Group sparse-based Taylor expansion method for liver pharmacokinetic parameters imaging of dynamic fluorescence molecular tomography

The forward problem of DFMT can be described by a set of coupled diffusion equations (Joshi et al 2006, Wang and Wu 2012), as shown in equation (1)

$$\begin{eqnarray}&&\left\{\begin{array}{l}-{\rm{\nabla }}[{D}_{x}(r){\rm{\nabla }}{{\rm{\Phi }}}_{x}(r,t)]+{\mu }_{{ax}}(r){{\rm{\Phi }}}_{x}(r,t)={\rm{\Theta }}\delta (r-{r}_{s},t)(r\in {\rm{\Omega }})\\ -{\rm{\nabla }}[{D}_{m}(r){\rm{\nabla }}{{\rm{\Phi }}}_{m}(r,t)]+{\mu }_{{am}}(r){{\rm{\Phi }}}_{m}(r,t)={{\rm{\Phi }}}_{x}(r,t)\eta {\mu }_{{af}}(r,t)(r\in {\rm{\Omega }})\end{array}\right.,\end{eqnarray} \tag{ 1 }$$

where Ω and δ(r) represent imaging region and the Dirac function, respectively, and Θ is a calibration factor, which accounts for the unknown gain of the DFMT system. The subscripts x and m indicate the excited and emitted progressing, respectively. D_x and D_m represent optical diffusion coefficient of excited and emitted progressing, respectively. μ_ax and μ_am represent absorption coefficient of excited and emitted progressing, respectively. η μ_af (r, t) represents the unknown fluorescence yield distribution to be reconstructed.

Solving this equation by the finite element method yields the following linear relationship between the photon flow rate distribution Φ(t) in the biological tissue during the emission of the fluorescent light source and the distribution vector X(t) of the fluorophores concentration in the organism (Schweiger et al 1995), t is the time variation of fluorophores concentration

$$\begin{eqnarray}&&{MX}(t)={\rm{\Phi }}(t),\end{eqnarray} \tag{ 2 }$$

where M is the system matrix expressed as m*n, m is the number of measurements on the imaging mouse surface, and n is the total number of nodes in the mouse; X is an n*t dimensional matrix; and Φ is an m*t dimensional matrix.

Our proposed method is mainly aimed at monitoring metabolism in the liver. The distribution of fluorophores X(t) will change due to metabolism in organs and tissues. This change can be described by the biexponential model based on compartmental modeling (Cuccia et al 2003, Milstein et al 2005, Alacam and Yazici 2009)

$$\begin{eqnarray}&&{X}_{i}(t)=-{A}_{i}\exp \left(-{\alpha }_{i}t\right)+{B}_{i}\exp \left(-{\beta }_{i}t\right),\end{eqnarray} \tag{ 3 }$$

where $A={\left[{A}_{1},\cdots ,{A}_{n}\right]}^{T}$, $B={\left[{B}_{1},\cdots ,{B}_{n}\right]}^{T}$, $\alpha ={\left[{\alpha }_{1},\cdots ,{\alpha }_{n}\right]}^{T}$, $\beta ={\left[{\beta }_{1},\cdots ,{\beta }_{n}\right]}^{T}$. And i (i = 1, ..., n) is employed to denote the spatial locations of nodes when a domain is discretized into n nodes. The parameters A (a.u.), B (a.u.), α (min⁻¹) and β (min⁻¹) are the pharmacokinetic parameters describing the metabolic process of indocyanine green (ICG) (Gurfinkel et al 2000), when the gain of the DFMT system is unknown, the pharmacokinetic parameters A and B (a.u.) have arbitrary unit (a.u.) (Zhang et al 2013). α (min⁻¹) and β (min⁻¹) are the uptake and excretion rates of fluorophores, which have physiological significance for quantitative evaluation of organ function (Shinohara et al 1996, Gurfinkel et al 2000).

