Summary of findings

We compared the $$$k_{w}$$$ obtained using conventional means as well as QTMnet in simulated brain diffusion-weighted arterial spin labeling images. The QTMNet method gives 3.7% NRMSE in quantification on $$$k_{w}$$$ compared to 31.2% for the traditional model.

Introduction

Blood-brain-barrier (BBB) water exchange rate ($$$k_{w}$$$) is an emerging biomarker of the BBB dysfunction due to the aging or aging-related disorders¹. Traditional tracer kinetic model based on single-pass approximation (SPA) ignores the tracer concentration exchange from extravascular extracellular space to the vascular space². However, ignoring the backflow of the tracer introduces a systematic error and the non-linearity of the SPA makes the inverse problem hard to solve. Here we use QTMNet, a biophysical-modeling-based deep learning method, to quantify $$$k_{w}$$$. We demonstrate that on the simulated brain ASL images, QTMNet achieves a 90% reduction in NRMSE compared to the traditional SPA model.

Methods

In QTMnet, a Unet is trained to learn the inverse mapping from diffusion-weighted arterial spin labeled (DW-ASL) to the water exchange rate ($$$k_{w}$$$). The training data for QTMnet consists of simulated data within a cube-shaped tissue of size 32x32x32mm3 obtained as follows. We first extract large vessels from an example MR angiography image and use constrained constructive optimization^3-5 to generate synthetic micro-vessel branches. The radius of mother and daughter vessels follows cubic rules: $$$r^3_M=r^3_{d1}+r^3_{d2}$$$⁴. Meanwhile, unlike the common plug flow assumption⁶, we applied the quadratic blood velocity profile $$$\textbf{u}(r)$$$⁷ which is more realistic for vessel flow. The tracer concentration profile is computed by solving the following transport equations:
$$\partial_t c_c(r,t)=\nabla\cdot(-\mathbf{u}(r) c_c(r,t))+k_w(r)(c_b(r,t)-c_c(r,t))-R_{1a}c_c(r,t)$$
$$\partial_t c_b(r,t)=-k_w(r)(c_b(r,t)-c_c(r,t))-R_{1b}c_b(r,t)$$
Here $$$c_b (r,t)$$$ and $$$c_c (r,t)$$$ are the tracer concentration inside brain tissue and capillary, respectively, $$$R_{1a}=0.6 s^{-1}$$$ and $$$R_{1b}=0.83 s^{-1}$$$ are the longitudinal relaxation rates of blood and brain tissue, $$$\mathbf{u}(r)$$$ is the velocity field, $$$\nabla$$$ is the gradient operator and $$$\partial_t$$$ is the derivative of time. $$$C(t)=\int_{V_c}c_c(r,t)dV+\int_{V_b}c_b(r,t)dV$$$ is the concentration of the voxel. $$$V_c$$$ and $$$V_b$$$ are the volume of the capillary and brain tissue, respectively. Finally, the tracer concentrations were summed over each voxel to obtain DW-ASL images. In the training, we minimize the L1 norm of the predicted $$$k_w$$$ from QTMNet $$$\Psi$$$ and the ground truth with regularization on $$$\nabla k_w$$$:
$$k_w(r)=argmin_{k_w}\|k_w(r)-\Psi(C(t))\|_1+\lambda\|\nabla k_w(r)\|_1$$
Conventional $$$k_w$$$ mapping was performed by fitting the simulated DW-ASL data to the signal model¹:
$$\Delta M_b=\frac{2CBF\epsilon M_0\beta}{\lambda}[\frac{e^{-(R_{1a}-R_{1b})ATT}}{R_{1b}}(e^{-R_{1b}(t-\delta)}-e^{-R_{1b} t})-\frac{e^{-(R_{1a}-\alpha)ATT}}{\alpha}(e^{-\alpha(t-\delta)}-e^{-\alpha t})]$$
$$\Delta M_c=-\frac{2CBF\epsilon M_0}{\lambda\alpha}e^{-(R_{1a}-R_{1b})ATT}(e^{-\alpha(t-\delta)}-e^{-\alpha t})$$
$$\Delta M(b)=\Delta M_b e^{-bD_b}+\Delta M_c e^{-bD_c}$$
Here, $$$\Delta M$$$ is the signal of the acquired ASL image; $$$\Delta M_b$$$ and $$$\Delta M_c$$$ are signals of tissue and capillary compartment, respectively. $$$\lambda = 1$$$ is the tissue/blood partition coefficient for the simulated data, and $$$\epsilon = 1$$$ is the tagging efficiency for simulated data. CBF is the cerebral blood flow, $$$\alpha = k_w+R_{1a}$$$ and $$$\beta = \frac{k_w}{k_w+R_{1a}-R_{1b}}$$$ are auxiliary parameters and $$$M_0$$$ is the proton density-weighted image. $$$\delta=1.5s$$$ is the labeling time, ATT is the arterial transit time for blood, and b is the b-value reflects diffusion gradient strength. $$$D_c=0.08 mm^2/s$$$ and $$$D_b=0.001 mm^2/s$$$ are the diffusion coefficients in the blood and brain tissue[8], respectively.
Given the DW-ASL images acquired with different b-value (0,10,20,50,100 $$$s/mm^2$$$) and different post-labeling delay (PLD=1200,1500,1800,2100 ms), The $$$k_w$$$ could be evaluated by minimizing the following cost function.
$$k_w(r)=argmin_{k_w}\sum_b\sum_{PLD}\|\Delta M(b,PLD)-f(k_w)\|_2^2+\lambda\|k_w(r)\|_2^2$$

Results

Figure 1 shows the simulated DW-ASL image frames at PLD=1200, 1500, 1800, 2100 ms under different b-values with the normalized concentration.
Figure 2a-c compares the $$$k_{w}$$$ ground truth of the simulated brain DW-ASL images with the quantitative results from the traditional SPA model and QTMNet, respectively. Figure 2d-e shows that QTMNet reduces the $$$k_{w}$$$ error compared to the traditional SPA models.
Figure 3a-d show the clinical images with $$$PLD=1200, 1500, 1800, 2100 ms$$$ and $$$b=0$$$. Figure 2e-f show the evaluation of $$$k_{w}$$$ using SPA model and QTMNet, respectively.

Conclusion

We demonstrate the feasibility of QTMNet method in $$$k_{w}$$$ quantification. The QTMNet method surpasses the traditional SPA model on the CFD-simulated DW-ASL data. The use of the biophysical CFD simulation contributes to the accurate $$$k_{w}$$$ mapping. Possible limitations of this work include selecting the $$$k_{w}$$$ values preset in the brain and cube generations. The inconsistency of the reported $$$k_{w}$$$ makes the value selection difficult and might introduce bias into the network.

Acknowledgements

We acknowledge the funding from National Institute on Aging (1 R01 AG080011-01A1) and National Institute of Biomedical Imaging & Bioengineering (1 R01 EB034755-01)

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