Application of Time Sampling in Brain CT Perfusion Imaging for Dose Reduction

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1 Application of Time Sampling in Brain CT Perfusion Imaging for Dose Reduction S. H. Lee a, J. H. Kim* a, b, K. G. Kim b, S. J. Park a, Jung Gi Im b a Interdisciplinary Program in Radiation Applied Life Science major; b Dept. of Radiology, Seoul National University College of Medicine, 28, Yongon-Dong, Chongno-Gu, Seoul, , Korea ABSTRACT The purpose of this study is to determine a stable sampling rate not to be affected by sampling shift for reducing radiation exposure with time sampling and interpolation in cerebral perfusion CT examination. Original images were obtained every 1 second for 40 time series from 3 patients, respectively. Time sampling was performed with sampling intervals (SI) from 2 to 10 seconds. Sampling shift was applied from +1 to SI-1 for each sampling rate. For each patient, 30 tissue concentration time-course data were collected, and arterial input curves were fitted by gamma-variate function. The sinc function was introduced for interpolation. Deconvolution analysis based on SVD was performed for quantifying perfusion parameters. The perfusion values through time-varying sampling and interpolation were statistically compared with the original perfusion values. The mean CBF values with increase of sampling interval and shift magnitude from the collected data had a wider fluctuation pattern centering around the original mean CBF. The mean CBV values had a similar tendency to the mean CBF values, but a relatively narrower deviation. The mean MTT values were fluctuated reversely to the trend of the mean CBF values. The stable sampling interval for quantifying perfusion parameters with lower radiation exposure was statistically acceptable up to 4 seconds. These results indicate that sampling shift limits sampling rate for acquiring acceptable perfusion values. This study will help in selecting more reasonable sampling rate for low-radiation-dose CT examination. Keywords: Perfusion CT, time-varying sampling, interpolation, deconvolution, singular value decomposition 1. INTRODUCTION Perfusion CT was presented in the late 1990s as an imaging modality to be used in patients with acute stroke to obtain reliable information about the location and size of brain ischemia in the early stage of brain infarction [1-3]. Acute evaluation of cerebral perfusion parameters is important for the determination of therapeutic plans or prognostic predictions of patients with cerebrovalscular disease [4, 5]. It is necessary to monitor the passage of contrast medium with continuous single-slice scans for practically one minute for acquiring reliable and useful parameters of cerebral perfusion. This results in too much radiation dose exposure to a patient, which is one of obstacles to apply the perfusion CT to the clinical situation with this procedure from being widespread. Thus, reduction of radiation dose is seriously required. The Nyquist-Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. The theorem is commonly called Shannon s sampling theorem. Sampling is the process of converting a continuous time signal into discrete time sequence. The theorem states that exact reconstruction of a continuous time baseband signal by convolution with the sinc function known as the interpolation function from its samples is possible if the signal is bandlimited and the sampling frequency is greater than twice the signal bandwidth called Nyquist frequency [6, 7]. This theory can be proposed as an approach for less CT scanning and dose reduction by undersampling in dynamic CT imaging. Undersampling is essentially sampling too slowly, or sampling rate below the Nyquist frequency for a particular signal of interest. Undersampling leads to aliasing and the original signal cannot be properly reconstructed. However, undersampling may be useful in a certain application such as dose reduction in order to protect a patient if the perfusion parameter values acquired from undersampling are diagnostically acceptable. Thus, our approach is not to reconstruct an original contrast agent concentration-time curve with more image scanning samples than in usual clinical dynamic CT imaging, but to approximate the resultant perfusion Medical Imaging 2007: Physics of Medical Imaging, edited by Jiang Hsieh, Michael J. Flynn, Proc. of SPIE Vol. 6510, 65102P, (2007) /07/$18 doi: / Proc. of SPIE Vol P-1

