Lippmann-Schwinger Integral Equation for Photon Migration
Diffuse optical tomography uses NIR light to investigate anomalies within the tissue that have different optical absorption and scattering parameters. As these perturbations of optical parameters are usually due to the hemoglobin concentration, DOT is sensitive to biological function and disease that are associated with hemoglobin concentration changes. Accordingly, it has been investigated as a promising non-invasive imaging tool, for example, for functional brain imaging and breast cancer detection (12, 13, 14, 15).
The inverse problem in DOT is to find the locations and geometric features of anomalies with their optical parameters (such as absorption and/or diffusion ), using a finite number of pairs of optical measurements on a part of its boundary . In particular, our DOT system is mainly interested in the absorption parameter changes due to the hemoglobin concentration changes represented by
whereas the diffusion parameters of the anomalies are indifferent from the background, i.e., , for all . Here,
represents the characteristic function of a domain, and the anomalies be compactly supported inside and have boundaries , respectively, such that is a simply connected domain.
If the medium has a characteristic of highly scattering interactions dominating over absorption, which is the case for the medium such as a cloud milk or biological tissue, then the light propagation can be modeled by the diffusion equation (12, 18):
where denotes the diffuse wave-number, , for , is the total photon density corresponding to the photon flux at the boundary from the th source distribution, is the extrapolation distance, and denotes the normal derivative in the direction of outward unit normal at .
Let be the background photon density for the homogeneous medium (in the absence of any perturbation) and be the scattered photon flux. Then, the th scattered photon flux measurement at the boundary for the absorption parameter perturbation is given by
where the measurement operator is given by
To facilitate further discussion, we define the forward mapping by collecting measurements with respect to all source positions, i.e.,
where denotes the detector locations at the boundary; see Supplementary Material for further details.
Note that the in (5) is a matrix of dimension of (no. of detectors) (no. of sources). (3) is the so-called Lippmann-Schwinger integral equation (derived in Supplementary material) that relates the scattered field to the perturbation in the absorption parameter. This construction of the multi-static data matrix was an essential part of the joint sparse recovery formulations (23, 24, 25, 26, 27). However, the measurement operator relating perturbation in the optical parameter involves the total optical photon density which, in turn, also depends on the unknown perturbation. This makes the inverse problem highly non-linear. Furthermore, due to the dissipative nature of diffusive wave and smaller number of measurements compared to the number of unknowns, resolving the image from DOT is a severely ill-posed problem (35). The basic idea of joint sparse recovery formulations (23, 24, 25, 26, 27) is then to decouple the nonlinear inverse problems in two consecutive steps, taking advantage of the fact that the optical perturbation does not change position during multiple illumination. In this paper, we further extend this idea to the point that the optical anomalies can be directly recovered from the measured multi-static data matrix .
Neural network for inverting Lippmann-Schwinger Equation
In this section, we derive a novel neural network to directly invert the forward operator (5). Specifically, let be the 3D distribution of the perturbation in the region of interest. If comes from a smoothly varying perturbation, then we can easily expect that the corresponding Fourier spectrum is mostly concentrated in a small number of coefficients, so there exists a spectral domain function such that
which can be equivalently represented in the discrete spatial domain by
where is called the annihilating filter (36). (6) implies the existence of a rank deficient block Hankel matrix, denoted by , constructed from and its rank is dependent on the Fourier domain sparsity level as rigorously shown in (37).
Let the block Hankel matrix
admit the singular value decompositionwhere and
denote the left and the right singular vector bases matrices, respectively,is the diagonal matrix whose diagonal components contain the singular values, and denotes its rank. If there exist two matrices pairs () and (, ) satisfying the conditions
where denote the range space of and represents a projection onto , then
with the coefficient matrix given by
One of the most important discoveries in (32) is that an encoder-decoder structure convolution layer is emerged from (9) and (8). Precisely, (8) is equivalent to the following paired encoder-decoder convolution structure:
where denotes the flipped version of the vector , i.e., its indices are reversed (37).
