Time domain X-ray luminescence computed tomography: numerical simulations

: X-ray luminescence computed tomography (XLCT) has the potential to image the biodistribution of nanoparticles inside deep tissues. In XLCT, X-ray excitable nanophosphors emit optical photons for tomographic imaging. The lifetime of the nanophosphor signal, rather than its intensity, could be used to extract biological microenvironment information such as oxygenation in deep tumors. In this study, we propose the design, the forward model, and the reconstruction algorithm of a time domain XLCT for lifetime imaging with high spatial resolution. We have investigated the feasibility of the proposed design with numerical simulations. We found that the reconstructed lifetime images are robust to noise levels up to 5% and to unknown optical properties up to 4 times of absorption and scattering coefficients.

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Laplace transform based time domain XLCT algorithm
In this paper, we apply the generalized pulse spectrum technique (GPST) method to convert the real domain optical diffusion forward model into its Laplace transform, following the similar approach described in Ref [20]. for time domain fluorescence molecular tomography is the Robin boundary condition coefficient, f R is the internal reflection coefficient at the boundary, and ∇ is the gradient coefficient. ( ) k S p r, is the source term which stands for the k-th X-ray beam illumination pattern and can be written as: where ( ) k T r is the X-ray intensity distribution, η is the quantum efficiency, and ( ) af μ r is the absorption coefficient. In Eq. (2), the light yield ( ) af ημ r and the lifetime ( ) τ r are the phosphorescent nanoparticle properties to be reconstructed. In XLCT, while an X-ray beam scans the object along a straight line, the X-ray beam intensity distribution along the scanning line follows the Beer-Lambert law. If we assume a uniform X-ray attenuation medium, ( ) k T r can be expressed as: where 0 T is the initial X-ray beam intensity, ( ) x μ r is the X-ray attenuation coefficient at the position r , and ( ) L r is the distance from X-ray beam start position to current position r .
Based on the finite element method (FEM), the forward model of the time domain XLCT can be expressed as [21]: where n d is the number of detector nodes, I is the total number of angular projections, J is the number of linear scan for each projection, and m is the finite element mesh node number.
Here, we define the intermediate quantity, being the shape functions and the unknowns. b is the measurement, and A is the system matrix that can be calculated as: is the sensitivity matrix where each row vector i Φ is solved by Eq. (1) when setting the detector node i to be 1. j T is the excitation vectors from X-ray beam illumination patterns. In XLCT, the excitation regions have the known locations along the X-ray beam and can be described as: 1, ( ) 0, j node s is within the X ray beam s otherwise The XLCT reconstruction can be solved like what used in FMT [22][23][24]. The solution of Eq. (4) can be obtained by minimizing the following regularized squared measurement misfit under the non-negativity constraint: where α is the regularization parameter and || ( ) is the L q norm term. In this paper, the majorization-minimization (MM) algorithm is applied to minimize the L 1 regularized mismatch between the measurements and the modeled values by updating the images iteratively. The details of the MM algorithm have been described elsewhere [23,24]. The two unknown distributions, the phosphorescent yield and the lifetime of the phosphor particles can be explicitly recovered from the images of x p x p p x p p x p ημ τ r r r r r r r r r r (8) where the transform factors are: In which, ( ) In this study, we use a pair of transform factors only.

