Assessment of Cerebral Blood Flow, Volume, and Mean Transit Time from Bolus-Tracking MRI Images: Theory and Practice - Advance Image Processing in Magnetic Resonance Imaging - page 582

and deconvolution reduces to a CBF and MTT parameter estimation problem.
However, this approach, by introducing strong assumptions on R(t) behavior, is
likely to introduce a bias in CBF estimates.
A more general model for R(t) was introduced by Ostergaard et al. [28], based
on a model of macrovascular transport and microvascular retention in the brain.
The model, originally introduced to describe tracer transport and retention in the
heart [29], describes the vasculature as a major feeding artery in series with 20 SVs
in parallel and allows to take into account delay and dispersion of the arterial input.
However, the vascular model was found sensitive to data noise level.
The model-independent approaches owe their name to the fact that they make
virtually no assumptions on the description of the unknown function to be decon-
volved. These methods are more powerful and less biased than the model-dependent
ones, but they have to deal with the ill posedness and ill conditioning of the
deconvolution problem. One of the simplest methods to solve the inverse problem
of Equation 19.18 is to use the convolution theorem of Fourier transform, which
states that the transform of two convolved functions equals the product of their
individual transforms:

FCBF Rt { ⋅ ( ) ⊗ C ( )} t = FC { ( )} t (19.27)

AIF VOI

From Equation 19.28, one obtains

=

"
$

FC {( )}

{( )}

t

VOI

CBF R t ⋅ () F

− 1

(19.28)

!

FC t

AIF

where F −1 denotes the inverse of the Fourier transform F . The Fourier transform
approach has the attraction of being theoretically very easy to implement and
insensitive to delays between the AIF and the tissue. However, its use is not
without problems, and discordant results have been reported in the literature. For
instance, Ostergaard et al. [13] showed that the Fourier transform approach biases
CBF, in particular, underestimating it in case of high flow. They also showed that
Fourier transform approach has an inherent problem in arriving at the actual CBF
when the residue function has discontinuities. On the other hand, other researchers
found satisfactory estimates of CBF in comparison with other, more sophisticated
deconvolution techniques [30].
Another method to solve Equation 19.18 is to resort to a linear algebraic
approach. More precisely, assuming that tissue and arterial concentrations are
measured at equidistant time points, t i t so that CBF·R(t)

is reasonably approximated by a staircase function, a discrete version of
Equation 19.18 can be written in matrix form

= t i−1 + ∆ t, and choosing ∆

j

∑

C () t ≈ CBF ⋅ ⋅ ∆ t C () ( t ⋅ R t − t ) (19.29)

VOI j AIF i j i

i=0