2 parameter values from undersampling of CT images and its interpolation to original perfusion parameter values and to determine an acceptable range of sampling rate. The undersampling corresponds to the purpose of lower radiation exposure and the interpolation to approximation to original perfusion values. A consideration in image undersampling is sampling shift effect [8]. When a radiologist performs perfusion CT imaging to a patient, the initial timing for image scanning can be changed to some degree for each examination. This initial timing variation causes the sampling shift, which affects perfusion parameter values. This effect can be severer with decrease of sampling rate. Thus, considering the sampling shift effect is inevitable in order to determine a more stable sampling rate in undersampling. Perfusion CT is a method for the quantification of cerebral blood flow (CBF), cerebral blood volume (CBV), and mean transit time (MTT) on the basis of dynamic bolus tracking data. In order to obtain quantification of tissue hemodynamic parameters such as CBF, CBV and MTT from perfusion CT, the arterial input function (AIF) has to be determined in the first stage [9-11], [14]. The AIF can be acquired non-invasively from arterial pixels in imaged slices [9], [12-14]. K. Murase et al. investigated that the threshold range should be carefully considered when quantifying CBF using deconvolution analysis based on singular value decomposition (SVD), and emphasized the importance of the optimal threhsolding [15]. In this study, optimal thresholding with time-varying sampling and its interpolation from the AIF and the tissue concentration-time curves was performed. Also, the resultant CBF, CBV and MTT values from time-varying sampling and interpolation were compared with them acquired from original clinical perfusion imaging protocol. The purpose of this study was to statistically determine a stable sampling rate not to be affected by sampling shift with timevarying sampling and its interpolation for reducing radiation exposure in cerebral perfusion CT examination. 2. METHODS 2.1 Hypothesis Several hypotheses were set up in order to provide direction for this study and to help to clearly understand our investigation focused. Time-varying image undersampling is performed based on a dynamic contrast-enhanced CT images acquired through usual clinical perfusion imaging protocol. Undersamping is performed to the limit of the maximum sampling time within the predefined total imaging period. Forward time sampling shift is only considered. The initial pre-contrast image is not applied to sampling shift in order to fix the delay time up to the initial contrast enhancement. 2.2 Imaging and data acquisition Perfusion CT imaging was performed at out institution with a multi-detector row helical CT scanner (Lightspeed; GE Medical Systems, Milwaukee, Wis, USA). Immediately after a non-enhanced transverse scanning of the brain from three patients, helical contrast material-enhanced scanning was performed every 1 second for 40 seconds at a single-slice level, respectively. The time-dependent concentrations of the contrast agent in the input artery (C AIF (t)) and the thirty rectangular volumes of interest (VOI) (C VOI (t)) in the brain tissue were collected as quantitative data for each patient. 2.3 Time sampling of the AIF and the tissue concentration-time curves The AIF and the tissue concentration time curves were uniformly sampled by non-shift sampling with time intervals from 2 to 10 seconds within total imaging period, respectively. In addition to, forward time sampling shift was applied from 1 to sampling interval (SI) 1 of shift magnitude (SM) for each sampling rate. 2.4 Curve fitting of the AIF The time-dependent concentrations of the original and the sampled AIFs were modeled as gamma variate function of the first and the second passages of the contrast agent concentration by C++ curve fitting code for each sampling rate. Proc. of SPIE Vol P-2

