Enhancement of the localization and quantitative performance of fluorescence molecular tomography by using linear nBorn method

Lichao Lian; Yong Deng; Wenhao Xie; Guoqiang Xu; Xiaoquan Yang; Zhihong Zhang; Qingming Luo

doi:10.1364/OE.25.002063

1. Introduction

Hybrid fluorescence molecular tomography and X-ray computed tomography (FMT-XCT) can noninvasively resolve the three-dimensional (3D) spatial distributions of fluorescent markers associated with molecular and cellular functions [1,2]. With the development of the reconstruction methods [3] and the free-space FMT system [4], the reconstruction quality of FMT has been greatly improved and this further promotes the application of FMT in the early recognition of cancers in mice [5], the study of disease growth [6], and drug development [7].

The localization and quantitative performance are two important indicators for evaluating the reconstruction quality of FMT. They also determine the applications and future development directions of FMT. Because of this, improving the localization and quantitative performance of FMT is considered as very important in FMT development. Conventional FMT reconstruction using only emission data is not practical, because it is nearly impossible to accurately measure the source strength and the instrumentation response functions that impact the detected intensity of signals [8,9]. The small errors accumulated by the system during this process produce a declining reconstruction quality. The use of nonlinear normalized Born ratio (nBorn) method [10] greatly improves the image reconstruction. It may not minimize model mismatch and measurement bias, but as it normalizes the emission data with the excitation data, it helps eliminate some unknown experimental factors and to some extent weaken the effects of absorption heterogeneities on image reconstruction [11].

When compared with the image reconstruction using only emission data, the nonlinear nBorn method greatly improves the reconstruction quality. Nevertheless, FMT is still far from reaching its potential of accurately visualizing biological information with a high localization and quantitative accuracy. It is because the inverse problem of FMT is severely ill-posed [12] and even small fluctuations in data can lead to large perturbations in the solution. Thus the quality of the reconstructed image is limited by measurement noise [13], the complexity of the physiological environment, and the absorption and scattering properties of the samples [14]. Though utilizing functional and structural a priori information obtained from diffuse optical tomography (DOT) and XCT helps improve the image reconstruction quality [15], the process of obtaining the functional and structural information is extremely difficult. For example, it is difficult for the XCT system to get the structural information of the soft tissues that have almost the same X-ray absorption coefficient [16] and the inverse problem of DOT is also severely ill-posed [17] which results in inaccurate functional information. All these difficulties will lead to unsatisfying localization and quantitative performance of FMT. In addition, as the nonlinear nBorn method normalizes the acquired fluorescence signals with the excitation data nonlinearly, the relative distributions of the normalized measurement are different from those of the emission data. When compared with the reconstruction using only emission data, what the nonlinear nBorn method does is equivalent to introducing excitation noise to the emission data. The introduction of measurement noise also makes it difficult to quantify the signal-to-noise ratio of the resulting normalized measurements [18] and to find the true fluorescence distribution. Finally, it can result in worse localization and quantitative performance of FMT.

Some of the linear nBorn methods normalize the emission data with the excitation data at a fixed detector position [8] or using the maximum excitation data at one detector [10]. Different from the nonlinear nBorn method, the relative distributions of the normalized measurements they construct are the same as those of the emission data. However, these methods rely heavily on the stability of the excitation data at a single reference point, which puts the reconstruction quality under the influence of the measurement noise in the excitation data and therefore the reconstruction quality can be sometimes worse than that using the nonlinear nBorn method.

To alleviate these limitations and improve the localization and quantitative performance of FMT, we present a novel linear nBorn method in this study, which, to our knowledge, has not been reported previously. In this method, the emission data for each projection is normalized with the average excitation data of all detectors. It offers advantages of eliminating the experimental factors, the unknown source strength, and the coupling losses, which can also be achieved by using the nonlinear nBorn method. Moreover, the relative distributions of the normalized measurements are consistent with those of the emission data and the acquired excitation data are averaged which is effective in suppressing excitation noise. To study the localization and quantitative performance of the linear nBorn method in the presence of high scattering and absorption heterogeneities, we used a free-space FMT-XCT system and demonstrated the linear nBorn method using phantom and in vivo mice experiments. In the phantom experiments, different configurations of phantoms heterogeneities with different scattering and absorption coefficients were constructed. We compared the results of linear and nonlinear nBorn methods, which are reconstructed using both the homogeneous and the matched heterogeneous optical coefficients. In the in vivo experiment, we implanted three glass tubes with known concentration of DiR solution in the lower abdominal cavity of a euthanized mouse and the images were reconstructed using homogeneous optical coefficients. The reconstruction results were compared with the results of using the nonlinear nBorn method. The results indicated that the linear nBorn method can effectively localize and quantify the fluorescent targets and has better localization and quantitative performance than the nonlinear nBorn method.

