Abstract Body:

Objectives:  

Many techniques have been proposed for deriving the blood input function without invasive arterial sampling in the quantification of glucose kinetics in human brain dynamic Fluorine-18 fluorodeoxyglucose positron emission tomography (dFDG-PET) imaging. However, many of these techniques require strong historical information about the blood input function for a certain target population, decreasing the generalizability of these methods. The technique proposed in this work utilizes the assumption that the dynamic PET scan can be perfectly modeled by the Patlak plot to determine an unscaled blood input function and Ki map.

Materials and Methods:  

Twelve normal subjects underwent dynamic FDG PET imaging while 25 arterial blood samples were simultaneously collected during imaging. These arterial samples were corrected for noise via the 7-parameter model [1].  We determined that the Patlak assumption [2] was satisfied in each patient’s brain after a t* of 11.0 minutes, and computed each patient’s ground truth Ki map accordingly. 

The time activity curve (TAC) for each voxel in the dynamic image can be considered as an n-dimensional vector. In this context, a TAC follows the Patlak plot if and only if this TAC is a linear combination of the blood input function and the integral of the blood input function [3]. Therefore, the set of possible TACs which follow the Patlak assumption, given some blood input function, forms a 2-dimensional subspace of the original n-dimensional vector space. In this way, the process of determining the kinetic parameters for a certain TAC is equivalent to projecting this TAC onto this subspace and determining that projection’s coordinates with respect to the basis formed by the blood and the integral of the blood. In view of this interpretation of the Patlak plot, our technique first determines the plane which best fits the dynamic data using Principal Component Analysis (PCA) [4] (Figure 1a) and then finds the basis {u, v} in this subspace which minimizes the difference between v and the antiderivative of u. As numerical estimations of the antiderivative are susceptible to noise, we model the vectors in our subspace and determine the integral analytically (Figure 1c). We also scale and shift this antiderivative to match v so that we can ignore the magnitude of u and v during minimization (Figure 1b, 1d). Furthermore, in our loss function, we divide the error between v and the antiderivative of u by the error between v and its model approximation so that noisier choices for v aren’t discouraged (Figure 1e). Bounding u and v so that both are positive and so that v is the sum of a positive amount of the first component and a negative amount of the second component, this optimization problem is quite reasonable and yields vectors u and v which are close to the blood and the integral of the blood when scaled (Figure 1f). We take the model approximation of u as our predicted blood input function to obtain our predicted Ki map.

Results:   

We scaled our predicted Ki maps so that the mean predicted Ki and the mean ground truth Ki matched. We found an average root mean squared error (RMSE) of 7.424e-4±2.511e-4, an average mean absolute percentage error (MAPE) of 2.157±0.7976, and an average structural similarity index (SSIM) of 0.9923±0.006016 between our predicted and ground truth maps across all twelve patients.

Conclusion: 

While not often viewed as a prior, the assumption that some dynamic data follows a kinetic model gives strong information. In the case of the Patlak plot, we can use this assumption to estimate an unscaled blood input function and unscaled Ki map.

Image/Figure:
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Image/Figure Caption:

(a) For visualization purposes, we reduce each TAC to a 3-dimensional vector consisting of the mean of the first two data points (after t*), the mean of the next four data points, and the mean of the remaining data points. The left plot shows one thousand randomly chosen TACs from Normal1’s brain versus the plane generated by the blood input function and its integral. The right plot shows the best-fitting plane and the two vectors, u0 and v0, which explain the most variance in the data. (b) We parameterize the two proposed basis vectors, u and v, with two angles, theta and phi, via the equations on the bottom. In this case, we take theta as 0.8 radians and phi as -0.2 radians. The left plot is a visualization of where these vectors lie on the plane defined by u0 and v0 while the right plot shows each vector’s TAC, truncated to beyond t*. (c) Instead of approximating each vector in our space individually, we approximate the basis vectors for our space and linearly combine them accordingly. The model we use is shown below the left plot and is motivated by integrating the 7-parameter model at late time points. The left plot shows the approximations for u0 and v0 (u0 hat and v0 hat), and the right plot shows the resulting approximations for u and v (u hat and v hat). (d) The left plot shows the raw antiderivative (with initial value zero) versus v, and the right plot shows the scaled and shifted antiderivative. (e) The given equation is the loss function that we wish to minimize. (f) On the left, we plot the loss function over theta and phi. On the right, we plot the u hat and the antiderivative of u hat obtained by minimizing the loss function, individually scaled (and shifted in the case of the antiderivative) to match the blood and integral of blood, versus the blood and integral of blood.

Author

William Terrell
University of Virginia