3 2.5 Interpolation of the fitted AIF and the sampled tissue concentration-time curves The fitted AIFs and the sampled tissue concentration-time curves were interpolated by convolution operation with the sinc function, respectively. The reconstruction functions are given as followings with initial conditions, C iaif ( t 0 ) = 0 and CiVOI( t 0) = 0 by zero-padding. π ( tj ti) sin τ () () π ( tj ti) i= 0 τ N 1 CiAIF tj = CfAIF ti π ( tj ti) sin τ () () (1) π ( tj ti) i= 0 τ N 1 CiVOI tj = CsVOI ti CfAIF( t ): the fitted AIF CiAIF( t ): the interpolated AIF CsVOI( t ): the sampled tissue time-concentration CiVOI( t ): the interpolated tissue time-concentration N : the total number of interpolated frames τ : the sampling time interval t i : the time of the ith frame 2.6 Deconvolution and quantification of hemodynamic parameters According to the indicator dilution theory [16] for intravascular contrast agent, the C VOI (t) is given as followings. 1 VOI N h j AIF i j i i= 0 kc () t = CBF t C () t Rt ( - t ) (2) The constant k h (=(1 - H LV )/(1 - H SV )) is a correction constant for the difference in hematocrit in the large blood vessels (H LV ) and small blood vessels (H SV ) of the brain [9], and was set to 1 in this study. R(t) is the residue function which is the relative amount of contrast agent in the VOI in an idealized perfusion experiment. The above equation can be formulated as a matrix equation. where CAIF( t0) AIF( 1) AIF( 0)... 0 A C t C t = CAIF( tn-1) CAIF( tn-2)... CAIF( t0) c = Ab (3) ( 0) ( 1) R t b CBF R t = t... Rt ( N-1) and CVOI( t0) VOI( 1) c C t = kh... CVOI( tn-1) The matrix A is able to be expressed as the product of an N ⅹ N column-orthogonal matrix U, an N ⅹ N diagonal matrix W with the singular values and the transpose of N ⅹ N orthogonal matrix V [17]. The SVD constructs the matrices so that the inverse of A can be represented as followings. A -1 = V[diag(1/w i )]U T (5) (4) Proc. of SPIE Vol P-3

4 Where w i are the diagonal elements of W and U T is the transpose of U. The vector b can be calculated as the following expression. b = V[diag(1/w i )](U T c) (6) The vector b contains the elements of the residue function R(t). In this study, this R(t) was modeled by a single exponential decay. The difference between R(t) and the modeling function was calculated in all threshold range under maximum of w i. The singular value,, w i corresponding to thresholding with the minimum difference between them was selected as the optimal threshold value. The CBF values was calculated by the maximum of R(t) (R(t) = 1) through optimally thresholding. The tissue concentration-time curves were reconstructed by convolution operation of the first passage of the AIF and the R(t). The CBV values were calculated using the first passage of the fitted AIF and the reconstructed tissue time-concentration as the following expression. C C iffaif( i) rfvoi( i) CBV = N 1 CrfVOI() ti i= 0 N 1 kh CiffAIF ti i= 0 () t : the AIF interpolated with the first passage of the fitted AIF t : the tissue time-concentration reconstructed by convolution of the first passage of the fitted AIF and the R(t) The MTT values were calculated based on the central volume principle which states that MTT is given by the tissue blood volume divided by the tissue blood flow as followings. MTT = CBV CBF (7) (8) 2.7 Statistical Analysis The original perfusion parameter values and the perfusion values calculated from the time-varying sampling and interpolation were compared each other by independent samples t-test in the SPSS version 13.0 statistical analysis software package (SPSS Inc., Chicago, Illinois, USA), respectively. The p value <0.05 was regarded as being statistically significant difference. 3. RESULTS Table 1. The p values from independent samples t test on the comparison of perfusion parameter values. Sampling Interval (SI, sec), Patient 1 Patient 2 Patient 3 Shift Magnitude (SM, sec), Total Number of Scans (TNS) CBF CBV MTT CBF CBV MTT CBF CBV MTT SI :2, SM: 0, TNS: SI :2, SM: 1, TNS : SI :3, SM: 0, TNS: SI :3, SM: 1, TNS: SI :3, SM: 2, TNS: SI :4, SM: 0, TNS: SI :4, SM: 1, TNS: SI :4, SM: 2, TNS: Proc. of SPIE Vol P-4