This encoder-decoder representation of the signal is ideally suitable for inverting Lippmann-Schwinger integral equation for our DOT imaging problem. First, we can simply choose . Then, by defining an inversion operator for the forward operator in (5) and substituting in (10), the encoder-decoder structure neural network is re-written as
where could be further processed coefficient from to remove noise, i.e.
for some filter . Finally, by choosing appropriate number of the filter channel , we can control the dimension of the reconstruction image manifold.
The corresponding three-layer network structure is illustrated in Fig. 1(b). Here, the network consists of a single fully connected layer that represents , two paired 3D-convolutional layers with filters and
, and the intermediate 3D-convolutional layer for additional filtering. To enable learning from the training data, we used the hyperbolic tangent function (tanh) as an activation function for the fully connected layer and two convolutional layers (C1 and C2), whereas the last convolutional layer (C3) was combined with rectified linear unit (ReLU) to ensure the positive value for the optical property distribution. Therefore, the network is designed such that the inversion of the Lippmann-Schwinger equation is performed to lead the resulting optical parameters in low-dimensional manifold structure.
Compared to the conventional model-based approaches, the proposed deep network has many advantages. First, the inversion of the Lippmann-Swinger equation is fully data-driven such that we do not need any explicit modeling of the acquisition system and boundary conditions. Second, unlike the explicit regularization in the model-based approaches, the low-dimensional manifold structure of the desired optical distribution is embedded as convolutional layers that are also learned from the data. This is why the proposed neural network is felicitous to provide a robust reconstruction. This will be substantiated through extensive experiments.
DOT Hardware System
Fig. 2 shows the schematics of the frequency domain diffuse optical tomography (DOT) system, which we used in this study. The DOT system has been developed at the Korea Electrotechnology Research Institute (KERI) to improve diagnostic accuracy of the digital breast tomosynthesis (DBT) system by combining the DOT system with the DBT system for joint breast cancer diagnosis(38, 39). After its development, the system has been installed and been under clinical test at Asan Medical Center (AMC) (40). The DOT system briefly consists of four parts: light source, optical detector, optical probe, and data acquisition and controller. The light source has three fiber pigtailed laser diode modules with 785 nm, 808nm and 850 nm. 70 MHz RF signal is simultaneously applied to these light sources using bias-T, RF splitter and RF AMP. Two optical switches are used to deliver light to 64 specific positions in the source probe. During optical switching time, one-tone modulation light photons reach 40 detection fiber ends after passing an optical phantom and are detected simultaneously by 40 avalanche photodiodes (APD) installed in the home-made signal processing card. The DOT system uses an In-phase(I) and quadrature(Q) demodulator to get amplitude and phase of the signal in the signal processing card. The 40 IQ signal pairs are simultaneously acquired using data acquisition boards.
Next, to analyze the performance of the proposed approach in controlled real experiments, two biomimic phantoms with known inhomogeneity locations are created (see Fig. 4). The first phantom is made of polypropylene containing a vertically oriented cylindrical cavity that has 20 diameter and 15 height. The cavity is filled with the acetyl inclusion which has different optical properties to the background. For the second phantom, we used a custom-made open-top acrylic chamber (175mm x 120mm x 40mm) and three different sized knots (5mm, 10mm, 20mm diameter) for the mimicry of a tumor-like vascular structure. The knots were made using thin polymer tube (I.D 0.40mm, O.D 0.8mm diameter) and were filled with the rodent blood that was originated from the abdominal aorta of Sprague-Dawley rat that was under 1 to 2% isoflurane inhalation anesthesia. The chamber was filled with the completely melted pig lard and the medium was coagulated at room temperature for the imaging scan.
The mouse colon cancer cell line MC38 were obtained from Scripps Korea Antibody Institute(Chuncheon, Korea) and the cell line was cultivated in Dulbecco’s modified Eagle’s medium (DMEM, GIBCO, NY, US) supplemented with 10% Fetal bovine serum(FBS, GIBCO) and 1x Antibiotic-Antimycotic(GIBCO). For the tumor-bearing mice, 5x106 cells were injected subcutaneously into the right flank region of C57BL/6 mice aging 7-9 weeks (Orient Bio, Seongnam, Korea). Animal hairs were removed through trimming and waxing. Anesthesia was applied during the imaging scanning with an intramuscular injection of Zoletil and Rumpun (4:1 ratio) in normal saline solution. Mice were placed inside of the custom-made 80mm x 80mm x 30mm open-top acrylic chamber that had a semicircle hole-structure on the one side of the chamber for the relaxed breathing. A gap between the semicircle structure and the head was sealed with the clay. The chamber was filled with the water/milk mixture as 1000:50 ratios. All experiments associated with this study were approved by Institutional Animal Care and Use Committees of Asan Medical Center (IACUC no. 2017-12-198).