2. Materials and methods

2.1 Forward model of FMT

Photon propagation in biological tissues is modeled by two coupled diffusion equations with Robin boundary conditions [10]. The sample surfaces are obtained by XCT. The emission and excitation field detected by the CCD camera can be written as:

U^{f} (r_{d}, r_{s}) = Q_{E}^{λ_{fluo}} Θ_{f}^{fluo} \int d^{3} r Θ_{\det} (r_{d}) g_{0}^{λ_{fluo}} (r_{d}, r) \frac{v}{D^{λ_{fluo}}} n (r) g_{0}^{λ_{exc}} (r, r_{s}) Θ_{src} (r_{s}) T^{λ_{fluo}},

U^{e} (r_{d}, r_{s}) = Q_{E}^{λ_{exc}} Θ_{f}^{exc} Θ_{\det} (r_{d}) g_{0}^{λ_{exc}} (r_{d}, r_{s}) Θ_{src} (r_{s}) T^{λ_{exc}},

where

n (r) = η μ_{a}^{fluo} (r)

denotes the set of fluorescence parameters to be reconstructed, which is directly related to the fluorophore concentration.

η

is the quantum yield of the fluorescent target,

μ_{a}^{fluo} (r)

is the absorption of the fluorescent target,

D^{λ_{fluo}}

is diffusion coefficient at the emission wavelength, v is the speed of light in the medium,

Θ_{src} (r_{s})

is the intensity of the source projected on the object surface,

Θ_{f}^{fluo}

(or

Θ_{f}^{exc}

) is the attenuation of the filter at the emission (or excitation) wavelength,

Θ_{\det} (r_{d})

represents the detectors gains,

Q_{E}^{λ_{fluo}}

(or

Q_{E}^{λ_{exc}}

) is the detector quantum efficiency at the emission (or excitation) wavelength,

T^{λ_{fluo}}

(or

T^{λ_{exc}}

) is the product of exposure time and electron-multiplying gain of CCD at the emission (or excitation) wavelength,

g_{0}^{λ_{exc}} (r, r_{s})

describes the propagation of the light from the source to the fluorescent targets and

g_{0}^{λ_{exc}} (r, r_{s})

from the fluorescent targets to the detector.

Using the linear nBorn method, the emission data for each projection is normalized with the average excitation data of all detectors. According to Eq. (1) and Eq. (2), the linear nBorn field $U^{LnB} (r_{d}, r_{s}^{k})$ at a point $r_{d}$ induced by the kth source at $r_{s}^{k}$ can be calculated by the following integral equation:

U^{LnB} (r_{d}, r_{s}^{k}) = \frac{U^{f} (r_{d}, r_{s}^{k})}{U_{LnB}^{e} (r_{d}, r_{s}^{k})} = α_{0} \frac{\int d^{3} r g_{0}^{λ_{fluo}} (r_{d}, r) n (r) g_{0}^{λ_{exc}} (r, r_{s}^{k})}{\sum_{j=1}^{j=n} g_{0}^{λ_{exc}} (r_{d}^{j}, r_{s}^{k}) / n},

where

α_{0} = \frac{Q_{E}^{λ_{fluo}}}{Q_{E}^{λ_{exc}}} \frac{Θ_{f}^{fluo}}{Θ_{f}^{exc}} \frac{v}{D^{λ_{fluo}}} \frac{T^{λ_{fluo}}}{T^{λ_{exc}}}

is the calibration factor depending on system characteristics (i.e., fibers attenuation factors, laser power, CCD system gains, etc.). It can simply be determined experimentally by the measurement of a fluorescent target with known concentration, n is the number of the detectors corresponding to the kth source.

2.2 Image reconstruction and reconstruction quality evaluation indicators

The normalized measurements can be written as $U_{mea}^{LnB} = \frac{U^{f}}{U_{LnB}^{e}}$ . Then the FMT problem can be formulated as the following linear matrix equation [19]:

U_{m e a}^{L n B} = W x,

where x is the vectorized unknown fluorescence distribution and W is the forward matrix that characterizes photons propagation from each source to each sample volume element and then to each detector. It is clear that the linear nBorn method can also eliminate the experimental factors, the unknown source strength, and the coupling losses, similar to the nonlinear nBorn method. The difference lies in the fact that the relative distributions of the normalized measurements constructed by the linear nBorn method are consistent with those of the emission data, and the acquired excitation data are averaged, which is effective in suppressing excitation noise.