5 SI :4, SM: 3, TNS: SI :5, SM: 0, TNS: * SI :5, SM: 1, TNS: , SI :5, SM: 2, TNS: SI :5, SM: 3, TNS: SI :5, SM: 4, TNS: SI :6, SM: 0, TNS: SI :6, SM: 1, TNS: SI :6, SM: 2, TNS: * SI :6, SM: 3, TNS: SI :6, SM: 4, TNS: SI :6, SM: 5, TNS: * SI :7, SM: 0, TNS: * SI :7, SM: 1, TNS: * SI :7, SM: 2, TNS: * SI :7, SM: 3, TNS: * * * SI :7, SM: 4, TNS: * * SI :7, SM: 5, TNS: * SI :7, SM: 6, TNS: * * * 0.030* SI :8, SM: 0, TNS: * ,012* 0.019* * SI :8, SM: 1, TNS: * * 0.038* SI :8, SM: 2, TNS: * * 0.036* SI :8, SM: 3, TNS: * SI :8, SM: 4, TNS: * * SI :8, SM: 5, TNS: * * * SI :8, SM: 6, TNS: * * 0.012* * SI :8, SM: 7, TNS: * 0.018* * SI :9, SM: 0, TNS: * * SI :9, SM: 1, TNS: * 0.005* * * 0.001* * SI :9, SM: 2, TNS: * 0.022* 0.006* 0.002* * * SI :9, SM: 3, TNS: * * 0.003* * 0.003* SI :9, SM: 4, TNS: * * 0.000* 0.046* 0.000* SI :9, SM: 5, TNS: * * 0.014* 0.000* SI :9, SM: 6, TNS: * * SI :9, SM: 7, TNS: * * * SI :9, SM: 8, TNS: * * * * SI :10, SM: 0, TNS: * * * * 0.037* Proc. of SPIE Vol P-5

SI :10, SM: 1, TNS: 5 0.180 0.432 0.034* 0.073 0.429 0.000* 0.005* 0.662 0.000* SI :10, SM: 2, TNS: 5 0.002* 0.032* 0.263 0. 283 0.107 0.333 0.000* 0.665 0.000* SI :10, SM: 3, TNS: 5 0.000* 0.000* 0.008* 0.

013* SI :10, SM: 6, TNS: 5 0.229 0.011* 0.343 0.000* 0.986 0.000* 0.000* 0.067 0.000* SI :10, SM: 7, TNS: 5 0.451 0.006* 0.241 0.016* 0.381 0.031* 0.000* 0.000* 0.000* SI :10, SM: 8, TNS: 5 0.000* 0.017* 0.

6 SI :10, SM: 1, TNS: * * 0.005* * SI :10, SM: 2, TNS: * 0.032* * * SI :10, SM: 3, TNS: * 0.000* 0.008* 0.000* * 0.000* * SI :10, SM: 4, TNS: * 0.007* 0.041* 0.000* * 0.019* 0.007* 0.000* SI :10, SM: 5, TNS: * * 0.000* * 0.001* * SI :10, SM: 6, TNS: * * * 0.000* * SI :10, SM: 7, TNS: * * * 0.000* 0.000* 0.000* SI :10, SM: 8, TNS: * 0.017* 0.000* * 0.000* 0.000* 0.000* 0.000* SI :10, SM: 9, TNS: * * 0.011* 0.000* 0.000* * 0.000* * represents statistically significant difference (p<0.05). Table 1 shows the results from independent samples t-test to compare with the original perfusion parameters the ones calculated from interpolated images for each sampling interval and shift magnitude in 3 patients. The results from patient 1 to 3 started to represent statistically significant differences from the sampling intervals of 7, 5 and 7 seconds, respectively. Therefore, when considering the statistical results from given data, a stable sampling interval not to be affected by sampling shift can be suggested as from 2 to 4 seconds except the original sampling interval, 1 second. riiiiiiiiiiiiiiiiiiii.i!ii iiihilohi (a) (b) (c) Fig. 1. (a) Mean CBF, (b) mean CBV and (c) mean MTT values for each sampling interval and shift magnitude from 3 patients Figure 1(a) shows the mean CBF values on the thirty tissue VOIs for each sampling interval and shift magnitude from patient 1 to 3. The mean CBF values represent wider fluctuation patterns with increase of sampling interval and shift magnitude centering around the original CBF value. Figure 1(b) shows the mean CBV values, which has a similar tendency to the mean CBF values with a relatively narrower deviation. Figure 1(c) shows the mean MTT values, which were fluctuated reversely to the trend of CBF. Approximation to the original perfusion values is not only achievable with short sampling intervals, but also with moderate sampling intervals depending on an irregular fluctuation cycle and width of the ones with increase of sampling interval and shift magnitude. However, approach to a kind of perfusion parameter does not guarantee the allowance of other perfusion parameters at the level of a long sampling interval because the fluctuation cycles and widths of perfusion parameters with increase of sampling interval and shift magnitude are significantly different one another. Proc. of SPIE Vol P-6