To determine the maximally usable source-detector distance, we measured signal magnitude according to the source-detector distances. We observed that the signals were above the noise floor when the separation distance between the source and the detector was less than 51 (). Therefore, instead of using measurements at all source-detector pairs, we only used the pairs having source and detector less than 51 apart. This step not only enhanced the signal-to-noise ratio (SNR) of the data but also largely reduced the number of parameters to be learned in the fully connected layer. For example, in the source-detector configuration of the biomimic phantom, the number of source-detector pairs reduced from 2560 to 466. This decreased the number of the parameters to train from 137,625,600 to 25,105,920, which is an order difference. To match the scale and bias of the signal amplitude between the simulation and the real data, we multiplied appropriate calibration factor to the real measurement to match the signal envelope from the simulation data. We centered the data cloud on the origin with the maximum width of one by subtracting the mean across every individual data and dividing it with its maximum value. To deal with the unbalanced distribution of nonzero values in the 3D label image and to prevent the proposed network learning a trivial mapping (rendering all zero values), we weighted the non-zero values by multiplying a constant according to the ratio of the total voxel numbers over the non-zero voxels.
Neural network training
In order to test the robustness of the deep network in real experiments and to obtain a large database in an efficient manner, the training data were generated by solving (2) using finite element method (FEM) based solver NIRFAST (see, e.g., (41, 42)). The finite element meshes were constructed according to the specifications of the phantom used in each experiment (see Table 1). We generated 1500 numbers of data by randomly adding up to three spherical heterogeneities of different sizes (having radius between 2 to 13 ) and optical properties in the homogeneous background. The optical parameters of the heterogeneities were constrained to lie in a biologically relevant range, i.e., two to five times bigger than the background values. The source-detector configuration of the data is set to match that of real experimental data displayed in Table 1
. To make the label in a matrix form, FEM mesh is converted to the matrix of voxels by using triangulation-based nearest neighbor interpolation with an in-built MATLABfunction. The number of voxels per each dimension used for each experiment can be found in Table 1. To train the network for different sizes of phantoms and source-detector configurations, we generated different sets of training data and changed the input and output sizes of the network, accordingly. The specifications of the network architecture are provided in Table 2. Note that the overall structure of the network remained the same except the specific parameters.
|# of sources||# of detectors||FEM mesh||
|# of voxels per xyz dimensions (resolution)|
|Mouse (with tumor)||12,288||63,426||0.0045||0.3452|
The input of the neural network is the multi-static data matrix of pre-processed measurements. To perform convolution and to match its dimension with the final output of 3D image, the output of the fully connected layer is set to the size of the discretized dimension for each phantom. All the convolutional layers were preceded by appropriate zero padding to preserve the size. We usedas a non-linear activation function for every layer except the last convolutional layer where ReLU is used. In the network structure for polypropylene phantom, for example, the first convolutional layer convolves 16 filters of with stride 1 followed by . The second convolutional layer again convolves the filter of with stride 1 followed by ReLU (Table 2).
. Training runs for up to 120 epochs with early stopping if the validation loss has not improved in the last 10 epochs. To prevent overfitting, we added a zero-centered Gaussian noise with standard deviation
and applied dropout on the fully connected layer with probability. We used a GTX 1080 graphic processor and i7-6700 CPU (3.40 GHz). The network took about 380 seconds for training. Since our network only used the generated simulation data for training, it suffered from the noise which cannot be observed from the synthetic data and overfitting due to lack of complexity. However, by carefully matching the signal envelops of the simulation data to those of the real data and tuning the parameters of the modules such as dropout probability and the standard deviation of the Gaussian noise, we could achieve the current network architecture which performs well in various experimental situations. We have not imposed any other augmentation such as shifting and tilting since our input data is not in the image domain but in the measurement domain which is unreasonable to apply such techniques. Every 3D visualization of the results is done by using ParaView (45).
