d) compared with the normal ROI (Fig. 7c). Residue function with CPI0, dispersed residue function, also showed broadening (higher dispersion) in infarcted ROI (Fig. 7d) compared with the normal ROI (Fig. 7c). R10 maps for CPI and CPIþVTF, and a map of the sharpness parameter (S) from the gamma VTF for CPIþVTF are shown in Figure 8 for the patient before and after CEA. The CPI method showed the area of infarction with very high R10 values (Fig. 8b), along with increased values elsewhere in the left hemisphere mainly in the MCA territory. CPIþVTF showed similar areas of higher values in R10 (Fig. 8c) but values in left MCA territory were much reduced compared with the higher values seen in CPI results. The dispersion coefficient (S) indicated a higher amount of dispersion in left MCA territory (arrow in Fig. 8d). Post-CEA perfusion analysis showed decrease in R10 across the brain for both CPI (Fig. 8f) and CPIþVTF (Fig. 8g) and S (Fig. 8h) with CPIþVTF methods. Figure 9 shows the delay and p estimation from the patient, the sum of delay and p with CPI0 and CPIþVTF were found to closely resemble the delay estimated by the CPI method, as will be described below. DISCUSSION Bolus dispersion is challenging to quantify in DSC-MRI, and more problematic in clinical population such as cerebrovascular patients (19,40,46). In the current work, we developed a means to assess dispersion and how it contributes to error in: (i) the CBF estimation and (i) the tissue residue function. We found that in presence of dispersion the accuracy in CBF estimation is considerably improved with dispersion corrected CPI methods and also allowed to separate tissue residue function from effects of dispersion with greater accuracy at least at low

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FIG. 9. (top) Estimated delay with CPI, CPI0 and CPIþVTF; (middle) p and (bottom) delayþp. Delay estimate with CPI appears similar to the delayþ p estimate with other two methods, highlighting the ambiguity in delay and p estimation during the deconvolution process. Data from a 76 years old male with ICA stenosis (left ¼ 75%, right¼ 20%) scheduled for left carotid endarterectomy. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

to medium level of dispersion. CPIþVTF method corrected the residue function for dispersion and showed variation in tissue residue function in in vivo data from patients with atherosclerotic diseases. Calamante (40) showed that dispersion reduces the accuracy in CBF estimation when using the standard SVD methods by as much as 70% (46). Similar findings were observed in this study, where oSVD was found to underestimate absolute CBF by a large amount (CBFratio ¼ 0.57– 0.77). However, modified methods designed to specifically handle dispersion (VMþVTF, CPIþVTF and CPI0) were more accurate in absolute CBF estimation (CBFratio¼0.94– 1.04). CPIþVTF was most accurate in CBF estimation at low to medium levels of dispersion whereas CPI0 was better at high level of dispersion. However, due to the practical difficulties with quantifying CBF in absolute units (16), relative CBF measurements are often used in the clinical evaluation of patients with vascular abnormalities, by making a comparison of the CBF in a region in the contralateral hemisphere. For our simulations, relative CBF measurements were best performed by oSVD (rCBFratio¼0.67–0.91) over a range of simulated dispersion; which was followed closely by CPI0 (with rCBFratio¼0.63–0.91). However, the better performance of oSVD for rCBF quantification is primarily a result of serendipitous underestimation of CBF both in the presence and absence of dispersion. It is clear from the other results that oSVD does not inherently handle dispersion well and is substantially affected when it comes to absolute quantification, leading to a poor characterization of the shape of the true residue function. For rCBF calculation, it is therefore apparent that methods that explicitly correct for dispersion do not necessarily offer a direct advantage, and their major role is when the shape of the residue function is of interest or when absolute quantifi-