original SI: 2, SM: 0 SI: 2, SM: 1 SI: 3, SM: 0 SI: 3, SM: 1 SI: 3, SM: 2 SI: 4, SM: 0 SI: 4, SM: 1 SI: 4, SM: 2 SI: 4,

SM: 3 SI: 6, SM: 4 SI: 6, SM: 5 SI: 7, SM: 0 SI: 7, SM: 1 SI: 7, SM: 2 SI: 7, SM: 3 SI: 7, SM: 4 SI: 7, SM: 5 SI: 7,

SM: 0 SI: 9, SM: 1 SI: 9, SM: 2 SI: 9, SM: 3 SI: 9, SM: 4 SI: 9, SM: 5 SI: 9, SM: 6 SI: 9, SM: 7 SI: 9, SM: 8 SI: 10,

7 original SI: 2, SM: 0 SI: 2, SM: 1 SI: 3, SM: 0 SI: 3, SM: 1 SI: 3, SM: 2 SI: 4, SM: 0 SI: 4, SM: 1 SI: 4, SM: 2 SI: 4, SM: 3 SI: 5, SM: 0 SI: 5, SM: 1 SI: 5, SM: 2 SI: 5, SM: 3 SI: 5, SM: 4 SI: 6, SM: 0 SI: 6, SM: 1 SI: 6, SM: 2 SI: 6, SM: 3 SI: 6, SM: 4 SI: 6, SM: 5 SI: 7, SM: 0 SI: 7, SM: 1 SI: 7, SM: 2 SI: 7, SM: 3 SI: 7, SM: 4 SI: 7, SM: 5 SI: 7, SM: 6 SI: 8, SM: 0 SI: 8, SM: 1 SI: 8, SM: 2 SI: 8, SM: 3 SI: 8, SM: 4 SI: 8, SM: 5 SI: 8, SM: 6 SI: 8, SM: 7 SI: 9, SM: 0 SI: 9, SM: 1 SI: 9, SM: 2 SI: 9, SM: 3 SI: 9, SM: 4 SI: 9, SM: 5 SI: 9, SM: 6 SI: 9, SM: 7 SI: 9, SM: 8 SI: 10, SM: 0 SI: 10, SM: 1 SI: 10, SM: 3 SI: 10, SM: 4 SI: 10, SM: 5 SI: 10, SM: 6 SI: 10, SM: 7 SI: 10, SM: 2 SI: 10, SM: 8 SI: 10, SM: 9 Fig. 2. CBF images for each sampling interval (SI, sec) and shift magnitude (SM, sec) from patient 2 original SI: 2, SM: 0 SI: 2, SM: 1 SI: 3, SM: 0 SI: 3, SM: 1 SI: 3, SM: 2 Proc. of SPIE Vol P-7 SI: 4, SM: 0 SI: 4, SM: 1