Baseline algorithm for comparison
The baseline iterative algorithm was developed using the Rytov iterative method similar to (21). Specifically, after estimating the bulk optical properties as an initial guess, the Lippmann-Schwinger equation is linearized using Rytov approximation, and the associated penalized least squares optimization problem with penalty was solved. The optical properties are updated until the algorithm converges, and the background Green’s functions are updated accordingly so that the penalized least squares solutions are obtained using newly updated Green’s function at each iteration. We set the convergence criterion if the reconstructed optical parameter at current iteration has not improved in the last two iterations. Unless an initial guess is not closer, the algorithm generally converged in six to ten iterations and each iteration took approximately 40 seconds, which makes total working time less than 10 minutes.
To validate our deep learning approach in a quantitative manner, the reconstruction experiments from numerical phantom were first performed, and some representative results are shown in Fig. 3. Here, we used the network trained using the biomimic phantom geometry (see Table 2). The ground truth images are visualized with binary values to show the location of virtual anomalies clearly. To demonstrate the robustness of the algorithm under various situations, we randomly chose the data with different anomaly sizes and locations, with distinct z-locations. Our proposed network successfully resolved the anomaly locations.
Next, to analyze the performance of the proposed approach in controlled real experiments, two biomimic phantoms with known inhomogeneity locations are created (see Fig. 4). The first phantom is made of polypropylene containing a vertically oriented cylindrical cavity that has 20 diameter and 15 height. The cavity is filled with the acetyl inclusion which has different optical properties to the background. The schematic illustrations of the phantoms are shown with their specifications in Fig. 4. Then, we obtained the measurement data using our multi-channel system. The reconstructed 3D images from the conventional iterative method and our proposed network are compared. For the phantom with a single inclusion, both the conventional iterative method and our network accurately reconstructed the location of optical anomalies (Fig. 4 (top row)). The contrast is also clearly seen in the DBT image. The second phantom is made of lards to mimic a condition similar to the human breast. We inserted tubes inside the phantom and made three knots of different sizes. After solidifying the lards, the tubes were filled with blood to mimic a tumor-like vascular structure. In the DBT image, the locations of the knotted tubes are barely seen, as pointed out by the black arrows. In this phantom case, the reconstructed image using the iterative method exhibited globally distributed artifacts, and suffered from artifacts even after thresholding out the values smaller than the 60% of the max value (Fig. 4 (bottom row)). Since the artifacts had higher values, which dominate the signal from the inclusions, we manually stopped the algorithm at iteration three to prevent the smoothing regularization from erasing the real signal. In contrast, the reconstructed image using the proposed network finds three inclusions without those artifacts showing a better contrast than DBT image. Furthermore, it is remarkable that our reconstruction has a good contrast so that the knots are easily seen, whereas it was hard to distinguish the virtual tumors from the background using the DBT image.
Finally, we performed in-vivo animal experiments using a mouse (Fig. 5 and Fig. 6). In order to get the scattered photon density measurements, we recorded the data with and without the mouse placed in the chamber, which is filled with the water/milk mixture of ratio. Optical scattering data from animal experiments were collected using the single-channel system. Fig. 5 and Fig. 6 shows the reconstructed images of the mouse with and without tumor. Both the conventional and the proposed methods recovered high values at the chest area of the mouse. However, the iterative reconstruction finds a big chunk of high values around the left thigh of the mouse, which is unlikely with the normal mouse (Fig. 5). In contrast, our proposed network shows a high along the spine of the mouse where the artery and organs are located. Furthermore, in the mouse with tumor case, our network finds a high values around the right thigh, where the tumor is located (Fig. 6). The lateral view of our reconstructed images also match with the actual position of the mouse, whose head and body are held a little above the bottom plane due to the experiment setup.