As the inverse problem of FMT is severely ill-posed, iterative reweighted L1 regularization (IRL1) is used in this study to solve the inverse problem, and the solution $x$ is updated as follows:

x_{k + 1} = \underset{x \geq 0}{\arg} min \frac{1}{2} {‖ W x - U_{m e a}^{L n B} ‖}_{2}^{2} + λ {‖ M_{k} x ‖}_{1},

where λ is the regularization parameter and M is a diagonal weight matrix. The diagonal elements of M are updated as:

{(m_{i i})}_{k + 1} = \frac{1}{| {(x_{i})}_{k} | + α},

where i ranges over the solution

x

,

α

is a stable parameter, and k is the iteration count. Equation (5) was solved under the split Bregman framework and transformed into the following Bregman iteration:

x_{k + 1} = min_{x} \frac{1}{2} {‖ W x - U_{m e a}^{L n B} ‖}_{2}^{2} + \frac{μ}{2} {‖ d_{k} - M_{k} x - b_{k} ‖}_{2}^{2},

d_{k + 1} = min_{d} λ {‖ d ‖}_{1} + \frac{μ}{2} {‖ d - M_{k} x - b_{k} ‖}_{2}^{2},

b_{k + 1} = b_{k} + (M x_{k + 1} - d_{k + 1}),

where μ is a constant regularization parameter and d and b are intermediate vectors introduced in the calculation process. Previous studies verified that the iterative reweighted L1 regularization effectively reduces the artifacts in the reconstruction results, even when fewer fluorescence projection images have been used [19].

To validate the localization performance of the linear nBorn method, we utilize the position error (PE) [20] in this study. The PE is defined as:

PE= {‖ P_{r} - P_{0} ‖}_{2},

where P₀ is the actual location of the fluorescent target and P_r is the centroid of the reconstruction result.

The quantitative performance of the linear nBorn method is evaluated by calculating the coefficient of determination R² of the linear fitting for the average reconstructed FMT intensity against the true DiR concentration is used to evaluate.

2.3 Experimental apparatus

A free-space FMT-XCT system (as shown in Fig. 1) was used to perform the experiments. The XCT system included an X-ray source (UltraBright, Oxford Instruments, Oxfordshire, UK) and a flat-panel X-ray detector (PaxScan2520V, Varian Medical Systems, Palo Alto, CA, USA). The FMT system consists of a 748-nm continuous-wave diode laser (B&W Tek, Newark, DE, USA) and an electron-multiplying CCD (EMCCD) camera (DU-897, Andor Technology Ltd, Belfast, UK). The sample was placed on the rotation stage. The laser spot was scanned at multiple positions on the sample surface by a dual-axis galvanometer scanner. An EMCCD at the opposite end of the laser acquired the fluorescence and excitation projection images. FMT and XCT raw data were obtained separately over 360° by rotating the rotation stage. The model of the phantom used in the phantom study is shown in Fig. 1(b). 1% intralipid without ink filled in a 30-mm-diameter cylinder glass cup was used as the background medium. The region marked in dark gray was 10 mm in height and was used for FMT reconstruction. H1 and H2 were the two heterogeneous-solution container with a diameter of 10 mm and their center were at (–7.9 mm, –0.4 mm) and (7.5 mm, –1.9 mm) respectively. The fluorescent targets were three transparent 2-mm-diameter glass tubes T1, T2, and T3 filled with different concentrations of DiR solution (excitation wavelength = 748 nm and emission wavelength = 780 nm). The center of T1 was at (–0.9 mm, –6.7 mm), T2 was at (7.0 mm, 5.2mm), and T3 was at (–6.9 mm, 6.0 mm). They were placed outside H1 and H2 to avoid influence on the optical coefficients of the heterogeneities H1 and H2.

Fig. 1 (a) Schematic of the experimental system. (b) The model of the phantom used in the phantom study.

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2.4 Experimental setup

To study about the localization and quantitative performance of the linear nBorn method in the presence of high scattering and absorption heterogeneities, phantom and in vivo experiments were performed. In the phantom experiments, for both the nonlinear nBorn method and the linear nBorn method, the images were reconstructed using both the homogeneous optical coefficients (that were the same as the background optical coefficients) and the matched heterogeneous optical coefficients. The homogeneous optical coefficients were always used in FMT reconstruction because in most cases we cannot get the true distributions of optical coefficients. As a concession, homogeneous absorption and scattering coefficients were used. When the true optical coefficients of the sample were known and used in the inverse problem, they offer improved performance. However, before we get the true optical coefficients, we should first use MRI or XCT to get the structural information of the samples and then perform DOT or look up the optical parameters of different organs in the published papers. Yet the inverse problem of DOT is also severely ill-posed which results in inaccurate functional information. Therefore, we usually refer to the published papers to get the optical coefficients. We note that due to the individual differences, the optical coefficients that we referred to the published papers might be inaccurate when compared with the true values. In the in vivo experiment, we chose the lower abdominal cavity of the mice as the reconstruction region and homogeneous optical coefficients were used in the inverse problem, because it is hard for our XCT system to get the boundary of the organs in the lower abdominal cavity.

