Improved Vascular Transport Function Characterization in DSC-MRI via Deconvolution with Dispersion-Compliant Bases

Marco Pizzolato^{1}, Rutger Fick^{1}, Timothé Boutelier^{2}, and Rachid Deriche^{1}

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 Function^{1,2}.
This severely affects the estimation of hemodynamic parameters^{3} 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 technique^{4} 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
Bases^{5} (DCB), and subsesquently fit the CPI+VTF^{4} model to the
obtaned effective residue function.

We perform DCB deconvolution representing $$$R^{eff}(t)$$$ on a
sampling grid $$$t_1,t_2,..,t_M$$$ as^{5}
$$R_{DCB}(t) = \Theta(t-\tau) \sum_{n=1}^{N} [a_n + b_n (t-\tau)]
e^{-\alpha_n (t-\tau)}$$
with order $$$N$$$ ($$$6$$$ here), $$$\tau,a_n,b_n$$$ unknown an
$$$\alpha_n$$$ predefined^{5}. 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 literature^{4}
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 function^{1}.The tissular concentration $$$C_{ts}(t)$$$ was
generated as $$$C_{ts}(t)=C_a\otimes[R\otimes{VTF}(t)](t)$$$ with
bi-exponential^{5} $$$R(t)$$$. Three ground-truth VTF models
were used^{4}: gamma (GDK), exponetial and log-normal. For each, three
dispersion levels were tested: low, medium, high^{4}. 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 added^{5} 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 DCB^{5}, CPI+VTF^{4} and oSVD^{6} 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).

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.

1. Calamante et al., “Delay and dispersion effects in dynamic susceptibility contrast mri: simulations using singular value decomposition,” Magn Reson Med, vol. 44(3), pp. 466–473, 2000

2. Calamante et al., “Estimation of bolus dispersion effects in perfusion mri using image-based computational fluid dynamics,” NeuroImage, vol. 19, pp. 341–353, 2003.

3. Willats et al., “Improved deconvolution of perfusion mri data in the presence of bolus delay and dispersion,” Magn Reson Med, vol. 56, pp. 146156, 2006.

4. Mehndiratta et al., “Modeling and correction of bolus dispersion effects in dynamic susceptibility contrast mri: Dispersion correction with cpi in dsc-mri,” Magn Reson Med, vol. 72, pp. 17621774, 2013.

5. Pizzolato et al., “Perfusion mri deconvolution with delay estimation and non-negativity constraints,” in 12th International Symposium on Biomedical Imaging (ISBI). IEEE, 2015, pp. 1073–1076.

6. Wu et al., “Tracer arrival timinginsensitive technique for estimating flow in mr perfusionweighted imaging using singular value decomposition with a blockcirculant deconvolution matrix,” Magn Reson Med, vol. 50(1), pp. 164–174, 2003.

Relative absolute errors boxplots for the effective blood flow $$$BF^{eff}$$$ (peak of the effective residue function) computed with oSVD^{6}, CPI+VTF^{4 }and DCB^{5}. **Left.** combined results when gamma (GDK), exponential and log-normal dispersion kernels are used as ground-truth for the Vascular Transfer Function (VTF). **Right.** only the gamma kernel is used.

Relative absolute errors boxplots for the effective mean transit time $$$MTT^{eff}=BV/BF^{eff}$$$ (with $$$BV$$$ the blood volume) computed
with oSVD^{6}, CPI+VTF^{4} and DCB^{5}. **Left.**
combined results when gamma (GDK), exponential and log-normal
dispersion kernels are used as ground-truth for the Vascular Transfer
Function (VTF). **Right.** only the gamma kernel is used.

Relative absolute errors boxplots for the time-to-maximum $$$T2MAX$$$ of the effective residue function computed
with oSVD^{6}, CPI+VTF^{4} and DCB^{5}. **Left.**
combined results when gamma (GDK), exponential and log-normal
dispersion kernels are used as ground-truth for the Vascular Transfer
Function (VTF). **Right.** only the gamma kernel is used.

Estimation of the $$$BF,p,s,$$$ parameters with CPI+VTF deconvolution^{4} (purple) and with the fitting of the CPI+VTF model on the the effective residue function obtained with DCB deconvolution^{5} (DCB+VTF). The ground-truth data was generated with the gamma dispersion kernel (GDK). Results shown for low, medium, high and all dispersion levels^{4}.

Maps of the actual blood flow $$$BF$$$ and of the parameters $$$p,s$$$ characterizing a gamma Vascular Transport Function obtained after fitting of the CPI+VTF model on the the effective residue function estimated with DCB deconvolution^{5} (DCB+VTF). Maps reveal an infarcted region in the right hemisphere particularly evident in the $$$s$$$-map.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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