Numerical simulation studies
To validate our proposed time domain XLCT imaging system and algorithms, we have performed numerical simulation cases using a three-target phantom. To simulate the proposed imaging system, we took measurements using one optical fiber bundle. For the simulation studies, we used a 10 mm long cylindrical phantom with a diameter of 13 mm. The optical properties of the phantom were set to be ( ) In numerical simulations, we adopted a normalized X-ray beam intensity. Therefore, the X-ray intensity at the entry to the phantom 0 ( ) T was assumed to be equal to 1. The X-ray attenuation coefficient was 1 0.0214 x mm μ − = in the phantom. Then, the X-ray intensity along the X-ray beam in the phantom is given by the following equation: where [0, 13] L ∈ was the distance from one side to another side of the phantom. All the three-targets had a diameter of 0.4 mm and a height of 6 mm and were embedded in the phantom. The positions of the targets are shown in Fig. 2, in which we can see that the target center-to-center distance (CtCD) was 0.8 mm. For numerical study, we set the phosphor particle concentration to be 1 mg/mL in targets and 0 mg/mL (no phosphors) in the background. Targets are divided in two groups: Group #1 including T1 and T2; Group #2 including T3. The fiber bundle was placed at 3 mm under the phantom top surface. The relative position of fiber bundle to the phantom was fixed. During the experiments, the fiber bundle and the phantom translated and rotated together. We used a focused X-ray beam to scan the phantom at a depth of 5 mm. The focused X-ray beam diameter and the linear scan step size were set to be 100 μm. We used six angular projections with an angular step size of 30 degrees. For each projection, we had 130 measurements. The numerical measurements were generated from the forward mode nodes, 153,05 was added to
Center-todistance error where r Dist an respectively. half maximum Dice simil reconstructed where r ROI is intensities are to 100%, the b

Result of
The scanned pixel size of interpolated to applied using Figure 3 Table 2. From the dotted blue line in Fig.  3(B), line profiles are plotted in Fig. 3(C). From the FWHM, we calculated reconstructed target size of 0.4249 mm with a target size error (TSE) of 6.23% and 0.4222 mm with a TSE of 5.54% for phosphorescence yield and lifetime, respectively. In addition, the CtCD of yield and lifetime are 0.7769 and 0.7742 mm with errors of 2.89% and 3.23%, and DICE were evaluated to be 89% and 87.62% for yield and lifetime, respectively. Based on our results, the lifetime and yield of multiple targets deeply embedded inside tissues can be successfully reconstructed simultaneously using time resolved data. To see how robust the proposed time domain XLCT imaging algorithm is to different measurement noises from 1% to 10%, we added different Gaussian white noises onto the numerical measurements and ran the reconstruction with a fixed projection number of 6. We have calculated the image qualify metrics from both the reconstructed lifetime images and yield images as listed in Table 3. Reconstruction using lifetime data with noise levels of 1% and 2% resulted in equivalent CtCD of 0.7769 mm with 2.89% error. At 5% noise level, the CtCD was 0.7755 mm with 3.06% error. At the largest noise level of 10%, the CtCD was evaluated at 0.9515 mm with 18.94% error. DICE coefficients remained larger than 86% in noise levels 1%, 2%, and 5%. For noise level of 10%, the DICE coefficient decreased to 27.53%. The reconstructed yield images resulted in similar DICE coefficients to those utilizing lifetime images, but the lowest DICE coefficient was 85.14% at the greatest noise level. The DICE coefficients at noise levels 1%, 2% and 5% did not fluctuate significantly from 86%. In addition, the lowest CDE of 1.17% occurred at 1% noise level while the greatest CDE of 27.64% occurred at 5% noise level. At noise levels 2% and 10%, CtCD were generated at 0.5844 mm with 26.95% error and 0.6187 mm with 22.66% error respectively. Additionally, we reconstructed images of lifetime and yield with measurement data at different number of projections with a fixed noise level of 2% as shown in Fig. 4. From these images, we calculated the image quality metrics as listed in Table 4. The reconstructed lifetime images performed best with measurement data of projections 6 and 12 considering their high DICE coefficients of 87.62% and 86.15% and low CDE of 3.07% and 1.63%, respectively. From the reconstructed yield images, we see that DICE coefficients did not vary significantly with different projection numbers. The greatest DICE coefficient was calculated with 6 projections at 86.32%. The closest CtCD to the actual value was evaluated with 24 projections at 0.8016 mm with 0.20% error, while the greatest CtCD error arose from 3 projections at 1.0560 mm with 32.00% error. Lastly, we have investigated the robustness of the reconstructed lifetime and yield images to the unknown optical properties. In the forward model, we have set the absorption coefficient of 0.0072 mm −1 and the reduced scattering coefficient of 0.72 mm −1 . In the reconstruction model, we have changed the optical properties to be 2 times absorption coefficient (2*µ a ), 4 times absorption coefficient (4*µ a ), and 2 times absorption coefficient plus 2 times reduced scattering coefficient (2*µ a and 2* µ s '). In these studies, we had a fixed noise level of 2% and a fixed projection number of 6. From both the reconstructed lifetime and yield images, we have calculated the image qualify metrics as listed in Table 5, from which we see that the reconstructed lifetime images seem independent of the optical properties of the object being imaged with the same CtCD of 0.7769 mm with 2.89% error and same DICE coefficient of 86.50%, which are very close to the numbers we got with the true optical properties. From Table 5, we also see that the reconstructed yield images resulted in slightly different metrics with respect to different optical properties. The lowest CDE of 1.52% occurred at 2*µ a and 2*µ s ' with a CtCD of 0.7879 mm, and the greatest CDE of 27.81% occurred at 2*µ a with a CtCD of 0.5775 mm. DICE coefficients remained relatively constant at 86% for 2*µ a , and 4*µ a while the greatest DICE coefficient was generated from 2*µ a and 2*µ s ' at 87.05%.