Expanding the double-exponential model in equation (3) with Taylor formula, we can get

$$\begin{eqnarray}\begin{array}{rcl}{X}_{i}(t) & = & -{A}_{i}\displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{1}{k!}{\left(-{\alpha }_{i}t\right)}^{k}+{B}_{i}\displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{1}{k!}{\left(-{\beta }_{i}t\right)}^{k}\\ & = & \varphi \cdot T,\end{array}\end{eqnarray} \tag{ 4 }$$

where $\varphi =\left[{A}_{i}-{B}_{i},{A}_{i}{\alpha }_{i}-{B}_{i}{\beta }_{i},{A}_{i}{\alpha }_{i}^{2}-{B}_{i}{\beta }_{i}^{2},\cdots ,{A}_{i}{\alpha }_{i}^{k}-{B}_{i}{\beta }_{i}^{k}\right]$$=f\left(\left[{A}_{i},{B}_{i},{\alpha }_{i},{\beta }_{i}\right]\right)$, $T={\left[-1,t,-\tfrac{1}{2}{t}^{2},\cdots ,\tfrac{-1}{k!}{\left(-t\right)}^{k}\right]}^{T}$.

In the following method introduction, k is Taylor expansion order (k = 3, 4...). We will verify which order is the optimal choice to balance time, computational cost and reconstruction accuracy through experiments. At this time, combined with equations (2) , (3) and (4), we convert the problem into the following form

$$\begin{eqnarray}&&M\varphi =\tilde{{\rm{\Phi }}}(t)={\rm{\Phi }}(t)/T.\end{eqnarray} \tag{ 5 }$$

Based on equation (5), combined with the typical DFMT problem, we can construct the following objective function

$$\begin{eqnarray}&&\arg \mathop{\min }\limits_{X(t)}\parallel M\varphi -\tilde{{\rm{\Phi }}}(t){\parallel }_{F}^{2}+\lambda \parallel L\varphi {\parallel }_{F}^{2}\end{eqnarray} \tag{ 6 }$$

L is the weight matrix obtained from anatomical structure (Schulz et al 2009).

Considering the metabolism of ICG by other organs, we define fluorophores concentration as a linear combination of liver region fluorophores concentration and other region fluorophores concentration

$$\begin{eqnarray}&&X(t)={X}_{1}(t)+{X}_{2}(t)\end{eqnarray} \tag{ 7 }$$

among them, X₁ represents the fluorescence yield in the liver, and X₂ represents the total fluorescence yield in all organs outside the liver and muscles of mouse, X₁ and X₂ are all n*t dimensional matrix

$$\begin{eqnarray}&&\varphi (t)={\varphi }_{1}(t)+{\varphi }_{2}(t)\end{eqnarray} \tag{ 8 }$$

among them, φ₁ represents the distribution of pharmacokinetic parameters within the liver, and φ₂ represents the total distribution of pharmacokinetic parameters in all organs and tissues outside the liver of mouse. φ, φ₁ and φ₂ are all the n*(k+1) matrix, and k is Taylor expansion order.

Considering the prior information of histomorphology and the group sparsity of fluorescence yield distribution in liver region, we combine equations (6) and (8) to construct the following objective function

$$\begin{eqnarray}&&\arg \mathop{\min }\limits_{{\varphi }_{1},{\varphi }_{2}}\frac{1}{2}{\parallel M\left({\varphi }_{1}+{\varphi }_{2}\right)-\tilde{{\rm{\Phi }}}(t)\parallel }_{F}^{2}+{\lambda }_{1}{\parallel L{\varphi }_{1}\parallel }_{2,1}+{\lambda }_{2}{\parallel L{\varphi }_{2}\parallel }_{F}^{2}.\end{eqnarray} \tag{ 9 }$$

To solve equation (9), we use alternating optimization based on AGD (Qiao et al 2019). The objective function can be split into the following two sub problems

$$\begin{eqnarray}&&\arg \mathop{\min }\limits_{{\varphi }_{1}}\displaystyle \frac{1}{2}{\parallel M{\varphi }_{1}-\left(\tilde{{\rm{\Phi }}}(t)-M{\varphi }_{2}\right)\parallel }_{F}^{2}+{\lambda }_{1}{\parallel L{\varphi }_{1}\parallel }_{2,1}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\arg \mathop{\min }\limits_{{\varphi }_{2}}\displaystyle \frac{1}{2}{\parallel M{\varphi }_{2}-\left(\tilde{{\rm{\Phi }}}(t)-M{\varphi }_{1}\right)\parallel }_{F}^{2}+{\lambda }_{2}{\parallel L{\varphi }_{2}\parallel }_{F}^{2}\end{eqnarray} \tag{ 11 }$$

and λ₁ and λ₂ are regularization parameter of L_2,1 norm regularization term and Frobenius norm regularization term, respectively.