cation of CBF is required. Moreover, when dispersion in the patients under study is only expected to be low-tomedium, the results from Figure 4 suggest that quantification error are actually comparatively smallest for the case of absolute quantification with CPIþVTF. Given that various methods to scale DSC-MRI measurements to absolute units have recently been introduced (47–49), absolute quantification may become more widespread in clinical studies, increasing the benefits of deconvolution methods that account for dispersion. It is already known that the broadening in the DSC time course caused by dispersion might be interpreted as an increase in the MTT if the analysis does not account for dispersion (19). This source of error propagates to the CBF estimation, leading to underestimation. In this case, the tissue response function would appear to be broader and to decay more slowly (an effect of apparent increase in MTT) compared with the true response function. Such effects were observed in our study when the analysis was performed with the original implementations of VM and CPI that cannot accommodate dispersion (Fig. 5a), with the estimated residue functions having slower decay when compared with those simulated. CPIþVTF was found to perform better in such cases (Fig. 5b), as the dispersion can now be accommodated (to some extent) in the estimated VTF. For the in vivo data, the residue function estimated by CPI in the infarcted ROI was found to be broader and with a slower decay (Fig. 7) for the patient with atherosclerotic disease compared with a healthy scenario. This was also seen in the R10 map (Fig. 8). The residue function when assessed with CPIþVTF in the same infarcted ROI decayed faster than the CPI estimate; this would suggest the presence of dispersion, as suggested by the simulation study. Higher values of S (dispersion

Dispersion Correction with CPI in DSC-MRI

coefficient) in the left MCA territory (Fig. 8) with CPIþVTF were further evidence of dispersion in this region. The CPIþVTF results suggest that, while there was a contribution from dispersion, there was still some variation in residue function associated with the infarcted tissue, as could be clearly seen by a broader residue function with CPIþVTF in an infarcted ROI compared with a more rapid decay in the normal ROI (Fig. 7c,d). This is reasonable because changes in the microvasculature and thus the residue function would be expected in infarcted tissue. The wider region of a slower residue function decay observed in the left MCA territory with CPI was absent from the results obtained with CPIþVTF, suggesting that there were no substantial observable changes in the microvasculature in this region, but that the effect was primarily due to dispersive effects (Fig. 8). It was also later observed in the post carotid endarterectomy perfusion analysis (Fig. 8f–h), where these dispersive effects were not observed with either of the methods (CPI and CPIþVTF). For in vivo data, neither the actual residue function shape nor the effects of dispersion (VTF) are known a priori, particularly in pathological conditions. It is challenging to completely separate the effects of macrovascular dispersion from the microvascular residue function because it is only their combined effect that can be observed in the final measured voxel-wise time course. To study the variation in residue function in pathology we took the approach of fixing the dispersion contribution by a gamma dispersion kernel. Several other functions have been used in the literature to model dispersion (14,16,19,36,40,46), the choice of gamma here was based on recent work characterizing dispersion using arterial spin labeling (38). An exponential dispersion kernel has previously been used for DSC (19,38); it represents a special case from the family of gamma dispersion kernels, with p 5 0. To test the sensitivity of the proposed method, with the gamma kernel, to variations in the underlying dispersion process as well as variations in the residue function, further simulations were performed with multiple shapes for the residue function (exponential, linear, Lorentzian) under different conditions of dispersion (low, medium, and high) with three dispersion kernels (GDK, EDK, and LNDK). The results in Table 2 and Supplementary Tables 2–3 show that CPIþVTF was reasonably able to accommodate variation in both microvascular residue function and macrovascular bolus dispersion. As might be expected mismatch between the “true” dispersion process and the VTF used for analysis will result in errors in the residue function, but these were seen to be small, possibly reflecting the flexibility afforded by the choice of a gamma kernel. The results also demonstrated the considerable improvement in estimation of residue function shape even in presence of dispersion (at low to medium level) using the CPIþVTF method compared with alternatives. The CPIþVTF approach might thus can provides a method to avoid the strict model-based assumptions for the residue function with a plausible approximation of the dispersion kernel while still offering a smooth interpretable residue function estimate. Using CPIþVTF a better estimation of absolute CBF was achieved in the