(1) Phantom study

In this experiment, T1, T2, and T3 were filled with 0.5, 1.0, and 2.0 μM DiR solution, respectively. Five distinct sets of experiments were performed and each with different scattering and absorption heterogeneities. In all five sets, 1% intralipid solution without ink was used as the background medium. India ink can be added to intralipid without changing its scattering properties and the concentration of intralipid will not change the absorption properties of the solution [21]. The configurations of heterogeneous-solution in H1 and H2 and their true optical coefficients in experimental conditions (I)–(V) are shown in Table 1. In this study, the laser source scanned 9 positions along the z axis with a step of 1.5 mm at each of the 24 rotation angles (15° for each angle). The excitation and fluorescence projection images were acquired separately with the corresponding filters. Then the XCT data were acquired using a tube voltage of 50 kVp and a tube current of 0.8 mA.

Table 1. Five sets of experiments and their configurations and the true optical coefficients of the heterogeneous-solution

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(2) Animal study

In the in vivo experiment, the lower abdominal cavity of the mice, where complex mixed scattering and absorption heterogeneities exist, was chosen as the reconstruction region. 7-week-old female Balb/c mice were euthanized and then had their hair shaved off. We implanted three glass transparent tubes, filled with different concentrations of DiR solution, in the lower abdominal cavity of the mice. All animal procedures are complied with the protocols approved by the Hubei Provincial Animal Care and Use Committee and with the experimental guidelines of the Animal Experimentation Ethics Committee of Huazhong University of Science and Technology.

All three tubes T1, T2, and T3 were 2 mm in diameter and were filled with 0.5, 1.0, and 2.0 μM DiR solution, respectively. In this study, the laser source scanned 9 positions along the z axis with a step of 1.5 mm at each of the 24 rotation angles (separated by 15° for each angle). The excitation and fluorescence projection images were acquired separately with the corresponding filters. In the inverse problem, the absorption and reduced scattering coefficients are $μ_{a}$ = 0.2 ${cm}^{- 1}$ and $μ_{s}'$ = 12 ${cm}^{- 1}$ for both the excitation wavelength of 748 nm and the emission wavelength of 780 nm [22]. Then the XCT data were acquired using a tube voltage of 50 kVp and a tube current of 0.8 mA.

3. Results and discussion

3.1 Reconstruction results of phantoms with and without absorption heterogeneities

(1) Image reconstruction using the homogeneous optical coefficients

In experiment (I), the heterogeneous-solution in H1 and H2 was 1% intralipid without ink which was the same as that of the background medium. In experiment (II), it was 1.0% intralipid & 20 ppm of ink and in experiment (III), it was 1.0% intralipid & 30 ppm of ink. The concentration of DiR solution in tubes T1, T2, and T3 were 0.5, 1.0, and 2.0 μM, respectively.

In this section, for both the nonlinear and the linear nBorn methods, the images were reconstructed using homogeneous optical coefficients which were the same as that of the background medium. The reconstruction results are shown in Fig. 2. Rows 1–3 showed the reconstruction results in experimental conditions (I)–(III), respectively. The first column indicated the 3D reconstruction results of using the nonlinear nBorn method in different experimental conditions. The second column showed the corresponding 2D middle-slice results of the 3D results. Similarly, the fourth and the third columns were the 3D and 2D results of using the linear nBorn method in different experimental conditions. In Fig. 2 we could see that in the three different experimental conditions, the reconstruction results corresponding to the nonlinear nBorn method showed serious trailing between T1 and T3. To give a quantitative description, we utilized the position error (PE) [20] in this study. The results are shown in Table 2. We could see that the results corresponding to the linear nBorn method had a better localization performance than those for the nonlinear nBorn method. These results are consistent with the results shown in Fig. 2.

Fig. 2 The reconstruction results of nonlinear and linear nBorn methods in experimental conditions (I)–(III) when homogeneous optical coefficients were used in the presence of absorption heterogeneities. [3D renderings were implemented using AMIRA software, FEI Company, Hillsboro, OR, USA.].

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Table 2. The PE and R² of the nonlinear and the linear nBorn methods in experimental conditions (I)–(III) when homogeneous optical coefficients were used in the presence of absorption heterogeneities.

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To compare the quantitative performance of these two methods, we calculated the relation between the average reconstructed FMT intensities of the fluorescent targets and the true DiR concentration. We chose three regions and each of them shared the same center with the corresponding fluorescent target. The area size of the cross-section of each region was about five times bigger than that of the true fluorescent target. The height of the volume was equal to that of the true fluorescent target. The quantitative results are shown in Fig. 3. We used R² to indicate the quantitative accuracy. The results are shown in Table 2. We could observe that the linear nBorn method had a better quantitative performance. Combining the above results, we concluded that the linear nBorn method has good localization and quantitative performance, which were better than the performance of the nonlinear nBorn method even when homogeneous optical coefficients were used in the presence of absorption heterogeneities.

