Synopsis

The blood flow quantification is of great importance in the diagnosis of many diseases. The current blood flow quantification methods are mainly based on the Kety’s method, which highly replies on the practically unmeasurable arterial input function (AIF). To void using the AIF as an input, we propose quantitative transport mapping (QTM) that quantitatively models blood flow in tissue according to the underlying biophysics of the transport equation. The inverse problem of QTM is to reconstruct the velocity field of the blood transportation using the time-resolved 4D tracer concentration data. As an example, with the approximation of kidney microvascular network, the renal blood flow (RBF) can be quantified by QTM.

Introduction

Theory

$$$\hspace{1cm}{\bf\text{1.Biophysical model for QTM}}\hspace{5mm}$$$QTM reconstructs a blood velocity field$$$\,{\bf u}({\bf r})\,$$$ from tracer concentration data $$$c({\bf r},t)\,$$$according to the local mass conservation law. Assuming that the velocity$$$\,{\bf u}({\bf r})\,$$$is constant in time, the Fokker-Planck equation holds:

$$\frac{\partial c}{\partial c}=-\nabla\cdot(c{\bf u})-\gamma c+\nabla\cdot(D\nabla c),\hspace{3cm}(1)$$

where$$$\,\gamma\,$$$is the concentration decay rate and$$$\,D\,$$$the diffusion coefficient. Eq.1 describes the convection, decay, and diffusion effects of the tracer in the blood microvasculature. The values of$$$\,\gamma\,$$$and$$$\,D\,$$$depend on the tracer type. In this study we assume $$$\,D=0\,$$$ and $$$\gamma=1/T_1$$$ when using arterial spin labeling (ASL).

$$$\hspace{1cm}{\bf\text{2.Voxelization of QTM model}}\hspace{5mm}$$$Because the voxel size is much larger than the typical size of the micro vessels, Eq.1 should be integrated over a voxel before fitting. For a a voxel location$$$\,{\boldsymbol\xi}=(\xi_x,\xi_y,\xi_z)\,$$$ we obtain

$$J(C({\boldsymbol \xi},t), M^R,M^A,M^S,U,V,W)=0,\hspace{3cm}(2)$$

with

$$\hspace{-7cm}J=\frac{C({\boldsymbol\xi},t)-C({\boldsymbol\xi},t-\delta_t)}{\delta_t}+\lambda C({\boldsymbol\xi},t)$$

$$\hspace{-2cm}+\frac{C({\boldsymbol \xi},t)M^R({\boldsymbol \xi})U({\boldsymbol \xi})-C(\xi_x-1,\xi_y,\xi_z)M^R(\xi_x-1,\xi_y,\xi_z)U(\xi_x-1,\xi_y,\xi_z)}{V_m({\boldsymbol\xi})}$$

$$+\frac{C({\boldsymbol \xi},t)M^A({\boldsymbol \xi})V({\boldsymbol \xi})-C(\xi_x,\xi_y-1,\xi_z)M^A(\xi_x,\xi_y-1,\xi_z)V(\xi_x,\xi_y-1,\xi_z)}{V_m({\boldsymbol\xi})}$$

$$\hspace{-2cm}+\frac{C({\boldsymbol \xi},t)M^S({\boldsymbol \xi})U({\boldsymbol \xi})-C(\xi_x,\xi_y,\xi_z-1)M^S(\xi_x,\xi_y,\xi_z-1)W(\xi_x,\xi_y,\xi_z-1)}{V_m({\boldsymbol\xi})},$$

where$$$\,C({\boldsymbol\xi})\,$$$is the concentration averaged over the voxel and $$$V_m({\boldsymbol\xi})$$$ the blood volume, $$$M^R,\,M^A\,$$$and$$$\,M^S\,$$$the total vessel areas that intersects with the Right, Anterior, and Superior voxel surfaces, respectively. The latter four parameters are called the microvasculatural parameters (MP) below. $$${\bf U}=(U,V,W)\,$$$is the apparent blood velocity field to be reconstructed.

