The arterial input, Ca(t), and tissue volume-of-interest, CVOI(t), concentration functions are related to R(t) through the convolution equation. )( )( )( tR. tC. CBF.
Removing CBF Artifacts Introduced during SVD Deconvolution M. R. Smith1, H. Lu1, S. Trochet2, R. Frayne1 1
University of Calgary, Calgary, Alberta, Canada, 2Ecole Nationale Superieure des Telecommunications de Bretagne, Plouzane, Bretagne, France
Synopsis Cerebral blood flow (CBF) estimates can be obtained by deconvolving the tissue concentration curve with the arterial input function (AIF) using singular value decomposition (SVD). An under-estimation of the CBF occurs, from a broadening distortion of the residue function, R(t), when the number of eigenvalues used in the SVD solution is limited to control noise instabilities. The standard SVD algorithm does not correctly handle this distortion, especially when the arterial and tissue delay (ATD) is small. A reformulated SVD is proposed to handle the R(t) distortions, and is shown to also remove other previously reported delay artifacts.
Introduction The tissue residue function, R(t), is used to assess regional CBF. The arterial input, Ca(t), and tissue volume-of-interest, CVOI(t), concentration functions are related to R(t) through the convolution equation CVOI (t ) = CBF • C a (t ) ⊗ R (t ) where ⊗ denotes convolution. Singular-value decomposition (SVD)1 is a common approach for estimating R(t) from experimental Ca(t) and CVOI(t) measurements. Noise in the R(t) estimate can be controlled through the threshold parameter, PSVD, used to manipulate the number of eigenvalues used in the SVD estimate.1,2 As the threshold PSVD increases in size, fewer eigenvalues are included in the SVD deconvolution solution, the residue function is broadened and distorted (see Fig. 1). The distortion leads to a decrease in the CBF estimate, the maximum of R'(t) = CBF•R(t). The underestimation can be seen in Fig. 2 (closed circles) for large ATD. A substantial oscillatory artifact signal is now present prior to the expected arrival time of the leading edge of the residue function. An oscillatory characteristic is introduced into the CBF estimate2 since the standard SVD algorithm is unable to handle this early arrival artifact (Fig. 2, closed circles, ATD of 0 s to 4 s delays). For ATD < 1 s, CBF estimates become significantly distorted. These distortions have been observed using invivo CBF estimates by manually manipulating the ATD.3
Method In this paper we introduce a reformulated SVD algorithm that takes into account the distortion artifact introduced by the use of the noise controlling SVD parameter, PSVD. Let t1 and t1 + ATD be the respective arrival times of the AIF and tissue signals, with TS the arrival time of the first non-zero distorted residue function component. CBF estimates can be obtained by determining the maximum value of the residue function R′[n ∆T EXPT ] , determined from a reformulated SVD deconvolution matrix equation
∆T EXPT [Ca] t≥t1 (R’) t≥TS = (CVOI) t ≥t1+TS with N x N matrix [Ca] t≥t1 =
C a [t1] C a [t1 + ∆TEXPT ]
C a [t1 + ( N
M
T
T
0 C a [t1]
M
− 1)∆TEXPT ] C a [t1 + ( N − 2)∆TEXPT ]
K 0 K 0 O M K C [t1] a
and N x 1 vectors (R’) t≥TS = (R’[TS], R’[TS + ∆TEXPT ], …, R’[TS +(N-1) ∆TEXPT ]) , where indicates transpose, and (CVOI) t ≥t1+TS = (CVOI[t1+TS], CVOI[ t1+TS + ∆TEXPT ], …, CVOI[t1 +TS + (N-1) ∆TEXPT ])T with ∆TEXPT the sample interval.
Results The use of the reformulated SVD algorithm leads to CBF estimates (Fig. 2 open circles) that are not affected by the oscillatory early arrival R(t) signal produced by limiting the number of eigenvalues in the SVD solution. In addition, in agreement with convolution shift theory, the new CBF estimates are not affected by the manipulation of the relative arterial-tissue signal delay.
Conclusion Use of the reformulated SVD (rSVD) restores the expected symmetry between time and frequency domain techniques used to determine CBF estimates. It has been shown that the CBF estimates produced by the time-domain rSVD (Fig. 2 open circles) and frequency domain FT (triangles) algorithms satisfy the delay independence requirement of the convolution shift theory, while estimates from the standard SVD algorithm do not (closed circles).
References 1. Østergaard L et al., Magn. Reson. Med. 1996; 36: 715-725. 2. Calamante et al., Magn. Reson. Med, 2000:44:466-473. 3. Wu, O et al., Proc. ISMRM 2002:660.
Fig. 1. The deconvolved R(t), is broadened as PSVD increases because the number of eigenvalues used in the SVD solution decreases. CBF estimates change if the deconvolution algorithm can not handle the signal components prior to R(t)’s leading edge.
Proc. Intl. Soc. Mag. Reson. Med. 11 (2003)
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Fig. 2. The CBFMEASURED / CBFTRUE ratio determined by rSVD (open circles) and FT (triangles) deconvolution have ATD independence not shown by standard SVD (closed circles) deconvolution.