Fig. 3 Average reconstructed FMT intensity as a function of the true DiR concentration when homogeneous optical coefficients were used in the presence of absorption heterogeneities. (a), (b), and (c) are the quantitative results in experimental conditions (I)–(III), respectively. The reconstructed values were normalized to the maximum value.

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(2) Image reconstruction using the matched heterogeneous optical coefficients

In this section, we analyzed the experiments in conditions (I), (II), and (III). For both the nonlinear and the linear nBorn methods, the images were reconstructed using the matched heterogeneous optical coefficients. The reconstruction results are shown in Fig. 4. Rows 1–3 showed the reconstruction results in experimental conditions (I)–(III), respectively. The first column was the 3D reconstruction results of using the nonlinear nBorn method in different experimental conditions. The second column was the corresponding 2D middle-slice results of the 3D results. Similarly, the fourth and the third columns were the 3D and 2D results of using the linear nBorn method in different experimental conditions. In Fig. 4, we could see that in the three different experimental conditions, the reconstruction results corresponding to the nonlinear nBorn method showed serious trailing between T1 and T3. Their corresponding PEs are shown in Table 3. We could see that the results corresponding to the linear nBorn method had a better localization performance than the nonlinear nBorn method.

Fig. 4 The reconstruction results of nonlinear and linear nBorn methods in experimental conditions (I)–(III) when the matched heterogeneous optical coefficients were used in the presence of absorption heterogeneities. [3D renderings were implemented using AMIRA software.]

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Table 3. The PE and R² of the nonlinear and the linear nBorn methods in experimental conditions (I)–(III) when the matched heterogeneous optical coefficients were used in the presence of absorption heterogeneities.

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Figure 5 shows the relation between the average reconstructed FMT intensities of the fluorescent targets and the true DiR concentration. The corresponding R² is shown in Table 3. We could observe that the linear nBorn method had a better quantitative performance. Combining the above results, we concluded that the linear nBorn method has good localization and quantitative performance which were better than those of the nonlinear nBorn method when the matched heterogeneous optical coefficients were used in the presence of absorption heterogeneities. When compared with the results using homogeneous optical coefficients in the same experimental condition, the improvements in localization and quantitative performance are not significant. In the same experimental condition, the linear nBorn method achieves better localization and quantitative performance than the nonlinear nBorn method, which means that the linear nBorn method can improve the reconstruction quality of FMT.

Fig. 5 Average reconstructed FMT intensity as a function of the true DiR concentration when the matched heterogeneous optical coefficients were used in the presence of absorption heterogeneities. (a), (b), and (c) are the quantitative results in experimental conditions (I)–(III), respectively. The reconstructed values were normalized to the maximum value.

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3.2 Reconstruction results of phantoms with and without scattering heterogeneities

(1) Image reconstruction using the homogeneous optical coefficients

In experiment (I), the heterogeneous-solution in H1 and H2 was 1% intralipid without ink which was the same as that of the background medium. In experiment (IV), the heterogeneous-solution in H1 and H2 was 1.5% intralipid without ink and in experiment (V), it was 2.0% intralipid without ink.

In this section, for both the nonlinear and the linear nBorn methods, the images were reconstructed using homogeneous optical coefficients which were the same as that of the background medium. The reconstruction results are shown in Fig. 6. Rows 1–3 showed the reconstruction results in experimental conditions (I), (IV), and (V), respectively. The first column indicated the 3D reconstruction results of using the nonlinear nBorn method in different experimental conditions. The second column showed the corresponding 2D middle-slice results of the 3D results. Similarly, the fourth and the third columns were the 3D and 2D results of using the linear nBorn method in different experimental conditions. In Fig. 6 we could see that in the three different experimental conditions, the reconstruction results corresponding to the nonlinear nBorn method showed serious trailing between T1 and T3. Their corresponding PEs are shown in Table 4. We could see that the results corresponding to the linear nBorn method had a better localization performance than the nonlinear nBorn method.

Fig. 6 The reconstruction results of nonlinear and linear nBorn methods in experimental conditions (I), (IV), and (V) when homogeneous optical coefficients were used in the presence of scattering heterogeneities. [3D renderings were implemented using AMIRA software.]

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Table 4. The PE and R² of the nonlinear and the linear nBorn methods in experimental conditions (I), (IV), and (V) when homogeneous optical coefficients were used in the presence of scattering heterogeneities.