Discussion and conclusions
In this study, we have, for the first time, proposed a time domain XLCT system design, forward model and reconstruction algorithm. We plan to generate high frequency pulsed Xray photons by using an optical chopper. The linear GPST method was applied to solve the optical diffusion forward model problem. In three-component target numerical simulations, three targets were successfully reconstructed. Furthermore, phosphorescence yield and lifetime distributions have been recovered simultaneously with good accuracy and resolution. Time domain methods usually take a relatively long scanning time. In the future experimental system, several TCSPC modules can be employed to achieve multiple channel detection to reduce measurement time.
From the numerical simulation studies, we found that the proposed time domain XLCT imaging is robust to measurement noise level up to 5% and to unknown optical properties. In particular, the reconstructed lifetime images are independent of the optical properties, which is a major advantage of the lifetime imaging due to the difficulty in estimating the optical properties of tissues.
We applied a 20% of maximum threshold when we calculated the DICE coefficient. It is also possible to apply a threshold of 10% of maximum. We have obtained similar DICE coefficients for all cases except the cases with the noise level of 5%. We believe it is feasible to use either 10% or 20% threshold in the future because the measurement noise should be less than 5%.
In this study, we have also used a pair of transform factors as shown in Eq. (8). In the future, more transform factor pairs will be applied from the measured pulses, which should result in better image quality due to more measurement data as input for the reconstruction algorithm.
It is worth noting that the proposed time domain XLCT algorithm is a generic reconstruction algorithm that works for nanophosphors with a lifetime ranging from picoseconds to microseconds. In the numerical simulation studies, we arbitrarily picked up the lifetime of 0.4 and 0.6 nanoseconds, which are enough to validate our algorithm. However, the experimental system as described in Fig. 1 might only work for measuring the lifetime less than hundreds of nanoseconds. For measuring nanophosphors with lifetimes around a few nanoseconds, we need an X-ray source such as free electron laser which can generate X-ray pulses as short as a fraction of nanosecond [26].
We noticed that the DICE dropped slightly when the measurement projection number increased as shown in Table 4. To figure out the reason, we have performed another set of numerical simulations with only one large target (5 mm in diameter) at different projection numbers with a constant noise level. With the large target, we did not observe this issue. A possible explanation is that our tdXLCT reconstruction is based on a finite element mesh and there are partial volume issues due to the small target size. We have also found that the reconstructed image quality could be improved substantially with more detectors instead of one detector as described in our simulations.
The proposed algorithm is a generic reconstruction algorithm that works well for both two-dimensional (2D) imaging and three-dimensional (3D) imaging. Due to the computational cost, we have demonstrated it with a 2D imaging. For 3D imaging, we can scan mice slice by slice of the region of interest.
In summary, we have proposed a time domain XLCT imaging framework and have performed a set of numerical simulation studies to validate its feasibility, which will guide the future design of a time domain XLCT imaging system.

Funding
National Institutes of Health (R01EB026646).

Disclosures
The authors declare that there are no conflicts of interest related to this article.