The two sub-problems can be generalized to the following form

$$\begin{eqnarray}&&\arg \mathop{\min }\limits_{{\varphi }^{* }}\displaystyle \frac{1}{2}{\parallel M{\varphi }^{* }-\tilde{{\rm{\Phi }}}{\left(t\right)}_{\mathrm{new}}\parallel }_{F}^{2}+\lambda \parallel L{\varphi }^{* }{\parallel }_{P}^{2}\end{eqnarray} \tag{ 12 }$$

where φ^* represents φ₁ or φ₂, $\tilde{{\rm{\Phi }}}{\left(t\right)}_{\mathrm{new}}$ represents $\tilde{{\rm{\Phi }}}(t)-M{\varphi }_{2}$ in equation (10), or $\tilde{{\rm{\Phi }}}(t)-M{\varphi }_{1}$ in equation (11). P represents L_2,1-norm or F-norm.

AGD algorithm is used to solve this problem (Qiao et al 2019)

$$\begin{eqnarray}&&\left\{\begin{array}{l}{s}^{j}={\varphi }^{j}+\tfrac{{\eta }_{j-2}-1}{{\eta }_{j-1}}\left({\varphi }^{j}-{\varphi }^{j-1}\right)\\ {\varphi }^{j+1}=\mathrm{prox}\left({s}^{j}-{\rm{\nabla }}\phi \left({s}^{j}\right)/{\gamma }^{j}\right)\\ {\eta }_{j}=\left(1+\sqrt{1+4{\eta }_{j-1}^{2}}\right)/2\end{array}\right.\end{eqnarray} \tag{ 13 }$$

where ${\gamma }^{j}=\max \left(2{\gamma }^{j},{\parallel M\left({\varphi }^{j}-{s}^{j}\right)\parallel }_{F}/{\parallel {\varphi }^{j}-{s}^{j}\parallel }_{F}\right)$, $\phi =\tfrac{1}{2}{\parallel M\varphi -\tilde{{\rm{\Phi }}}{\left(t\right)}_{\mathrm{new}}\parallel }_{F}^{2}$. j is iterations of alternating optimization.

When φ satisfying the condition is calculated, we can use φ to solve pharmacokinetic parameters A, B, α, β, that is, to solve the following equations

$$\begin{eqnarray}&&\left\{\begin{array}{l}A{\alpha }^{k-3}-B{\beta }^{k-3}=\varphi (:,\,k-2)\\ A{\alpha }^{k-2}-B{\beta }^{k-2}=\varphi (:,\,k-1)\\ A{\alpha }^{k-1}-B{\beta }^{k-1}=\varphi (:,\,k)\\ A{\alpha }^{k}-B{\beta }^{k}=\varphi (:,\,k+1)\end{array}\right..\end{eqnarray} \tag{ 14 }$$

Finally, the distribution of pharmacokinetic parameters A, B, α, β in mouse can be obtained. Flow chart of the GSTE method is shown in figure 1. And more details about alternating iterative solution based on AGD are described in section S2 of supplementary data.

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In the numerical simulation, the digimouse Atlas (Dogdas et al 2007) is employed to construct a 3D simulation model. As shown in figure 2(a), The 3D digital mouse is divided into muscle, liver, kidneys, and lungs with specific optical parameters (Ale et al 2012). Their relevant optical properties are shown in table S1 of the supplementary data, where the excitation wavelength is 650 nm and the emission wavelength is 670 nm. The excitation points, represented by the black points, are located one transport mean free path beneath the surface of mouse. As shown in figure 2(c), the fluorescence signals are acquired from the other side of the laser illumination, within a 120° field of view (FOV). As shown in figure S1 of supplementary data, the imaging small animal needs to be continuously rotated for multiple circles (Liu et al 2010a), in order to monitor the metabolic process of fluorescent probe in the body. In each circle, eight projections are acquired during the rotational process of the imaging small animal, and each projection can produce lots of measurement points. We will concatenate the eight measurements matrices from the 8 views along the direction of matrix column as the fluorescence signals measurements at one time point. Figure 2(b) shows the fluorescent probe concentration curve, simulating metabolic processes of fluorescent probe in the mouse within 180 minutes. These fluorescent probe concentration curves are generated based on equation (3) and the corresponding pharmacokinetic parameters in table 1. In figure 2(d), we present the ground truth of the parametric images A (a.u.), B (a.u.), α (min⁻¹) and β (min⁻¹) of the digital mouse obtained according to table 1 (Hillman and Moore 2007, Chen et al 2011) at z = 16.5 mm.