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The average reconstructed FMT intensities of the fluorescent targets against the true DiR concentrations are shown in Fig. 7. The corresponding R² which represented the quantitative accuracy is shown in Table 4. We could observe that the linear nBorn method had a better quantitative performance. Combining the above results, we concluded that the linear nBorn method has good localization and quantitative performance which were better than those of the nonlinear nBorn method even when homogeneous optical coefficients were used in the presence of scattering heterogeneities.

Fig. 7 Average reconstructed FMT intensity as a function of the true DiR concentration when homogeneous optical coefficients were used in the presence of scattering heterogeneities. (a), (b), and (c) are the quantitative results in experimental conditions (I), (IV), and (V), respectively. The reconstructed values were normalized to the maximum value.

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(2) Image reconstruction using the matched heterogeneous optical coefficients

In this section, we analyzed the experiments in experimental conditions (I), (IV), and (V). For both the nonlinear and the linear nBorn methods, the images were reconstructed using the matched heterogeneous optical coefficients. The reconstruction results are shown in Fig. 8. Rows 1–3 showed the reconstruction results of experiments (I), (IV), and (V), respectively. The first column indicates the 3D reconstruction results of using the nonlinear nBorn method in different experimental conditions. The second column shows the corresponding 2D middle-slice results of the 3D results. Similarly, the fourth and the third columns were the 3D and 2D results of using the linear nBorn method in different experimental conditions. In Fig. 8 we could see that in the three different experimental conditions, the reconstruction results corresponding to the nonlinear nBorn method showed serious trailing between T1 and T3. Their corresponding PEs are shown in Table 5. We could see that the results corresponding to the linear nBorn method had a better localization performance than those of the nonlinear nBorn method.

Fig. 8 The reconstruction results of nonlinear and linear nBorn methods in experimental conditions (I), (IV), and (V) when the matched heterogeneous optical coefficients were used in the presence of scattering heterogeneities. [3D renderings were implemented using AMIRA software.]

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Table 5. The PE and R² of the nonlinear and the linear nBorn methods in experimental conditions (I), (IV), and (V) when the matched heterogeneous optical coefficients were used in the presence of scattering heterogeneities.

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Figure 9 shows the relation between the average reconstructed FMT intensities of the fluorescent targets and the true DiR concentration. The corresponding R² is shown in Table 5. From Table 5 we could observe that the linear nBorn method had a better quantitative performance. From this section, we found that the linear nBorn method has good localization and quantitative performance which were better than those of the nonlinear nBorn method when the matched heterogeneous optical coefficients were used in the presence of scattering heterogeneities. When compared with the results using homogeneous optical coefficients in the same experimental condition, obviously, the matched heterogeneous optical coefficients give a better quantitative performance. In the same experimental condition, the linear nBorn method achieves better localization and quantitative performance than the nonlinear nBorn method, which means that the linear nBorn method can improve the reconstruction quality of FMT.

Fig. 9 Average reconstructed FMT intensity as a function of the true DiR concentration when homogeneous optical coefficients were used in the presence of scattering heterogeneities. (a), (b), and (c) are the quantitative results in experimental conditions (I), (IV), and (V), respectively. The reconstructed values were normalized to the maximum value.

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3.3 In vivo experiment results

Figure 10(a) shows the positions of the fluorescent targets in the lower abdominal cavity of the mouse. The concentration of DiR solution in tubes T1, T2, and T3 were 0.5, 1.0, and 2.0 μM, respectively. Region between two white circles was used for FMT reconstruction. In this in vivo experiment, the images were reconstructed using homogeneous optical coefficients. Figures 10(b) and 10(c) are the 3D reconstruction results corresponding to the nonlinear and the linear nBorn methods, respectively. In Fig. 10(b) we could see that a serious trailing between T2 and T3 and we cannot clearly recognize these two fluorescent targets, while in Fig. 10(c) we can clearly recognize three fluorescent targets. Their PEs are shown in Table 6. We can see that the linear nBorn method has a better localization performance than the nonlinear nBorn method.

Fig. 10 The localization and quantitative results of in vivo experiments. (a) The relative positions of the fluorescent targets T1, T2,and T3 in the lower abdominal cavity of the mouse. (b) The 3D reconstruction result of the fluorescent targets with the nonlinear nBorn method. (c) The 3D reconstruction result of the fluorescent targets with the linear nBorn method. [Coordinate system was defined by D (dorsal), V (ventral), Cr (cranial), Cd (caudal), L (left), and R (right). 3D renderings in (b)–(c) were implemented using AMIRA software.]

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Table 6. The PE and R² of the nonlinear and the linear nBorn methods in the in vivo experiment.

