As described in Chapter 24, the tracer kinetic model used to quantify perfusion is given by a mathematical convolution expression. This model applies to both dynamic susceptibility contrast magnetic resonance imaging (DSC-MRI) and perfusion computed tomography (CT; Nb: In fact, the convolution model is also applicable to arterial spin labeling perfusion MRI—see Chapter 31); the main difference relates to the way the concentration of the contrast agent is measured. Therefore, most of the statements described in this chapter regarding DSC-MRI are also relevant to CT perfusion quantification.

Quantification of perfusion requires measurement of the arterial input function (AIF), which is the function describing the input of the contrast agent to the tissue (see Chapter 26). Perfusion is then obtained by inverting the convolution expression (a mathematical process known as “deconvolution”), which removes, from the tissue concentration time course, the temporal spread contribution associated with the AIF. This chapter focuses on (a) the practical issues regarding this deconvolution step, (b) the factors that need to be taken into account, and (c) how it can be used to obtain absolute measurements of perfusion and other hemodynamic parameters.

The measured tissue tracer concentration depends not only on cerebral blood flow (CBF) but also on the way the contrast agent enters the tissue. In fact, two tissue areas (or two patients) could have exactly the same tissue concentration curves but very different perfusion status: In one case, the bolus could arrive very fast and with a narrow shape but then have a slow clearance owing to severe hypoperfusion; in the other case, the bolus could arrive very slowly and with a spread shape but could be cleared very quickly owing to normal perfusion or hyperperfusion relative to the previous patient. The two situations are obviously very different, but one cannot distinguish them based on the tissue concentration changes over time alone. To differentiate them, the contribution of each patient’s AIF must be taken into account; this is the role of the deconvolution step. Because of its mathematical nature, deconvolution is often regarded as an obscure step by clinicians and nontechnical users. Although many technical aspects can indeed be very complex, the general concepts underlying the deconvolution step can be easily understood by a simple analogy: The audience attending a lecture. (See Geek Box 1 for a more technical description of deconvolution.)

Audience analogy: For an ideal 60-minute lecture, the audience number would be a step function (Fig. 25.1A): Everyone present until the 60-minute mark, and no audience afterward. In practice, however, the audience is more likely described by Fig. 25.1B: Some may leave the room before the end, and some others may stay on to talk with the speaker. Importantly, the shape of this function is an intrinsic property of the lecture, a measure of its “quality”: The worse the lecture, the less people will remain in the audience, with a step function (Fig. 25.1A) representing a very good lecture (and a delta function at time t = 0 the extreme of a very bad lecture!). It is important to note, however, that Fig. 1A and B assume an “instantaneous” audience into the room; that is, everyone arrived at t = 0. In practice, this is never the case, and some arrive late, after the lecture has started. To investigate this effect, consider a lecture theater with one person at every door counting the number of people entering the room. After the doors are opened at t = 0, the total number of people entering the room would look like the function in Figure 25.1C (i.e., most people enter early, although some will continue to enter at later times). If now the number of people remaining in the audience is plotted (Fig. 25.1D), it will be a wider version of
Figure 25.1C, with some people staying until the end, but some others leaving (bored!) before the end. Contrary to Figure 25.1B, Figure 25.1D is no longer a sole measure of the “quality” of the lecture: Its shape reflects also the function describing the audience entrance into the room. To isolate the function solely describing the “quality” of the talk (i.e., Fig. 25.1B, which could only be measured in an ideal situation of an “instantaneous” audience entrance), the contribution of the function describing the audience entrance must be removed. In the perfusion imaging analogy, the theater corresponds to the tissue voxel, the audience corresponds to the contrast agent concentration, the function describing the audience entrance (Fig. 25.1C) corresponds to the AIF, the number of people remaining in the room (Fig. 25.1D) corresponds to the tissue concentration, and the function describing the “quality” of the talk for an ideal “instantaneous” audience (Fig. 25.1B) corresponds to the so-called impulse response function. Finally, the deconvolution in perfusion imaging corresponds to calculating from Figure 25.1D the function describing the “quality” of the talk (Fig. 25.1B) by removing from Figure 25.1D the contribution from the audience entrance (Fig. 25.1C).