8 SI: 4, SM: 2 SI: 4, SM: 3 SI: 5, SM: 0 SI: 5, SM: 1 SI: 5, SM: 2 SI: 5, SM: 3 SI: 5, SM: 4 SI: 6, SM: 0 SI: 6, SM: 1 SI: 6, SM: 2 SI: 6, SM: 3 SI: 6, SM: 4 SI: 6, SM: 5 SI: 7, SM: 0 SI: 7, SM: 1 SI: 7, SM: 2 SI: 7, SM: 3 SI: 7, SM: 4 SI: 7, SM: 5 SI: 7, SM: 6 SI: 8, SM: 0 SI: 8, SM: 1 SI: 8, SM: 2 SI: 8, SM: 3 SI: 8, SM: 4 SI: 8, SM: 5 SI: 8, SM: 6 SI: 8, SM: 7 SI: 9, SM: 0 SI: 9, SM: 1 SI: 9, SM: 2 SI: 9, SM: 3 SI: 9, SM: 4 SI: 9, SM: 5 SI: 9, SM: 6 SI: 9, SM: 7 SI: 9, SM: 8 SI: 10, SM: 0 SI: 10, SM: 1 SI: 10, SM: 3 SI: 10, SM: 4 SI: 10, SM: 5 SI: 10, SM: 6 SI: 10, SM: 7 SI: 10, SM: 2 SI: 10, SM: 8 SI: 10, SM: 9 Fig. 3. CBV images for each sampling interval (SI, sec) and shift magnitude (SM, sec) from patient 2 original SI: 4, SM: 2 SI: 2, SM: 0 SI: 2, SM: 1 SI: 3, SM: 0 SI: 3, SM: 1 SI: 3, SM: 2 SI: 4, SM: 0 SI: 4, SM: 1 SI: 4, SM: 3 SI: 5, SM: 0 SI: 5, SM: 1 SI: 5, SM: 2 SI: 5, SM: 3 SI: 5, SM: 4 SI: 6, SM: 0 Proc. of SPIE Vol P-8

9 SI: 6, SM: 1 SI: 6, SM: 2 SI: 6, SM: 3 SI: 6, SM: 4 SI: 6, SM: 5 SI: 7, SM: 0 SI: 7, SM: 1 SI: 7, SM: 2 SI: 7, SM: 3 SI: 7, SM: 4 SI: 7, SM: 5 SI: 7, SM: 6 SI: 8, SM: 0 SI: 8, SM: 1 SI: 8, SM: 2 SI: 8, SM: 3 SI: 8, SM: 4 SI: 8, SM: 5 SI: 8, SM: 6 SI: 8, SM: 7 SI: 9, SM: 0 SI: 9, SM: 1 SI: 9, SM: 2 SI: 9, SM: 3 SI: 9, SM: 4 SI: 9, SM: 5 SI: 9, SM: 6 SI: 9, SM: 7 SI: 9, SM: 8 SI: 10, SM: 0 SI: 10, SM: 1 SI: 10, SM: 3 SI: 10, SM: 4 SI: 10, SM: 5 SI: 10, SM: 6 SI: 10, SM: 7 SI: 10, SM: 2 SI: 10, SM: 8 SI: 10, SM: 9 Fig. 4. MTT images for each sampling interval (SI, sec) and shift magnitude (SM, sec) from patient 2 Figure 2, 3 and 4 show CBF, CBV and MTT images for each sampling interval and shift magnitude from patient 2, respectively. The parametric mapping images confirm degradation through over- or underestimation with increase of sampling interval and shift magnitude, which reflect the tendencies corresponding to the former statistical results. The outstanding discovery is that perfusion parametric images acquired with relatively longer sampling intervals can be represented even more similarly to the original ones than with shorter sampling intervals as the case may be. 4. DISCUSSION This study indicates a possibility for quantifying the hemodynamic parameters such as CBF, CBV and MTT calculated from time-varying sampling and its interpolation, using deconvolution analysis based on SVD. The interpolation process with time sampling has to be performed for obtaining a better approximation to an original perfusion value. Moreover, to determine a more stable sampling rate not to be affected by sampling shift is also important to achieving a clinical implementation. The perfusion parameter values had a tendency to be fluctuated through wider swings around the original ones with increase of sampling time interval and shift magnitude. Thus, we can approximate to the original perfusion parameter values through applying sampling shift even with the sampling rate of a longer sampling interval. Proc. of SPIE Vol P-9