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Figure 11 shows the average reconstructed FMT intensities of the fluorescent targets as a function of the true DiR concentration. The corresponding R² is shown in Table 6. We could observe that the linear nBorn method has a better quantitative performance. This finding agrees with the phantom results. It suggests that the linear nBorn method gives better localization and quantitative performance than the nonlinear nBorn method even when homogeneous optical coefficients are used in the presence of mixed absorption and scattering heterogeneities.

Fig. 11 Average reconstructed FMT intensity as a function of the true DiR concentration when the homogeneous optical coefficients were used in the in vivo experiment.

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4. Conclusion

Nonlinear nBorn method helps to eliminate some unknown experimental factors and weaken the effects of small absorption heterogeneities on image reconstruction to some extent. However, as the nonlinear nBorn method normalizes the acquired fluorescence signals with the excitation data nonlinearly, it makes it difficult to quantify the signal-to-noise ratio of the resulting normalized measurements, and it introduces measurement noise of the excitation light to the normalized measurements. Therefore, the nonlinear nBorn method results in worse localization and quantitative performance. In this study, a novel linear nBorn method was presented. For each projection, the linear nBorn method normalizes the emission data with the average excitation data of all detectors linearly. It is a linear normalization method which keeps the relative distributions of the normalized measurements consistent with those of the emission data, and thus restrains the impact of excitation noise on the reconstruction results. To demonstrate the feasibility of this method, we performed the five phantom experiments by setting different scattering and absorption heterogeneities. Each was reconstructed using the nonlinear and the linear nBorn methods with the homogeneous and the matched heterogeneous optical coefficients. In the in vivo experiment, we implanted three glass tubes with different concentrations of DiR solution in the lower abdominal cavity of the mice. It was used to study the practicality of the linear nBorn method by using homogeneous optical coefficients in the presence of complex mixed absorption and scattering heterogeneities. All our phantom and in vivo mice studies show that the linear nBorn method achieves better localization and quantitative performance than the nonlinear nBorn method. Based on these characteristics, the linear nBorn method will further expand the application of FMT in many new areas.

From the results we conclude that: (1) both the nonlinear and the linear nBorn methods achieve better localization and quantitative performance with a homogeneous phantom than with heterogeneous ones; (2) with the same method, the use of matched optical coefficients gives a small improvement in localization and quantitative performance when compared with the use of homogeneous optical coefficients, in the presence of absorption heterogeneities. However, the use of matched optical coefficients gives an obvious improvement in quantitative performance in the presence of scattering heterogeneities. It suggests that both nonlinear and linear nBorn methods perform stable localization and quantitative performance in the presence of absorption heterogeneities, but for the scattering heterogeneities, the use of homogeneous optical coefficients will lead to degraded quantitative performance. This means that DOT or structural a priori information can be introduced to improve the quantitative accuracy of FMT; (3) the localization performance of the nonlinear and the linear nBorn methods is robust to the absorption and scattering heterogeneities, while the quantitative performance is more easily affected by the scattering heterogeneities than by the absorption heterogeneities; (4) in the same experimental condition, the linear nBorn method achieves better localization and quantitative performance than the nonlinear nBorn method; (5) from the in vivo experiment, we find that in the presence of mixed absorption and scattering heterogeneities, the linear nBorn method gives better localization and quantitative performance than the nonlinear nBorn method even when homogeneous optical coefficients are used.

For each projection, the linear nBorn method refers the emission data to the average excitation data of all detectors. Though the use of average excitation data can suppress the excitation noise to some degree, this makes the accuracy of the reconstruction results highly dependent on the accuracy of the emission data. When the emission data is disturbed by a strong emission noise, the localization and quantitative accuracy of FMT will decrease significantly. The homogeneous optical coefficients are used in many conditions. However, the proposed linear nBorn method does not provide satisfying quantitative performance in the presence of scattering heterogeneities, although it performs better than the nonlinear nBorn method. Functional and structural a priori information which is crucial to FMT reconstruction quality can be introduced by using multimodality schemes [23]. Another way is to set reference to the excitation data which is stable to scattering heterogeneities. This is feasible, because the detected signals are affected differently. That is to say, there exist some detectors that are less affected by the scattering heterogeneities. This reminds us that it may improve the quantitative accuracy when we set reference to the excitation data that is less affected by the scattering heterogeneities. In addition, as affected by the heterogeneities, the linear nBorn method may lead to a certain under- or overestimation of the reconstructed values. But this error can be reduced to some extent by using tri-modality FT-DOT-XCT system [12] or more accurate forward model [24].

In conclusion, we presented a linear nBorn method which is shown to have better localization and quantitative performance than the nonlinear nBorn method. In this method, the emission data for each projection is normalized with the average excitation data of all detectors. It keeps the relative distributions of the normalized measurements consistent with those of the emission data and helps to diminish the effect of excitation noise on FMT reconstruction. Future works will consider improving the quantitative accuracy of the linear nBorn method by using homogeneous optical coefficients in the presence of scattering heterogeneities. While this factor can be overcome, the localization and quantitative accuracy of FMT will be improved even when homogeneous optical coefficients are used in the presence of complex heterogeneities, and the linear nBorn method investigated in this work could be potentially beneficial for clinical imaging as well.