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The digital mouse model is discretized into a uniform tetrahedral mesh consisting of 6096 nodes and 30 123 tetrahedral elements for reconstruction in Amira (Amira, Visage Imaging, Australia). In order to verify the performance of this method, the following experiments are designed as follows: Firstly, different order of Taylor expansion are adopted for comparison to determine the order of Taylor expansion that achieves the best performance with the method. Secondly, different time intervals within a metabolic time of 180 min (10 800 s) are used to find the optimal fluorescence data collection interval for the best performance. Finally, GSTE method proposed in this article compared with direct method using graphics processing units (direct method (GPU)) (Chen et al 2016), indirect method (Zhang et al 2013), and TE method, respectively. Direct method (GPU) is a very reliably accurate method for reconstructing parametric images in DFMT state of art. Direct method (GPU) reduces the reconstruction time by ∼90% without a degradation of quality compared with the method proposed by predecessors. The speed of indirect method in reconstructing parametric images is fast, but its accuracy is far inferior to the direct method; TE method only use Taylor expansion framework, AGD algorithm and the L₂-norm as the regularization term of the objective function, without taking advantage of the group sparsity derived from the structural prior.

In order to verify the reconstruction accuracy and speed of our proposed GSTE method, time cost is recorded to evaluate reconstruction speed among the methods, and relative difference(RD) is calculated to evaluate the reconstruction accuracy among the methods. Each study is conducted five times to ensure stability of method. And we randomly add 13dB noise to each group of experiments

$$\begin{eqnarray}&&{RD}=\mathrm{sqrt}\left(\parallel {X}_{\mathrm{reconstruction}\ }-{X}_{\mathrm{true}\ }\parallel /\parallel {X}_{\mathrm{true}\ }\parallel \right).\end{eqnarray} \tag{ 15 }$$

In vivo experiments are conducted on 7 week old BALB/c nude mice (Keaoke Biotechnology Co., Ltd, Xi'an, China). All operations followed the Laboratory Animal Management and Welfare Ethics, Northwest University, China (Permit Number: NWU-AWC-20210901 M). To reduce the pain of mice during the experiment and reduce the data collection error, mice are anesthetized with 2% isoflurane–air mixture gas (RWD Life Science Co., Ltd, Shenzhen, China) throughout the experiment. As shown in figure 3(a), our experiment is mainly conducted in a multimodal optical imaging system. The mice are placed on a rotating turntable, and then, fluorescence images are taken. The excitation source is obtained by a continuous-wave semiconductor laser at 780 nm with a power of 450 mW, and fluorescence imaging is achieved using an 840 nm emission filter. Their relevant optical properties are shown in table S1 of the supplementary data, where the excitation wavelength is 780 nm and the emission wavelength is 840 nm. Then, we inject a prepared ICG solution (0.02 ml of 1.5 mg ml⁻¹) into the tail vein of mouse. All fluorescence images are composed of a 512 × 512 pixel, Recorded by a CCD camera cooled at −80°C (Andor, Belfast, Northern Ireland, UK). The exposure time of CCD is set to 0.1 s, and the total imaging time of DFMT is about 180 min. Each FMT fluorescence image is collected every second, we collected a total of 10 800 FMT images, taking 180 min in total.

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After collecting the fluoresence data, two doses of iohexol are injected into the mouse body, with an intraperitoneal injection of 0.3 ml each time, with an interval of 30 min between the two doses, to enhance the anatomical structure of the liver in micro-CT imaging. 30 min after the second injection, the mouse is immobilized and scanned using an x-ray source with a micro-focused cone beam (L9181-02, Hamamatsu Photonics, Hamamatsu Japan). In the x-ray scanning, the system is operated at a source voltage of 90 kV and power of 27W. An x-ray flat-panel detector (C7942CA-22, Hamamatsu Photonics, Hamamatsu Japan), employed in high-resolution CT imaging, is used to obtain a total of 600 x-ray projections with an interval of 0.6° and integrating time of 0.5 s. The 3D anatomical structures are segmented from the CT data (Dogdas et al 2007), Amira is used to segment the lungs, liver, kidneys and muscle, then, we use the segmentation results constructing a Laplacian matrix (Schulz et al 2009).