$$$\hspace{1cm}{\bf\text{3.Inverse problem of QTM}}\hspace{5mm}$$$The inverse problem of QTM is to solve for$$$\,{\bf U}\,$$$in Eq.2 using 4D data$$$\,C,$$$⁵

$$\min_{\bf U}\sum_{k=1}^N\|J(C({\boldsymbol \xi},t^k),M^R,M^A,M^S,U,V,W)\|^2+\alpha\|\nabla{\bf U}\|^2,\hspace{3cm}(3)$$

where$$$\,N\,$$$is the number of time frames,$$$\,\alpha\,$$$is the regularization parameter.

The apparent blood velocity magnitude is$$$\,\|{\bf U}\|=\sqrt{U^2+V^2+W^2}\,\,(mm/s).\,$$$The blood flow rate is

$$\hspace{-4cm}F({\boldsymbol\xi})=M^R({\boldsymbol\xi})U({\boldsymbol\xi})+M^R(\xi_x-1,\xi_y,\xi_z)U(\xi_x-1,\xi_y,\xi_z)$$

$$\hspace{-2cm}+M^A({\boldsymbol\xi})V({\boldsymbol\xi})+M^A(\xi_x,\xi_y-1,\xi_z)V(\xi_x,\xi_y-1,\xi_z)$$

$$\hspace{1cm}+M^S({\boldsymbol\xi})W({\boldsymbol\xi})+M^S(\xi_x,\xi_y,\xi_z-1)W(\xi_x,\xi_y,\xi_z-1)\,\,\,(mm^3/s).\hspace{1cm}(4)$$

Then the renal blood flow (RBF) is obtained as

$$RBF({\boldsymbol\xi})=\frac{60}{1000}\cdot\frac{100}{\rho|\Delta|}\cdot|F({\boldsymbol\xi})|\,\,\,(ml/100g/min),\hspace{3cm} (5)$$

where$$$\,|\Delta|\,$$$is the volume of voxel and$$$\,\rho\,$$$the density of tissue. Note that Eq.4 computes the difference between inflow and outflow of the voxel, which indicates the perfusion effect.

Methods

For 7 healthy subjects, we constructed the microvasculature of human kidney by using the kidney shape segmented from the T2w images and the information of length, diameter, and direction of the main artery from the angiogram. Guided by the shape and direction information, fractal bifurcation structure is built to mimic the renal microvasculature as follows⁶:

i. Microvasculature is bifurcation tree, whose radius follow the Murray’s law: $$$r_p^3=r_{d1}^3+r_{d2}^3.$$$

ii. Smallest 10 microns diameter from totally14 branch levels.

iv. Parental and two daughter branches are on the same plane.

v. Angle$$$\,\theta\sim N(\frac{\pi}{3},\frac{\pi}{10})\,$$$between two symmetric daughter branches at bifurcation.

vi. Length ratio of parental and daughter branches is$$$\,\frac{L_p}{L_d}=0.85$$$.

vii. Vasculature grows in the kidney region only.

The PMs were calculated from the constructed microvasculature. They were scaled such that the $$$V_m({\boldsymbol\xi})$$$ was approximately equal to the normal renal blood volume of $$$16\%$$$^7,8. Using the four post-labeling-delay (PLD=1025ms, 1525ms, 2025ms, 2525ms) ASL data as shown in Fig.1, the QTM is performed and the workflow of QTM is summarized in Fig.2.

Results

Discussion

The preliminary data in this work shows the feasibility of QTM to estimate renal blood flow based on a biophysically justified spatio-temporal model of blood flow and an assumed vessel microstructure based on known anatomy and vessel density. In 7 healthy subjects, the flow estimated using the proposed method fell within the range reported in the literature for healthy subjects. A major advantage of the proposed method is that identifying or assuming an AIF is no longer necessary. With an approximate and organ specific microvasculature, QTM can also be applied to other organs like the brain, liver, and breast.

Conclusion

Acknowledgements

References

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