Eirini Messaritaki^{1}, S Umesh Rudrapatna^{2}, Najmus S Iqbal^{1}, Hannah Furby^{1,2,3}, Emma Yhnell^{1}, Claudia Metzler-Baddeley^{2}, William P Gray^{1}, and Derek K Jones^{2,4}

Convection-enhanced drug delivery to the human brain is a promising method for treating neurodegenerative diseases and brain tumors. Accurate predictions of the drug concentration via computational fluid dynamics models are essential, and models not accounting for diffusion non-Gaussianity give predictions that are not in good enough agreement with experimental results. We use a fluid dynamics model recently presented in the literature, which accounts for diffusion non-Gaussianity, to calculate the differences with models that do not account for it, in data from pre-symptomatic Huntington’s disease patients and metastatic brain tumor patients. We make recommendations on the use of fluid dynamics models in the clinical setting.

Fluid dynamics
theory predicts^{9} that the difference in drug concentration $$$\Delta C$$$ between models that include
diffusion non-Gaussianity and those that do not evolves with time $$$t$$$ as:

$$\varphi \frac{ \partial( \Delta C)}{\partial{t}} + \nabla \cdot \Big ( \textbf{v}_\textrm{R} \Delta C \Big ) -\nabla \Big [ \varphi \textbf{R} \cdot \nabla( \Delta C) \Big ] = - \nabla \cdot \Big ( \Delta \textbf{v} \cdot C_\textrm{D} \Big ) + \nabla \cdot \Big [ \varphi( \textbf{R} - \textbf{D} ) \cdot \nabla C_\textrm{D} \Big ].$$

$$$\textbf{D}$$$ is the diffusion
tensor, $$$\textbf{R}$$$ the diffusion-displacement tensor incorporating the
diffusion propagator^{10} , $$$\varphi$$$ the porosity, $$$C$$$ the drug concentration
and $$$\textbf{v}$$$ the drug velocity. The subscripts R and D indicate quantities derived using
the $$$\textbf{R}$$$ and $$$\textbf{D}$$$ tensors respectively. The complex dependence of $$$\Delta C$$$ on several parameters necessitates individual-participant simulations in order to quantify it.

We acquired MRI data from participants who gave written, informed consent. All procedures were approved by the appropriate Ethics Committees. Pre-HD patients were scanned in a 3T Siemens Prisma scanner with 80mT/m gradients, while MBT patients were scanned in a 3T Siemens Connectom scanner with 300mT/m gradients. The parameters of the scans are listed in Figure 1.

CFD simulations were run for
72 hours of infusion, using finite element methods as described in ^{9}. Models were run with (a) the diffusion tensor and (b) the diffusion propagator, and the differences in the drug concentrations were computed. These differences were correlated with voxel-wise estimates of the fractional anisotropy (FA). For pre-HD
patients, the infusion point was in the striatum, which exhibits
abnormalities in pre-HD and is a potential treatment target. For MBT patients, the infusion point was
inside the tumor.

Pre-HD patients: At each brain location, the differences in predicted concentrations between the two models ranged between -56% and +29% of the maximum concentration (Figure 2). There were differences in the structures covered by the drug when using the diffusion-tensor representation compared to when using the diffusion-propagator representation (Figure 3).

MBT patients: The error in concentration ranged between -18% and +10% of the maximum concentration (Figure 4).

For all participants, the error in concentration was statistically significantly correlated (uncorrected p-values < 10^{-10})
with the mean FA of the diffusion tensor of the voxels traced by the drug. The correlation coefficients were about 0.40 for the pre-HD patients, and about 0.25 for the
MBT patients. Therefore, the higher the anisotropy of the treated tissue, the larger the differences in the concentration between the diffusion-tensor and the diffusion-propagator CFD models. This is expected given that the diffusion non-Gaussianity captured by the diffusion propagator is high in brain regions of high fractional anisotropy.

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Figure 1: MR scanning parameters.

Figure 2: Differences in concentration as a fraction of the maximum concentration $$$C_{\textrm{max}}$$$: $$$f = \frac{C_R - C_D}{C_{\textrm{max}}}$$$, for two representative coronal slices of one of the pre-HD patients. The section of the slices is shown in the small figure at the bottom. For this participant, the differences range from +0.25 to -0.40 of the maximum concentration.

Figure 3: Distribution volume for two representative coronal slices of one of the pre-HD patients. (a) diffusion-propagator CFD model, contour of 10% of $$$C_{\textrm{max}}$$$, (b) diffusion-tensor CFD model, contour of 10% of $$$C_{\textrm{max}}$$$, (c) diffusion-propagator model, contour of 90% of $$$C_{\textrm{max}}$$$, (d) diffusion-tensor model, contour of 90% of $$$C_{\textrm{max}}$$$. The volumes covered with 90% of $$$C_{\textrm{max}}$$$, which are the volumes that are considered treated by the drug, are very different for the two models. Structures that are predicted to be covered according to the diffusion-tensor CFD model, are not covered when diffusion non-Gaussianity is taken into account.

Figure 4: Differences in concentration as a fraction of the
maximum concentration: $$$f = \frac{C_R - C_D}{C_{\textrm{max}}}$$$, for
two representative coronal slices of one of the cancer patients. The
section of the slices is shown in the small figure at the bottom. For this participant, the
differences range from +0.1 to -0.18 of the maximum concentration.