Figure 3(b) shows mean fluorescence intensity on surface collected within 180 min, and the fitting curves are represented by green lines, reflecting the metabolic pattern of ICG in mouse. Five representative frames are selected at 5, 33, 63, 120, and 150 min. Figure 3(h1)–(h5) show the corresponding fluorescence images captured by the CCD camera. In addition, figure 3(i1)–(i5) show measured data mapped on the three-dimensional surface of the mouse torso.

Similarly to the numerical simulation, in vivo mouse model is discretized into a uniform tetrahedral mesh consisting of 6115 nodes and 28 917 tetrahedral elements for reconstruction. In order to verify the performance of GSTE method in vivo experiments, we use the optimal parameters obtained from numerical simulations (i.e. the optimal order of Taylor expansion and the optimal fluorescence data collection interval for 180 min data collection). Then, We compare the performance of GSTE method with direct method (GPU), indirect method, and TE method in vivo experiments. Each study is conducted five times to ensure stability of method.

4.1.1. Parameter selection experiments

To assess the impact of the Taylor expansion order and fluorescence data collection interval for 180 min data collection on the GSTE method, numerical simulations with different Taylor expansion orders and fluorescence data collection intervals are conducted to select the optimal Taylor expansion order and fluorescence data collection interval for 180 min data collection from reconstructed results. Figure 4(a) shows RDs and time cost of the pharmacokinetic parameters reconstructed by the GSTE method when the k value is set to 3, 4, 5, 6 and 7, respectively. It can be seen that with the increase of the order, the RDs of the reconstruction gradually decreases and the reconstruction time gradually increases. A compromise method is to choose a suitable k to balance reconstruction accuracy and time. We found that when k is bigger than 4, the reduction of RDs becomes less significant. However, the increase of reconstruction time is still obvious when k is bigger than 4. Therefore, an appropriate trade-off between reconstruction accuracy and time can be achieved when k = 4.

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Figure 4(b) shows the RDs and time cost of the pharmacokinetic parameters reconstructed by the GSTE method when the fluorescence data collection interval for 180 min data collection is set to 540, 360, 270, 216 and 180, respectively (i.e. the number of columns corresponding Φ(t) is 20, 30, 40, 50, 60). It should be noted that the RDs here is the mean of the RDs of the four pharmacokinetic parameters. It can be seen that when the fluorescence data collection interval is smaller, that is, the surface fluorescence measurement data is richer, the RDs of the pharmacokinetic parameters reconstruction is lower, and the time cost of reconstruction is higher. When the fluorescence data collection interval for 180 min data collection is less than 270, the RDs has been reduced to 0.06 and gradually stabilized. When the fluorescence data collection time interval is greater than 270, the increase of reconstruction time is still obvious. Therefore, considering the reconstruction time and reconstruction accuracy, the fluorescence data collection interval is set to 270 for 180 min data collection (i.e. the number of columns corresponding Φ(t) is set to 40).

4.1.2. Comparison of pharmacokinetic parameters reconstruction using different methods

In order to test the performance of GSTE method, after obtaining the optimal Taylor expansion order and the optimal fluorescence data collection interval for 180 min data collection, the GSTE method is compared with other methods on the basis of using these optimal parameters.