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presence of dispersion in simulations. CPIþVTF was also found to isolate the effects of dispersion and to estimate cerebral hemodynamics with a reasonable accuracy at low and medium levels of dispersion, which represents the most likely clinical scenario. All methods performed poorly at extreme levels of high dispersion, suggesting that this may well present a limit to what can be accommodated in perfusion analysis. To account for dispersion effects which cannot be parameterized, a nonparametric deconvolution method accommodating dispersion has also been proposed in this study in the form of CPI0. CPI0 performed within acceptable limits in CBF estimation but the residue function and dispersion remained inseparable. The estimated residue function has the combined information of the residue function and dispersion similar to the SVD solution, although with the benefit of no oscillations. Whereas the sum of delay and p (from the VTF) can be estimated with a reasonable degree of accuracy, the errors in the two parameters are highly correlated. This is not surprising because the effects of both delay and dispersion on the DSC signal are similar, both causing a temporal shift in the peak signal drop. This can be appreciated from the curves shown in Figure 3: while some of these curves appear to be delayed, they were simulated with dispersion but no delay. In reality, when combined with a VTF the delay parameter only characterizes the delay for the leading edge of the input function to arrive, whereas p effectively characterizes the delay introduced to the bulk of the input function by dispersion. Thus, in previous work with delay correction (10,26,31), where the effects of dispersion have been ignored, it is likely that the estimates of delay accounted (to some extent) for both effects. This means that the values used in simulation here are liable to be more extreme than reality. The ambiguity between delay and dispersion was not found to have a substantial impact on the CBF or residue function shape estimation for CPIþVTF; although it did impact the accuracy of estimation of the other dispersion parameter, s. In practice, a relationship between delay and the extent of dispersion might be expected (50), because both should increase with distance from AIF site to tissue voxel. Such a relationship could, in principle, help constrain the problem further, possibly increasing the robustness of methods to correct for dispersion effects. However, the exact relationship, if such does exist, remains to be determined. It might be expected that the CBF estimation could be improved, and the residue function and effects of dispersion separated, if a model-based approach for tissue vasculature is used, parameterizing both the residue function and effects of dispersion. However, the results of our simulations suggest that a model-based deconvolution (VMþVTF) cannot clearly distinguish the two, leading to errors in CBF estimation and a significant amount of uncertainty in both CBF and the residue function shape. It was observed that variations in the signal time course from the VMþVTF model were similar for changes either in the dispersion parameters or in the parameters of the VM transit time distribution, most probably because both are defined by gamma distributions. Thus, there would be substantial ambiguity when estimating either set of parameters.

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A metric based on the time for the residue function to decay to 10%, R10, has been used here to characterize the residue function in vivo. This is a recognition of the fact that, although it might be able to estimate the residue function, its interpretation still needs to be fully explored. R10 in particular measures changes in the tail of the residue function and is thus associated with long transit times through the voxel. Thus it may capture information beyond that already found in the MTT parameter. The clinical significance of this parameter is beyond the scope of this work. An alternative to dispersion correction is the use of methods that derive local AIFs (15,51–58), these would bypass the errors introduced using analytical dispersion kernels. However, they suffer from issues due to partial volume effects (12,13) and these are difficult to reduce. Additionally, it still remains to be validated whether these methods can reliably measure the local AIF. Both the CPI and its dispersion variant had a higher computational time over the oSVD and VM methods for the in vivo analysis (oSVD 3 min, VM and VMþVTF 15–20 min each, CPI and CPIþVTF 60 min for a 128  128 image). The higher number of parameters to be estimated in the CPI/CPIþVTF methods were the main source of increase in the computational time over oSVD and VM/VMþVTF. However, these values represent nonoptimized code and the methods (CPI/CPIþVTF) would be highly suitable for parallel processing and implementation in C/Cþþ. In summary, we describe a method to improve the accuracy of absolute CBF estimates and residue function when dispersion is present. The method developed does not fully resolve the ambiguity between the true residue function and dispersion; however, incorporating an additional VTF model did show benefits. A reasonable separation of tissue hemodynamics and bolus dispersion can now be achieved with this approach for low to medium levels of dispersion. It remains to be seen what strategies should be used, however, when there is very high levels of dispersion. The modified CPI approach appears to provide a useful tool for the analysis of DSC-MRI data in patients with vascular abnormalities.

ACKNOWLEDGMENTS We thank the Felix Scholarship at University of Oxford in funding this research work and Royal Commission for the Exhibition of 1851 (UK), the National Health and Medical Research Council (NHMRC) of Australia, the Australian Research Council (ARC) and the Victorian State Government for their support. MAC was employed by The Centre of Excellence in Personalized Healthcare funded by the Welcome Trust and EPSRC under Grant number WT088877/Z/09/Z. REFERENCES 1. Essig M. Protocol design for high relaxivity contrast agents in MR imaging of the CNS. Eur Radiol 2006;16(Suppl 7):M3–M7. 2. Essig M, Shiroishi MS, Nguyen TB, et al. Perfusion MRI: the five most frequently asked technical questions. AJR Am J Roentgenol 2012;200:24–34.

