Synopsis
Bolus dispersion phenomena affect the residue function computed via
deconvolution of DSC-MRI data. Indeed the obtained effective residue
function can be expressed as the convolution of the true one with a
Vascular Transport Function (VTF) that characterizes the dispersion.
The state-of-the-art technique CPI+VTF allows to estimate the actual
residue function by assuming a model for the VTF. We propose to
perform deconvolution representing the effective residue function
with Dispersion-Compliant Bases (DCB) without assumptions on the VTF,
and then apply the CPI+VTF on DCB results. We show that DCB improve
robustness to noise and allow to better characterize the VTF.Purpose
To improve robustness of dispersion kernel characterization in
DSC-MRI by means of Dispersion-Compliant Bases.
Introduction
The residual amount of tracer, i.e. the residue function $$$R(t)$$$
computed from deconvolution of the measured arterial $$$C_a(t)$$$ and
tissular $$$C_{ts}(t)$$$ concentrations, characterizes the tissue
perfusion.
However the actual arterial concentration may undergo dispersion.
This causes the effective residue function to reflect additional
vascular properties mathematically described by the convolution
$$$R^{eff}(t)=R(t)\otimes{VTF(t)}$$$ where VTF is the Vascular
Transport Function1,2.
This severely affects the estimation of hemodynamic parameters3 such
as the blood flow $$$BF$$$, corresponding to the peak of $$$R(t)$$$,
and the mean transit time $$$MTT=BV/BF$$$ with $$$BV$$$ the blood
volume. Indeed only effective parameters $$$BF^{eff}$$$ (peak of
$$$R^{eff}(t)$$$) and $$$MTT^{eff}$$$ are computed.
A recent state-of-the-art technique4 based on control point
interpolation, CPI+VTF, allows to recover the actual $$$R(t)$$$
assuming that it is convolved with a VTF described by a Gamma Dispersion
Kernel (GDK)
$$VTF(t,s,p)=GDK(t,s,p)=\frac{s^{1+sp}}{\Gamma(1+sp)}t^{sp}e^{-st}$$
where $$$s,p$$$ are unknown. This allows the estimation of the actual
$$$BF$$$4.
The estimation of $$$s,p,BF$$$ is not an easy task and requires a
non-linear optimization routine which results are sensitive to noise.
We propose to improve robustness and precision of some of the
estimates by performing deconvolution with Dispersion-Compliant
Bases5 (DCB), and subsesquently fit the CPI+VTF4 model to the
obtaned effective residue function.
Methods
We perform DCB deconvolution representing $$$R^{eff}(t)$$$ on a
sampling grid $$$t_1,t_2,..,t_M$$$ as5
$$R_{DCB}(t) = \Theta(t-\tau) \sum_{n=1}^{N} [a_n + b_n (t-\tau)]
e^{-\alpha_n (t-\tau)}$$
with order $$$N$$$ ($$$6$$$ here), $$$\tau,a_n,b_n$$$ unknown an
$$$\alpha_n$$$ predefined5. The solution was constrained
via quadratic programming to
$$$R(t_m)\ge0\forall{t_m\in[t_1,t_{M-1}]}$$$ and $$$R(t_M)=0$$$.
The CPI+VTF deconvolution technique was implemented as in literature4
with 12 control points and initial parameters $$$p,s$$$ for the
optimization routine $$$log2\pm2$$$ ($$$mean\pm{SD}$$$). In order to
decouple the influence of the estimation framework from the model the
estimation was performed non-linearly bounding parameters to
$$$mean\pm3SD$$$.
The CPI+VTF model was also fitted on the effective residue function
computed with DCB by minimizing
$$$||R_{DCB}(t)-{[CPI+VTF]}_{model}||^2$$$ over the control points,
time-instants separations, and parameters $$$BF,s,p$$$4.
We perform synthetic experiments generating $$$C_a(t)$$$ in
$$$[0:1:90]s$$$ as a gamma-variate function1.The tissular concentration $$$C_{ts}(t)$$$ was
generated as $$$C_{ts}(t)=C_a\otimes[R\otimes{VTF}(t)](t)$$$ with
bi-exponential5 $$$R(t)$$$. Three ground-truth VTF models
were used4: gamma (GDK), exponetial and log-normal. For each, three
dispersion levels were tested: low, medium, high4. Number 100
repetitions were generated for each combination of dispersion kernel,
level, $$$BF\in[5:10:65]ml/100g/min$$$, $$$MTT\in[2:4:18]s$$$, and
$$$delay\in[0,5]s$$$4 with noise added5 with
$$$SNR=50$$$4. For each repetition DCB and CPI+VTF deconvolutions
were performed, as well as the CPI+VTF model fitting on
$$$R_{DCB}(t)$$$, henceforth DCB+VTF.
We then proceed with the following experiments:
1. we compare DCB5, CPI+VTF4 and oSVD6 deconvolutions and calculate
the relative errors of the recovered effective parameters
$$$BF^{eff}$$$ (Fig. 1), $$$MTT^{eff}=BV/BF^{eff}$$$ (Fig. 2), and
time-to-maximum $$$T2MAX$$$ of the $$$R^{eff}$$$ (Fig. 3);
comparisons are performed on all of the ground-truth dispersion kernels
(left images) and just on the GDK (right images);
2. we compare estimates of $$$p,s,BF$$$ obtained with CPI+VTF and
DCB+VTF in case of GDK (Fig. 4);
3. we apply DCB+VTF on stroke MRI data and show maps of $$$p,s,BF$$$
(Fig.5).
Results
Results in Fig. 1,2,3-left show that DCB-based estimates of
$$$BF^{eff},MTT^{eff},T2MAX$$$ have sensibly lower relative error
than those obtained with CPI+VTF and oSVD. When the gound-truth
kernel is GDK (right columns) CPI+VTF sensibly improve. Still, DCB
perform comparably or better. DCB generally reduce errors and their
variability. In addition DCB results appears more stable than with
CPI+VTF with respect to the ground-truth kernel. Results in Fig. 4 show that
CPI+VTF and DCB+VTF render similar results for $$$BF,p$$$ but
DCB+VTF relevantly improves $$$s$$$ estimates at medium and high dispersion levels.
Maps in Fig. 5 well depict the infarcted area which is specially highlighted
in $$$s$$$-map.
Discussion
The use of DCB deconvolution renders a better estimation of the
effective hemodynamic parameters.
The DCB do
not assume any model for the dispersion kernel (VTF)
and can handle the exponential and log-normal kernels better than
CPI+VTF. The use of these functional bases improves robustness to
noise and reduces variability in the results (Figs. 1-3). This leads
to a relevant improvement also when the CPI+VTF technique is applied
directly on the DCB results (DCB+VTF), specifically in the
estimation of the $$$s$$$ parameter of the gamma kernel (Fig.
4). The same parameter
reveals to be very effective
in delimiting the infarcted area in
in-vivo results (Fig. 5).
Conclusion
Perfusion deconvolution of DSC-MRI data by means of
Dispersion-Compliant Bases (DCB) provides more robust results in
quantifying the effective residue function and improves subsequent
VTF assessments. This allows a better characterization of dispersion phenomena and a consequent deeper understanding of the vascular dynamic.
Acknowledgements
We thank Olea Medical and the PACA Regional Council for providing
grant and support.References
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