As discussed previously, deconvolution inherently requires a numerical inversion. Therefore, the operation 1/C_A(f) (for frequency-domain deconvolution) or the matrix inversion (for deconvolution in time domain) leads to a so-called ill-conditioned deconvolution if the components in question are very small or zero. The problem with the
matrix formulation can be more easily appreciated when the singular value decomposition (SVD)¹ is used for solving the matrix inversion. SVD factors the matrix AIF into three matrices: AIF = U.W.V^*, where U and V are orthonormal matrices, W is a rectangular diagonal matrix containing the singular values (σ_ij) in decreasing order, and ^* indicates conjugate transpose. Then, the inverse (for square matrices) or pseudoinverse (for rectangular matrices) of A can be determined as AIF⁺ = V.W⁺.U^*, where AIF⁺ is the sought (pseudo)inverse of the matrix AIF, and W⁺ is the inverse of diagonal matrix W. Per definition, the diagonal matrix W⁺ can straightforwardly be formed by replacing every diagonal entry σ_ii of W by its reciprocal (i.e., 1/σ_ii) and then by transposing this resulting matrix. The columns of U and V can be interpreted as basis functions (with increasing frequency for decreasing singular value index j): The SVD solution thus corresponds to a decomposition of the residue function in this basis function set. It therefore follows that the contribution from the high-frequency components associated with very small (or zero) σ_ii will be greatly amplified.

Similarly, computing 1/C_A(f) in the frequency-domain deconvolution is not meaningful if the components are very small or zero. In theory, the components of C_A(f) or W should be always nonzero because they represent a spectrum of a nonperiodic signal. In practice, however, some of these components can end up being zero (or very close to zero) because of noise. Computation of direct reciprocals of these components will result in infinite or nearly infinite magnitudes of components in the solution, leading ultimately to physically implausible solutions for r(t). More specifically, the components of C_A(f) and W that are mostly affected by noise are the high-frequency components, because for typical shapes of the bolus curve, the CTC contains only little power in the high frequencies.

Stabilizing the solution to obtain physically meaningful estimates for r(t), CBF, and mean transit time (MTT) requires treatment of the noise-corrupted components. Most typically, these components are ignored in deconvolution—an operation that is termed regularization. The resulting ř(t) (where ř denotes the regularized version of the solution) is then computed with zero values in locations of W⁺ or 1/C_A(f); that is, in locations where noise-corrupted components were originally present, r is approximated with zero-amplitude basis functions: CBF·ř = AIF⁺·c_T. Therefore, because CBF is estimated from the initial value of the measured IRF (see Eq. [6]), the regularization process [which distorts the shape of r(t)] will have an effect on the estimated perfusion values (see Fig. 25.3). In essence, truncated (or regularized) SVD acts like a low-pass filter. The harsher the regularization filter, the smoother the resulting residue function and thus the more difficult for the measured function to reflect sudden changes in the residue function (e.g., at t = 0). This, in turn, will affect the ability to correctly determine the point in time when the maximum of the residue function will be occurring.

There are two main approaches to solving the ill-posed deconvolution problem (Eq. [1]): parametric (model-dependent) deconvolution and model-independent deconvolution. In model-dependent deconvolution, the solution to Eq. [1] is constrained by assuming that r(t) is described by a mathematical function, for example, r(t) = exp(−t/MTT), to model a well-mixed compartment.¹ In this example, only the free parameter MTT needs to be determined. More complex models can include parameters for arterial delay and dispersion,^6,7 or by modeling the microvasculature dynamics (e.g., Mouridsen et al.⁸). Model-dependent deconvolution has the advantage of ensuring smooth monotonically decreasing estimates of r(t). on the other hand, it can be very sensitive to noise,⁹ and increasingly so for the flexible models containing a larger
number of free parameters. Furthermore, if the in vivo situation differs from the assumed model, there will be large systematic CBF errors.¹ For this reason, model-independent deconvolution approaches are generally preferred and more widely used. In this approach, both CBF and the residue function are considered unknowns (see the fourth Geek Box for a brief description of the most commonly used methods).