Mehndiratta et al. 3. Østergaard L, Weisskoff RM, Chesler DA, Gyldensted C, Rosen BR. High resolution measurement of cerebral blood flow using intravascular tracer bolus passages. Part I. Mathematical approach and statistical analysis. Magn Reson Med 1996;36:715–725. 4. Rosen BR, Belliveau JW, Vevea JM, Brady TJ. Perfusion imaging with NMR contrast agents. Magn Reson Med 1990;14:249–265. 5. Rempp KA, Brix G, Wenz F, Becker CR, Guckel F, Lorenz WJ. Quantification of regional cerebral blood flow and volume with dynamic susceptibility contrast-enhanced MR imaging. Radiology 1994;193: 637–641. 6. Zierler KL. Equations for measuring blood flow by external monitoring of radioisotopes. Circ Res 1965;16:309–321. 7. Zierler KL. Theoretical basis of indicator-dilution methods for measuring flow and volume. Circ Res 1962;10:393–407. 8. Jacquez J. Compartmental analysis in biology and medicine: kinetics of distribution of tracer-labeled materials. Amsterdam: Elsevier; 1972. 9. Østergaard L, Chesler DA, Weisskoff RM, Sorensen AG, Rosen BR. Modeling cerebral blood flow and flow heterogeneity from magnetic resonance residue data. J Cereb Blood Flow Metab 1999;19:690–699. 10. Wu O, Østergaard L, Weisskoff RM, Benner T, Rosen BR, Sorensen AG. Tracer arrival timing-insensitive technique for estimating flow in MR perfusion-weighted imaging using singular value decomposition with a block-circulant deconvolution matrix. Magn Reson Med 2003; 50:164–174. 11. Bjornerud A, Emblem KE. A fully automated method for quantitative cerebral hemodynamic analysis using DSC-MRI. J Cereb Blood Flow Metab 2010;30:1066–1078. 12. Van Osch MJP, van der Grond J, Bakker CJG. Partial volume effects on arterial input functions: shape and amplitude distortions and their correction. Journal of magnetic resonance imaging: J Magn Reson Imaging 2005;22:704–709. 13. Van Osch MJ, Vonken EJ, Bakker CJ, Viergever MA. Correcting partial volume artifacts of the arterial input function in quantitative cerebral perfusion MRI. Magn Reson Med 2001;45:477–485. 14. Calamante F, Gadian DG, Connelly A. Quantification of perfusion using bolus tracking magnetic resonance imaging in stroke: assumptions, limitations, and potential implications for clinical use. Stroke 2002;33:1146–1151. 15. Calamante F, Morup M, Hansen LK. Defining a local arterial input function for perfusion MRI using independent component analysis. Magn Reson Med 2004;52:789–797. 16. Calamante F. Arterial input function in perfusion MRI: a comprehensive review. Prog Nucl Magn Reson Spectrosc 2013;74:1–32. 17. Mouridsen K, Christensen S, Gyldensted L, Ostergaard L. Automatic selection of arterial input function using cluster analysis. Magn Reson Med 2006;55:524–31. 18. Østergaard L, Johannsen P, Host-Poulsen P, Vestergaard-Poulsen P, Asboe H, Gee AD, Hansen SB, Cold GE, Gjedde A, Gyldensted C. Cerebral blood flow measurements by magnetic resonance imaging bolus tracking: comparison with [(15)O]H2O positron emission tomography in humans. J Cereb Blood Flow Metab 1998;18:935–940. 19. Calamante F, Gadian DG, Connelly A. Delay and dispersion effects in dynamic susceptibility contrast MRI: simulations using singular value decomposition. Magn Reson Med 2000;44:466–473. 20. Ostergaard L, Sorensen AG, Chesler DA, Weisskoff RM, Koroshetz WJ, Wu O, Gyldensted C, Rosen BR. Combined diffusion-weighted and perfusion-weighted flow heterogeneity magnetic resonance imaging in acute stroke. Stroke 2000;31:1097–1103. 21. Simonsen CZ, Rohl L, Vestergaard-poulsen P, Østergaard L. Final infarct size after acute stroke: prediction with flow. Radiology 2002; 225:269–275. 22. Hudetz AG, Feher G, Weigle CG, Knuese DE, Kampine JP. Video microscopy of cerebrocortical capillary flow: response to hypotension and intracranial hypertension. Am J Physiol 1995;268:H2202–H2210. 23. Tomita Y, Tomita M, Schiszler I, Amano T, Tanahashi N, Kobari M, Takeda H, Ohtomo M, Fukuuchi Y. Moment analysis of microflow histogram in focal ischemic lesion to evaluate microvascular derangement after small pial arterial occlusion in rats. J Cereb Blood Flow Metab 2002;22:663–669. 24. Jespersen SN, Østergaard L. The roles of cerebral blood flow, capillary transit time heterogeneity, and oxygen tension in brain oxygenation and metabolism. J Cereb Blood Flow Metab 2012:264–277. 25. Ostergaard L, Jespersen SN, Mouridsen K, et al. The role of the cerebral capillaries in acute ischemic stroke: the extended penumbra model. J Cereb Blood Flow Metab 2013;33:635–648.

