Leili Riazy^{1}, Marc Olbrich^{2}, Tobias Schaeffter^{3}, Simone Fritschi^{4,5}, Thoralf Niendorf^{1}, and Jeanette Schulz-Menger^{4,5}

Hypertrophic Cardiomyopathy (HCM) is a common cause of sudden heart death in young adults. MRI is a powerful tool for the diagnosis and surveillance of this myocardial morphology as well as tissue injury. In clinical routine the assessment is mainly based on visual assessment or semi-quantification is increasingly used. Quantification of perfusion defect should be nowadays assessed with computational tools. We aim at quantifying differences in perfusion with a computational flow model that incorporates the vascular, as well as extracellular compartment, using the Damköhler Number $$$Da$$$. Areas of different perfusion in N=5 patients with HCM were fitted in $$$Da$$$ with model-derived curves with an overall error of 10.48%.

A 3D virtual phantom was implemented in MATLAB (The Mathworks, Natick, USA) using the “threed” toolbox for the finite element method provided by [5]. For the computational modeling of contrast agent flow, a 3D convection-diffusion partial differential equation was used[6]. It determines the amount of contrast agent $$$c$$$ in the vascular space $$$c^v$$$ and the extracellular space $$$c^e$$$ by $$ \frac{\partial c^v}{\partial t} + u^v \cdot \nabla c^v - \alpha\nabla^2 c^v + Da(1-\phi)(c^v-c^e)=\frac{AIF}{\phi} $$ $$ \frac{\partial c^e}{\partial t} - \beta\nabla^2 c^e - Da\phi(c^v-c^e)=0 $$ where $$$\alpha, \beta \in \mathbb{R}$$$ are diffusion weights, $$$Da \in \mathbb{R}$$$ is the Damköhler Number and together with $$$ \phi \in \mathbb{R}$$$ describe the exchange between vascular and extracellular space, $$$u^v:\mathbb{R}^3\rightarrow \mathbb{R}$$$ is the blood flow velocity and $$$AIF:\mathbb{R}^3\times\mathbb{R}\rightarrow \mathbb{R}$$$ is the arterial input function, which is assumed to be given by the Gaussian $$ \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{t-T_{peak}}{\sigma}\right)^2} ,\, \sigma \in \mathbb{R}_{+}, T_{peak}\in\mathbb{R}\,.$$ All parameters and functions are extensively discussed in [6]. The solution $$$c=(c^v,c^e)$$$ was numerically estimated for different Damköhler Numbers $$$Da \in \mathbb{R}$$$ using the finite element method on 3D tetrahedrons with a Hat-Function basis. A cube $$$[0, 10]^3$$$ was discretized $$$11 \times 11 \times 11$$$ with inflow node at $$$(2, 5, 5)$$$ and sampling point $$$(5, 5, 5)$$$.

N=5 patients
with HCM were scanned at 3.0T (MAGNETOM Verio, Siemens Healthcare, Erlangen,
Germany) using stress perfusion imaging (Saturation Recovery Sequence, Gradient Echo, TR=2.05 ms, TE=0.98 ms, FA=17°, FOV= (272×340)
mm2, matrix=160×128, slice thickness=6mm) with contrast-agent
application (0.2 mmol/kg body weight, Gadobutrolum). Three areas (as shown in figure
1)) with differences in perfusion were selected by an MR-experienced
cardiologist and their signal intensity-time curves were analyzed. The measured
perfusion curves are fitted with the numerically simulated ones using a
least-squares approach to determine the Damköhler
Numbers as a quantitative measure for
the exchange between the two compartments. Parameters ($$$ \alpha=1, \beta=2,
\sigma=5, \phi=0.14$$$) were assumed, as well as a unit blood flow velocity in
x-direction. Contrast agent concentrations for $$$Da \in [0.01, 1.5]$$$ were
calculated into a large database and used for fitting to the curves shown in
figure 1. The simulation data is given in arbitrary units and is therefore
uniformly scaled and shifted at the signal intensity peak to fit the measured
data. The central curves, i.e. the simulation for $$$Da=0.5$$$ and the red
measurement curve were used for this alignment.

[1] Maron et al., Cardiology patient pages. Hypertrophic cardiomyopathy. (2002) Circulation

[2] Jerosch-Herold et al., Magnetic resonance quantification of the myocardial perfusion reserve with a Fermi function model for constrained deconvolution. (1998) Med. Phys.

[3] Vallée et al., Quantification of myocardial perfusion by MRI after coronary occlusion. (1998) MRM

[4] Ishida et al., Quantification of myocardial blood flow using model based analysis of first-pass perfusion MRI: extraction fraction of Gd-DTPA varies with myocardial blood flow in human myocardium. (2011) MRM

[5] Jeff Borggaard, https://people.sc.fsu.edu/~jburkardt/m_src/threed/threed.html

[6] Cookson et al., A spatially-distributed computational model to quantify behaviour of contrast agents in MR perfusion imaging. (2014) Medical Image Analysis

[7] Utz et al., Contrast-dose relation in first-pass myocardial MR perfusion imaging. (2007) JMRI

Stress perfusion images of one HCM patient in the basal
slice. a) shows an image where perfusion deficit is manifested. b) shows
selected regions of different perfusion. c) shows the signal-time-course of the
selected regions in corresponding colors.

Simulation
results at sampling point $$$(5,5,5)$$$ for different values of $$$Da$$$. An
increase in $$$Da$$$ shows higher participation of extracellular contrast agent
concentration $$$c^e$$$ to the signal intensity.

Plot
of simulated contrast agent concentration for different values of $$$Da$$$.

Fitted
simulated signal intensity plotted against measured signal intensity at green,
red and blue colored regions specified in figure 1. Fitted value is given above
the image.