Deconvolution-Based CT and MR Brain Perfusion Measurement: Theoretical Model Revisited and Practical Implementation Details

Abstract

Deconvolution-based analysis of CT and MR brain perfusion data is widely used in clinical practice and it is still a topic of ongoing research activities. In this paper, we present a comprehensive derivation and explanation of the underlying physiological model for intravascular tracer systems. We also discuss practical details that are needed to properly implement algorithms for perfusion analysis. Our description of the practical computer implementation is focused on the most frequently employed algebraic deconvolution methods based on the singular value decomposition. In particular, we further discuss the need for regularization in order to obtain physiologically reasonable results. We include an overview of relevant preprocessing steps and provide numerous references to the literature. We cover both CT and MR brain perfusion imaging in this paper because they share many common aspects. The combination of both the theoretical as well as the practical aspects of perfusion analysis explicitly emphasizes the simplifications to the underlying physiological model that are necessary in order to apply it to measured data acquired with current CT and MR scanners.

Figure 1

CT perfusion parameter maps of cerebral blood flow (CBF), cerebral blood volume (CBV), mean transit time (MTT), and time-to-peak (TTP). The ischemic stroke lesion is marked with arrows.

Figure 2

Physiological model of the tissue perfusion. A blood cell can take several paths through the capillary bed. The variables are defined in Table 1.

Figure 3

Examples of the time-concentration curves c _art(t), c _voi(t), and c_ven(t) given in arbitrary units (a.u.). (b) Represents a zoomed view of (a) with a rescaled ordinate.

Figure 4

Examples of the distribution function h(t) of transit times (the mean transit time is 4 s) and the corresponding residue function r(t).

Figure 5

Perfusion parameters that are measured directly using the time-concentration curve. See Sections 3.5 and 4.1 for explanations (BAT: bolus arrival time, TTP: time-to-peak, FM: first moment, AUC: area under the curve).

Figure 6

Examples of measured time-attenuation curves in perfusion CT in (a) an arterial vessel and (b) in tissue. The time curves have been pre-processed by baseline subtraction and removal of the baseline time frames. The example data is measured at N = 35 discrete time points.

Figure 7

Least-squares solution vector k _ls of (38) using the example data from Figure 6. (k _ls)_j denotes the jth entry of the vector k _ls. The plot shown in (a) illustrates the strong oscillations of k _ls. The plot given in (b) shows the amplitude |k _ls| of this function on a logarithmic scale.

Figure 8

SVD analysis of the matrix A constructed from the example data shown in Figure 6. The plot displays the absolute values of the weighting factors (u _i ^T c)/σ _i and of their individual components |u _i ^T c| and σ _i on a logarithmic scale.

Figure 9

(a) Linear and (b) double logarithmic plot of the Tikhonov filter factor f _λ ^(tikh) as a function of the singular value σ ∈ [10⁻⁵, 1].

Figure 10

Deconvolution with Tikhonov regularization: (a) Regularized solution k _λ ^(tikh) for two different regularization parameters λ _rel and (b) maximum of k _λ ^(tikh) as a function of λ _rel . (k _λ ^(tikh))_j denotes the jth entry of the vector k _λ ^(tikh).