Dispersion Correction with CPI in DSC-MRI 26. Mehndiratta A, Macintosh BJ, Crane DE, Payne SJ, Chappell MA. A control point interpolation method for the non-parametric quantification of cerebral haemodynamics from dynamic susceptibility contrast MRI. Neuroimage 2013;64:560–570. 27. Mehndiratta A, MacIntosh BJ, Crane DE, Payne SJ, Chappell MA. Quantifying cerebral haemodynamics beyond CBF using control point interpolation deconvolution for DSC MRI. In Proceedings of the 21st Annual Meeting of ISMRM, Salt Lake City, Utah, USA, 2013. Abstract 3068. 28. Payne SJ, Mehndiratta A, Crane DE, MacIntosh BJ, Chappell MA. Quantification of cerebral haemodynamics beyond CBF using nonparametric CPI method for dynamic susceptibility contrast MRI. In Proceedings of the 26th International Symposium on Cerebral Blood Flow, Metabolism and Function, Shanghai, China, 2013. Abstract 391. 29. Mehndiratta A, Calamante F, MacIntosh BJ, Crane DE, Payne SJ, Chappell MA. Modeling the residue function in DSC-MRI simulations: Analytical approximation to in vivo data. Magn Reson Med 2013. doi: 10.1002/mrm.25056. 30. Vonken EP, Beekman FJ, Bakker CJ, Viergever MA. Maximum likelihood estimation of cerebral blood flow in dynamic susceptibility contrast MRI. Magn Reson Med 1999;41:343–350. 31. Mouridsen K, Friston K, Hjort N, Gyldensted L, Østergaard L, Kiebel S. Bayesian estimation of cerebral perfusion using a physiological model of microvasculature. Neuroimage 2006;33:570–579. 32. Zanderigo F, Bertoldo A, Pillonetto G, Cobelli Ast C. Nonlinear stochastic regularization to characterize tissue residue function in bolustracking MRI: assessment and comparison with SVD, block-circulant SVD, and Tikhonov. IEEE Trans Biomed Eng 2009;56:1287–1297. 33. Smith MR, Lu H, Trochet S, Frayne R. Removing the effect of SVD algorithmic artifacts present in quantitative MR perfusion studies. Magn Reson Med 2004;51:631–634. 34. Willats L, Connelly A, Calamante F. Improved deconvolution of perfusion MRI data in the presence of bolus delay and dispersion. Magn Reson Med 2006;56:146–156. 35. Calamante F. Artifacts and pitfalls in perfusion MR imaging. In: Gillard J, Waldman A, Barker P, editors. Clinical MR neuroimaging: diffusion, perfusion and spectroscopy. New York: Cambridge University Press; 2005. p 141–160. 36. Calamante F, Yim PJ, Cebral JR. Estimation of bolus dispersion effects in perfusion MRI using image-based computational fluid dynamics. Neuroimage 2003;19:341–353. 37. Calamante F, Thomas DL, Pell GS, Wiersma J, Turner R. Measuring cerebral blood flow using magnetic resonance imaging techniques. J Cereb Blood Fow Metab 1999;19:701–735. 38. Chappell MA, Woolrich MW, Kazan S, Jezzard P, Payne SJ, Macintosh BJ. Modeling dispersion in arterial spin labeling: validation using dynamic angiographic measurements. Magn Reson Med 2013;69:563–570. 39. Chappell MA, Groves AR, Whitcher B, Woolrich MW. Variational Bayesian inference for a nonlinear forward model. IEEE Trans Signal Process 2009;57:223–236. 40. Calamante F. Bolus dispersion issues related to the quantification of perfusion MRI data. J Magn Reson Imaging 2005;22:718–722. 41. Leenders KL, Perani D, Lammertsma AA, Heather JD, Buckingham P, Healy MJ, Gibbs JM, Wise RJ, Hatazawa J, Herold S. Cerebral blood flow, blood volume and oxygen utilization. Normal values and effect of age. Brain 1990;113:27–47. 42. Calamante F, Gadian DG, Connelly A. Quantification of bolus-tracking MRI: Improved characterization of the tissue residue function using Tikhonov regularization. Magn Reson Med 2003;50:1237–1247. 43. MacIntosh BJ, Sideso E, Donahue MJ, Chappell MA, G€ unther M, Handa A, Kennedy J, Jezzard P. Intracranial hemodynamics is altered