Figure 5(a) shows the time cost of the four methods after taking the logarithm. We can get such information from the figure: the GSTE method and TE method which use Taylor expansion framework is much faster than the other two methods, they can reduce the time consumption by ∼90%, which also shows the superiority of the strategy using Taylor expansion framework of temporal sequences signal. However, in figure 5(b), we can see that the pharmacokinetic parameters reconstruction accuracy of GSTE method and direct method (GPU) is much better than the other two methods. The pharmacokinetic parameters reconstruction accuracy of indirect method is worst. GSTE method reduces the RDs by ∼45% compared to TE in average and almost keep the same RDs with direct method (GPU). The detailed experimental data of time consumptions and RDs in numerical simulations are shown in table 2. In order to further compare the pharmacokinetic parameters reconstruction accuracy of GSTE method and direct method (GPU) in each organ, we listed the 3D parametric images of the RDs between the reconstructed results of GSTE method and the ground truth, as shown figure 5(c1)–(c4), and the cross-sectional images at Z = 16.5 mm of figure 5(c1)–(c4), as shown figure 5(d1)–(d4), the 3D parametric images of the RDs between the reconstructed results of direct method (GPU) and the ground truth, as shown figure 5(e1)–(e4), and the cross-sectional images at Z = 16.5 mm of figure 5(e1)–(e4), as shown figure 5(f1)–(f4). The results demonstrate that, the pharmacokinetic parameters reconstruction accuracy of GSTE method is relatively higher in organs, especially in liver, and the reconstruction accuracy of direct method (GPU) is relatively higher in muscle. Moreover, as shown in figure 5(b), pharmacokinetic parameters reconstruction accuracy in all organs and tissues of GSTE method and direct method (GPU) are almost identical. Therefore, considering both reconstruction accuracy and time cost, the GSTE method performs better than the other three methods.

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Table 2. Time consumptions and RDs of the different methods in numerical simulations.

	RD (A)	RD (B)	RD (α)	RD (β)	Time (s)
GSTE	0.032 ± 0.003	0.032 ± 0.002	0.070 ± 0.002	0.102 ± 0.005	10.6 ± 0.55
Direct method (GPU)	0.029 ± 0.001	0.033 ± 0.005	0.072 ± 0.005	0.096 ± 0.004	424.0 ± 3.67
Indirect method	0.124 ± 0.005	0.124 ± 0.004	0.168 ± 0.003	0.215 ± 0.004	526.8 ± 3.03
TE	0.050 ± 0.010	0.043 ± 0.009	0.151 ± 0.003	0.160 ± 0.012	10.6 ± 0.55

Similarly to the numerical simulation, in order to verify the performance of GSTE method in vivo experiments, we use the optimal parameters obtained from numerical simulations (i.e. the optimal order of the Taylor expansion and the optimal fluorescence data collection interval for 180 min data collection). Parameter selection experiments in vivo experiments are shown in figure S2 of the supplementary data.

Figure 6(a1)–(a4) shows 3D images of reconstructed results of Full Direct method, because Full Direct method is the most accurate method for reconstructing pharmacokinetic parameters state of art. Figure 6(b1)–(b4) shows Cross-sectional images of figure 6(a1)–(a4), (g) shows the time cost of the four methods after taking the logarithm. We can get such information from the figure: the method using Taylor expansion framework is much faster than the other two methods,they can reduce the time consumption by ∼90%, which also shows the superiority of the strategy using Taylor expansion framework of temporal sequences signal. However, in figure 6(h), we can see that the pharmacokinetic parameters reconstruction accuracy of GSTE method and direct method (GPU) is much better than the other two methods. The pharmacokinetic parameters reconstruction accuracy of indirect method is worst. GSTE method reduces the RDs by ∼42% compared to TE method in average and almost keep the same RDs with direct method (GPU). The detailed experimental data of time consumptions and RDs in vivo experiments are shown in table S3 of the supplementary data. In order to further compare the pharmacokinetic parameters reconstruction accuracy of GSTE method and direct method (GPU) in each organ, we listed the 3D parametric images of the RDs between the reconstructed results of GSTE method and the reconstructed results of Full Direct method, as shown figure 6(c1)–(c4), and the cross-sectional images at Z = 68 mm of figure 6(c1)–(c4), as shown figure 6(d1)–(d4), the 3D parametric images of the RDs between the reconstructed results of direct method (GPU) and the the reconstructed results of Full Direct method, as shown figure 6(e1)–(e4), and the cross-sectional images at Z = 68 mm of figure 6(e1)–(e4), as shown figure 6(f1)–(f4). The results demonstrate that, the pharmacokinetic parameters reconstruction accuracy of GSTE method is relatively higher in organs, especially in liver, and the reconstruction accuracy of direct method (GPU) is relatively higher in muscle. Therefore, considering both reconstruction accuracy and time cost, the GSTE method performs better than the other methods.

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