To assess in-vivo which vascular transfer function (VTF) better describes the brain vascular dynamic in DSC-MRI.

Dispersion is a physiological phenomenon present in DSC-MRI data and its characterization is fundamental to assess the reliability of hemodynamic parameters while potentially revealing pathological conditions¹. Dispersion affects the time-dependent residual amount of tracer $R(t)$ calculated for each voxel via deconvolution of the measured arterial $C_a(t)$ and tissular $C_{ts}(t)$ concentrations^2,3. Mathematically the computed effective residual amount is represented as the convolution $R^{eff}(t)=R\otimes{VTF}(t)$, where $VTF(t)$ is the probability density of the transit times $t$ from the arterial measurement location to the voxel where $C_{ts}(t)$ is measured. A recent state-of-the-art technique⁴, henceforth CPI+GDK, disentangles the contributions of VTF and $R(t)$ by estimating a set of control points whose interpolation is convolved to a gamma dispersion kernel (GDK) that is assumed as model for the VTF $$GDK(t,s,p)=\frac{s^{1+sp}}{\Gamma(1+sp)}t^{sp}e^{-st}$$ where $s,p$ are unknown. However the true shape of VTF is unknown, therefore the GDK assumption may not be true.

We propose instead to perform deconvolution with Dispersion-Compliant Bases⁵ (DCB) which make no assumptions on the VTF. We then implement variants of CPI+GDK for the well-known exponential (CPI+EDK) and log-normal (CPI+LNDK) dispersion kernels⁴, to see which one among the three better describes the DCB results in-vivo. We finally obtain maps of the brain showing for each voxel what is the dispersion kernel that better describes the data.

We perform DCB deconvolution on a sampling grid $t_1,t_2,..,t_M$ representing the effective $R(t)$ as⁵ $$R^{eff}_{DCB}(t) = \Theta(t-\tau) \sum_{n=1}^{N} [a_n + b_n (t-\tau)] e^{-\alpha_n (t-\tau)}$$ with order $N$, and $\tau,a_n,b_n$ unknown with $\alpha_n$ predefined⁵. Differently from literature the solution is constrained via quadratic programming to $R^{eff}_{DCB}(t_m)\ge0\forall{t_m\in[t_1,t_{M-1}]}$, and $R^{eff}_{DCB}(t_M)=0$. $N=6$ was found sufficient.

For the CPI-based deconvolution we extend CPI+GDK⁴ by substituting the VTF model with an exponential (EDK) and a lognormal (LNDK) kernels which are often considered in literature⁴ $$EDK(t,\theta)=\frac{1}{\theta}e^{-\frac{1}{\theta}}$$ $$LNDK(t,\mu,\sigma)=\frac{1}{t\sigma\sqrt{2\pi}}e^{-\frac{(\ln(t)-\mu)^2}{2\sigma^2}}$$ where $\theta,\mu,\sigma$ are unknowns. The CPI+GDK was implemented as in literature⁴ with 12 control points and initial parameters $p,s$ for the optimization routine $log2\pm2$ ($mean\pm{SD}$). Similarly the CPI+EDK and CPI+LNDK were initialized with $\beta=1s\pm2s$ and $\mu=-1\pm1,\sigma=1\pm1$, which corresponds to low dispersion⁴. Non-linear estimation was performed bounding parameters to $mean\pm3SD$. We then proceed with two steps.

1. We perform synthetic experiments showing that DCB deconvolution performs comparably or better than CPI+GDK/EDK/LNDK when the ground truth dispersion kernel is GDK, EDK or LNDK respectively. Data was generated according to literature^2,5 with $SNR=50$⁴. We tested three dispersion levels low, medium, high⁴, $BF\in[5:10:65]ml/100g/min$, $MTT\in[2:4:18]s$, delay$\in[0,5]s$⁴ (100 repetitions for each combination).

2. We apply DCB deconvolution on stroke MRI data to obtain the estimated $R^{eff}_{DCB}(t)$ for each voxel; we calculate the re-convolution $C_{ts}^{*}(t)=C_a\otimes{R^{eff}_{DCB}(t)}$; we perform deconvolution with the three CPI+ techniques on $C_a(t)$ and $C_{ts}^{*}(t)$ obtaining $R^{eff}_{GDK}(t)$, $R^{eff}_{EDK}(t)$ and $R^{eff}_{LNDK}(t)$; for each voxel we select the CPI+ technique providing the lowest $l^2$ reconstruction error of $C_{ts}^{*}(t)$, and the best estimates of the effective blood flow $BF^{eff}$ ($R^{eff}(t)$ peak) and mean transit time $MTT^{eff}=(Blood Volume)/BF^{eff}$ compared to DCB results.

Synthetic results in Fig. 1 show the comparisons of DCB with CPI+GDK/EDK/LNDK when the dispersion kernels are respectively GDK, EDK and LNDK. We note that $MTT^{eff},BF^{eff}$ estimates with DCB have lower or similar relative error. Results on real data in Fig. 2 show the clustering of a slice where each pixel is colored according to the CPI+ technique that better reproduces $MTT^{eff},BF^{eff}$ obtained with DCB and the $l^2$ reconstruction error. Fig. 3 reports the $BF^{eff}$ DCB-based map, which unveils an infarcted region in the right hemisphere.In-vivo results show that the presence of the GDK, EDK, LNDK dispersion kernels is widespread in the brain. Each clustered region looks structured and remains homogeneous among the tested parameters (see Fig.2 “all”). We note that the three dispersion kernels seem linked to as many tissue areas/types. Moreover GDK is mainly present in the infarcted region (see right hemisphere in Fig. 3). Synthetic results (Fig. 1) demonstrate that DCB allow to perform reliable deconvolution without assumptions on the shape of VTF. At the same time the use of DCB opens for a-posteriori VTF characterization, for instance with the three CPI+ based techniques.We present results suggesting that different brain regions and tissue conditions support different descriptions of the underlying dispersion kernel (VTF), highlighting the need of estimating the effective residue amount of tracer $R^{eff}(t)$ without a-priori assumptions on the VTF, that is the case of DCB deconvolution⁵. We further believe that successive dispersion kernel detection and characterization based on VTF models can constitute a new bio-marker to shed light on the tissue vascular dynamic.