13

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

by carotid artery disease and after endarterectomy: a dynamic magnetic resonance angiography study. Stroke 2011;42:979–984. Crane DE, Donahue MJ, Chappell MA, Sideso E, Handa A, Kennedy J, Jezzard P, Macintosh BJ. Evaluating quantitative approaches to dynamic susceptibility contrast MRI among carotid endarterectomy patients. J Magn Reson Imaging 2013;37:936–943. Park CS, Payne SJ. A generalized mathematical framework for estimating the residue function for arbitrary vascular networks. Interface Focus 2013;3:20120078. Calamante F, Willats L, Gadian DG, Connelly A. Bolus delay and dispersion in perfusion MRI: implications for tissue predictor models in stroke. Magn Reson Med 2006;55:1180–1185. Bonekamp D, Degaonkar M, Barker PB. Quantitative cerebral blood flow in dynamic susceptibility contrast MRI using total cerebral flow from phase contrast magnetic resonance angiography. Magn Reson Med 2011;66:57–66. Shin W, Horowitz S, Ragin A, Chen Y, Walker M, Carroll TJ. Quantitative cerebral perfusion using dynamic susceptibility contrast MRI: evaluation of reproducibility and age- and gender-dependence with fully automatic image postprocessing algorithm. Magn Reson Med 2007;58:1232–1241. Zaharchuk G, Straka M, Marks MP, Albers GW, Moseley ME, Bammer R. Combined arterial spin label and dynamic susceptibility contrast measurement of cerebral blood flow. Magn Reson Med 2010; 63:1548–1556. Mouannes-Srour JJ, Shin W, Ansari SA, Hurley MC, Vakil P, Bendok BR, Lee JL, Derdeyn CP, Carroll TJ. Correction for arterial-tissue delay and dispersion in absolute quantitative cerebral perfusion DSC MR imaging. Magn Reson Med 2012;68:495–506. Alsop D, Wedmid A, Schlaug G. Defining a local input function for perfusion quantification with bolus contrast MRI. In Proceedings of the 10th Annual Meeting of ISMRM, Honolulu, Hawaii, USA, 2003. Abstract 659. Duhamel G, Bazelaire C, Alsop D. Input functions from echoplanar hemodynamic studies using dynamic susceptibility contrast: a numerical model compared to in-vivo results. In Proceedings of the 11th Annual Meeting of ISMRM, Toronto, Canada, 2003. Abstract 2199. Willats L, Christensen S, Ma HK, Donnan GA, Connelly A, Calamante F. Validating a local arterial input function method for improved perfusion quantification in stroke. J Cereb Blood Flow Metab 2011;31:2189–2198. Duhamel G, Schlaug G, Alsop DC. Measurement of arterial input functions for dynamic susceptibility contrast magnetic resonance imaging using echoplanar images: comparison of physical simulations with in vivo results. Magn Reson Med 2006;55:514–23. Lorenz C, Benner T, Chen PJ, et al. Automated perfusion-weighted MRI using localized arterial input functions. J Magn Reson Imaging 2006;24:1133–1139. Knutsson L, Larsson E-M, Thilmann O, Stahlberg F, Wirestam R. Calculation of cerebral perfusion parameters using regional arterial input functions identified by factor analysis. J Magn Reson Imaging 2006; 23:444–453. Gruner R, Bjornara BT, Moen G, Taxt T. Magnetic resonance brain perfusion imaging with voxel-specific arterial input functions. J Magn Reson Imaging 2006;23:273–284. Christensen S, Calamante F, Hjort N, Wu O, Blankholm AD, Desmond P, Davis S, Ostergaard L. Inferring origin of vascular supply from tracer arrival timing patterns using bolus tracking MRI. J magn reson imaging 2008;27:1371–1381.