10 5. CONCLUSION The time-varying sampling and interpolation techniques enable to obtain more acceptable perfusion parameter values and image maps through selecting a stable sampling rate not to be affected by sampling shift in lower radiation dose perfusion CT examination. REFERENCES 1. M. Koenig, E. Klotz, B. Luka, D. J. Venderink, J. F. Spittler, and L. Heuser, Perfusion CT of the brain: diagnostic approach for early detection of ischemic stroke, Radiology. 209(1), (1998). 2. E. Klotz, M. König, "Perfusion measurements of the brain: using dynamic CT for the quantitative assessment of cerebral ischemia in acute stroke," Eur.J.Radiol.30(3), (1999). 3. D. G. Nabavi, A. Cenic, R. A. Craen, A. W. Gelb, J. D. Bennett, R. Kozak and T. Y. Lee, CT assessment of cerebral perfusion: experimental validation and initial clinical experience, Radiology. 213(1), (1999). 4. J. Astrup, B. K. Siesjo and L. Symon, Thresholds in cerebral ischemia - the ischemic penumbra, Stroke. 12(6), (1981). 5. A. J. Furlan and G. Kanoti, When is thrombolysis justified in patients with acute ischemic stroke? A bioethical perspective, Stroke. 28(1), (1997). 6. C. E. Shannon, Communication in the presence of noise, in Proc. IRE, 37(1), 10-21, (1949). 7. A. D. Wyner and S. Shamai, Introduction to Communication in the presence of noise by C. E. Shannon, Proc. IEEE, 86(2), , (1998). 8. R. N. Bracewell, Fourier Transform and its Applications, 3rd. ed., McGraw-Hill, New York, K. A. Rempp, G. Brix, F. Wenz, C. R. Becker, F. Guckel and W. J. Lorenz, Quantification of regional cerebral blood flow and volume with dynamic susceptibility contrast-enhanced MR imaging, Radiology., 193(3), , (1994). 10. T. Fritz-Hansen, E. Rostrup, H. B. Larsson, L. Sondergaard, P. Ring and O. Henriksen, Measurement of the arterial concentration of Gd-DTPA using MRI: a step toward quantitative perfusion imaging, Magn. Reson. Med., 36(2), , (1996). 11. E. Akbudak, R. E. Norberg and T. E. Conturo, Contrast-agent phase effects: an experimental system for analysis of susceptibility, concentration, and bolus input function kinetics, Magn. Reson. Med. 38(6), , (1997). 12. E. J. Vonken, M. J. van Osch, C. J. Bakker and M. A. Viergever, Measurement of cerebral perfusion with dualecho multi-slice quantitative dynamic susceptibility contrast MRI, J. Magn. Reson. Imaging., 10(2), , (1999). 13. K. Murase, K. Kikuchi, H. Miki, T. Shimizu and J. Ikezoe, Determination of arterial input function using fuzzy clustering for quantification of cerebral blood flow with dynamic susceptibility contrast-enhanced MR imaging, J. Magn. Reson. Imaging., 13(5), , (2001). 14. G. K. von Schulthess and J. Hennig, Functional Imaging, Lippincott Raven, Philadelphia, New York, K. Murase, M. Shinohara and Y. Yamazaki, Accuracy of deconvolution analysis based on singular value decomposition for quantification of cerebral blood flow using dynamic susceptibility contrast-enhanced magnetic resonance imaging, Phys. Med. Biol., 46(12), , (2001). 16. P. Meier and K. L. Zierler On the theory of the indicator-dilution method for measurement of blood flow and volume, J. Appl. Physiol., 6(12), , (1954). 17. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd. ed., Cambridge: Cambridge University Press, Proc. of SPIE Vol P-10

NIH Public Access Author Manuscript Neuroimage. Author manuscript; available in PMC 2012 July 1.

NIH Public Access Author Manuscript Published in final edited form as: Neuroimage. 2011 July 1; 57(1): 182 189. doi:10.1016/j.neuroimage.2011.03.060. Early Time Points Perfusion Imaging: Theoretical Analysis