Funding

Key Research and Development Program (2016YFA0201403); Science Fund for Creative Research Group (61421064); National Natural Science Fund (91442201); Fundamental Research Funds for the Central Universities (2016YXMS035).

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Experiment NO.	Background (reduced scattering & absorption coefficient /cm)	H1 (reduced scattering & absorption coefficient /cm)	H2 (reduced scattering & absorption coefficient /cm)
(I)	1.0% intralipid without ink (12 & 0.02)	1.0% intralipid without ink (12 & 0.02)	1.0% intralipid without ink (12 & 0.02)
(II)	1.0% intralipid without ink (12 & 0.02)	1.0% intralipid & 20 ppm of ink (12 & 0.12)	1.0% intralipid & 20 ppm of ink (12 & 0.12)
(III)	1.0% intralipid without ink (12 & 0.02)	1.0% intralipid & 30 ppm of ink (12 & 0.18)	1.0% intralipid & 30 ppm of ink (12 & 0.18)
(IV)	1.0% intralipid without ink (12 & 0.02)	1.5% intralipid without ink (20 & 0.02)	1.5% intralipid without ink (20 & 0.02)
(V)	1.0% intralipid without ink (12 & 0.02)	2.0% intralipid without ink (26 & 0.02)	2.0% intralipid without ink (26 & 0.02)

Method	Experiment NO.		PE(mm)		R²
Method	Experiment NO.	T1	T2	T3	R²
Nonlinear nBorn	(I)	1.26	0.50	3.74	0.9406
	(II)	1.47	0.66	4.91	0.8171
	(III)	1.63	1.53	4.45	0.7282
Linear nBorn	(I)	0.56	0.33	0.67	0.9976
	(II)	1.32	0.60	0.69	0.9205
	(III)	1.00	1.46	2.33	0.9649

Method	Experiment NO.		PE(mm)		R²
Method	Experiment NO.	T1	T2	T3	R²
Nonlinear nBorn	(I)	1.26	0.50	3.74	0.9406
	(II)	1.49	1.21	4.53	0.8907
	(III)	1.36	0.89	4.83	0.7568
Linear nBorn	(I)	0.56	0.33	0.67	0.9976
	(II)	0.44	0.60	1.22	0.9344
	(III)	0.69	0.77	1.36	0.9898

Methods	Experiment NO.		PE(mm)		R²
Methods	Experiment NO.	T1	T2	T3	R²
Nonlinear nBorn	(I)	1.26	0.50	3.74	0.9406
	(IV)	1.40	1.18	5.07	0.6055
	(V)	1.42	1.17	4.84	0.6141
Linear nBorn	(I)	0.56	0.33	0.67	0.9976
	(IV)	1.14	0.51	0.67	0.8204
	(V)	1.31	1.03	1.64	0.7972

Method	Experiment NO.		PE(mm)		R²
Method	Experiment NO.	T1	T2	T3	R²
Nonlinear nBorn	(I)	1.26	0.50	3.74	0.9406
	(IV)	1.28	1.50	4.63	0.8679
	(V)	1.28	1.44	4.50	0.9124
Linear nBorn	(I)	0.56	0.33	0.67	0.9976
	(IV)	0.76	0.59	0.72	0.9795
	(V)	1.25	1.20	1.34	0.9702

Enhancement of the localization and quantitative performance of fluorescence molecular tomography by using linear nBorn method

Abstract

1. Introduction

2. Materials and methods

2.1 Forward model of FMT

2.2 Image reconstruction and reconstruction quality evaluation indicators

2.3 Experimental apparatus

2.4 Experimental setup

(1) Phantom study

(2) Animal study

3. Results and discussion

3.1 Reconstruction results of phantoms with and without absorption heterogeneities

(1) Image reconstruction using the homogeneous optical coefficients

(2) Image reconstruction using the matched heterogeneous optical coefficients

3.2 Reconstruction results of phantoms with and without scattering heterogeneities

(1) Image reconstruction using the homogeneous optical coefficients

(2) Image reconstruction using the matched heterogeneous optical coefficients

3.3 In vivo experiment results

4. Conclusion

Funding

References and Links

Cited By

Figures (11)

Tables (6)

Equations (10)

Optics Express

		PE(mm)		R²
Method	T1	T2	T3	R²
Nonlinear nBorn	0.53	1.57	3.04	0.7747
Linear nBorn	0.48	1.10	2.00	0.8988