Ulin Nuha Abdul Qohar^{1}, Erik Andreas Hanson^{1}, Steven Sourbron^{2}, and Antonella Zanna Munthe-Kaas^{1}

^{1}Mathematics, University of Bergen, Bergen, Norway, ^{2}University of Sheffield, Sheffield, United Kingdom

In this study, we present a simulation framework capable of generating synthetic reference perfusion MRI data suitable for evaluation and comparison of tracer kinetic models. The framework consists of a graph-based contrast agent flow model with a vascular geometry and allows for controlled simulations with realistic structural and vascular parameters. We demonstrate the potential application of the proposed framework by performing a comparison between traditional pharmacokinetic models of varying complexity, by studying the effect of ROI size.

The proposed model is based on a coupling between components on various spatial scales. The observable vascular networks above 30-micron radius were described using graph model

$$ \sum_{j \in N_i} R_{ij}\gamma \left(P_i-P_j\right) = q_i,$$

where $$$N_i, R_{ij}, \gamma$$$, and $$$q_i$$$ are the connected neighbour nodes to node $$$i$$$, the vessel resistance, the capillary resistance constant(1 for arteries and veins) and the inlet/outlet flow to the node $$$i$$$ (zero for inner nodes), respectively.

The contrast agent was assumed to move passively with the bloodstream obtained from the graph model. The CA influx into a small distribution volume $$$\Omega_k$$$ was defined by the product of CA concentration ($$$C_k$$$) and blood flow influx ($$$\textbf{q}_k$$$). It is equivalent to the CA change rate in the distribution volume $$$\Omega_k$$$

$$\frac{d}{dt}\int_{\partial \Omega_k}Cd\textbf{x} = -\frac{1}{\Omega_k} \int_{\partial \Omega_k}C_k (\textbf{q}_k \cdot \textbf{n}) d\textbf{x},$$

where index $$$k$$$ represents a segment in arteries, veins and capillaries. A bolus injection in the arterial root vessel was simulated using a gamma variate function,

$$C_{AIF}(t)=C_0(t - t_0)A.e^{-(t-t_0)/B}$$

for constants $$$t_0 = 6$$$s; $$$A = 3; B = 1; C_0 = 1$$$mM.

For illustration, the approach was used to build a CFD model based on a frog tongue vasculature segmentation (Figure 1). The blood circulation was obtained by solving the model based on equation (1). A synthetic perfusion series with a total acquisition time of 120s was generated using the steps outlined above.

The images were analyzed using four models: Maximum Slope (MS), One-compartment Model (1CM), two-compartment Uptake (2CUM) and Exchange Model (2CXM)

Figure 4 shows the error in the PF values for each model with a local AIF. The MS model was systematically underestimating PF values. Both the 2C models and the 1C gave similar results for all ROIs, except in ROI 2. This is not unexpected since the model considers intravascular CA only and the kinetics is therefore essentially one-compartmental in nature. Figure 5 shows that the global AIF yields lower PF estimation, consistent with the effects of bolus dispersion.

PV estimation results from 2CXM and 2CUM with local AIFs were relatively accurate with the average error of 4% and 3.9%, respectively. The estimations were robust to AIF selection with the global AIF giving a slightly larger average error (4.7% and 4.4%).

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The segmented vascular structures and ROIs in the experiment: Each colour represents a different ROI. Several ROIs are overlapping (ROI 1,3, and 4; 2 and 10; and 5 and 11) for error analysis in similar areas with different ROI size. ROI 6 and 7 are the two regions with biggest error among 11 ROIs, and the blue region (ROI 8) has the most accurate estimation for both one and two-compartments model.

Simulated CA flow in the vascular system after bolus injection. The flow patterns exhibit a natural behaviour with a flow from the arterial network roots feeding the whole vasculature, spreading to the capillaries and a washout through the veins. The images were captured at 6, 8, 10, 15, 20, 25, 35, 50, 70 and 100 second respectively (from left to right).

The simulated CA signal without noise averaged over ROI 1. The curve indicates the CA concentration from 0s to 60s acquisition time, which shows a short CA delay after bolus injection. This data is resembling the perfusion signal in clinical imaging. Four tracer kinetic models were used for quantifying perfusion from the time curve using a local AIF. The signal fittings for all models were well-performed with good fitting. MS fitting reveals the CA curve's maximum slope location and the slope magnitude and the remaining fittings coincide and generate an identical result.

Plasma flow (PF) estimation results using four tracer kinetic models in 11 ROIs. The MS model is systematically underestimating PF, but generate the best estimation among the four models in terms of absolute values. The remaining models are, in general, overestimating PF and generate close to identical values, except for ROI 2. This reveals that the 2-compartment models are over-parameterized and not suited for estimations in our CA flow phantom. The proposed flow model only accounts for blood circulation in the intravascular structures.

Since both 2CXM and 2CUM estimation results were almost identic with the 1CM. Only 1CM and MS fitting were analyzed for comparison between local and global AIF. PF estimation using a global AIF is in general lower than for a local AIF. It provides smaller errors for 1CM (due to overestimating) and larger errors for MS conversely. These results may be caused by bolus dispersion and are related to the distance between the location of global and local AIF. The global AIF has a higher concentration peak and longer delay to the ROI, compared to local AIF.