Compared to the results of the conventional iterative method, our network showed a robust performance over the various examples. While the iterative reconstruction algorithm often imposed high intensities on spurious locations, such as the bottom plane of the phantom (Fig. 4 bottom row) and the big clutter at the left thigh of the normal mouse (Fig. 5 top row), our network found accurate positions with high values only at the locations where inclusions are likely to exist. Unlike the conventional method, which requires a parameter tuning for every individual case, it can infer from the measured data without additional pre- and post-processing techniques, which is desirable in the practical applications.
Note that our network had not seen any real data during the training nor the validation process. However, it successfully finds the inversion by learning only from the simulation data. Furthermore, even though we trained the network with sparse examples only, our network successfully finds both sparse (phantom, Fig. 4) and extended targets (mouse, Fig. 5 and Fig. 6) without any help of regularizers.
To further show that the proposed architecture is near-optimal, we performed ablation studies by changing or removing the components of the proposed architecture. Since our output is a 3D distribution, the network needs to find a set of 3D filters and . We observed that the network with 3D-convolution showed better axis resolution compared to the one using 2D convolution (Fig. 7), which is consistent with our theoretical prediction. One may suspect that the performance of the network has originated solely from the first layer since over 98% of the trainable parameters are from the fully connected layer. To address this concern, we tested the network with and without convolutional layers after the fully connected layer. We observed that the performance of our network deteriorated severely and it failed to train without the consecutive convolution layers. At least a single convolution layer with a single filter were needed to recover the accurate location of the optical anomalies (Fig. 7). However, paired encoder-decoder filter in the proposed network is better than just using a single convolution layer, which coincides with our theoretical prediction.
Finally, we observed that the network training become inaccurate if the activation functions do not match with the physical intuition (Fig. 8). Since the first fully connected layer should learn the non-linear inversion of Lippmann-Schwinger equation and the consecutive 3D-convolution layers are supposed to learn the local bases and their adjoints , it appeared unreasonable to put non-negativity by imposing a ReLU activation function at each layer. Therefore, we used tanh as an activation function for every layer except the last convolutional layer outputting the 3D distribution of absorption coefficients which are positive values. This gave the most stable and accurate training. On the other hand, if we change the activation functions of the fully connected layer and intermediate layers from tanh to ReLU, we found the network is hardly trained or the performance is degraded, which runs counter to the common sense in computer vision area that ReLU gives a robust training. Interestingly, if we only have ReLU for the 3D-convolutional layers, we could observe that the training is very unstable (Fig. 8). Only after the activation functions of both fully connected layer and the following 3D-convolution layer have tanh activation, the network starts to learn and shows a stable training (Fig. 8).
In this paper, we proposed a deep learning approach to solve the inverse scattering problem of diffuse optical tomography (DOT). Unlike the conventional deep learning approach,which tries to denoise or remove the artifacts from image to image using a black-box approach for neural network, our network was designed based on Lippmann-Schwinger equation to learn the complicated non-linear physics of the inverse scattering problem. Even though our network was only trained on the generated numerical data, we showed that the learned mapping is general over the real experimental data. The simulation and real experimental results showed that the proposed method outperforms the conventional method and accurately reconstructs the anomalies without iterative procedure or linear approximation. By using our deep learning framework, the non-linear inverse problem of DOT can be solved in end-to-end fashion and new data can be efficiently processed in a few hundreds of milliseconds, so it would be useful for dynamic imaging applications. Moreover, our new design principle based on deep convolutional framelets is a quite general framework to incorporate physics in network design, so we believe that the proposed network can be used for other inverse scattering problems from complicated physics.
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Appendix A Derivation of the Lippmann-Schwinger Equation for DOT
In this appendix, we derive the general form of the Lippmann-Schwinger equation if both absorption and scattering changes are present. Although the derivation of the Lippmann-Schwinger equation is already available in the literature(20, 21, 22), the derivation is mainly based on physical intuition rather than mathematical rigor. Therefore, we provide here a more rigorous mathematical derivation. Let the optical parameter distribution within the domain of interest be described by
where represents the characteristic function of a domain . We define the perturbations in optical parameters by
Then, the total photon density satisfies the transmission problem
where for any function and .
As for mathematical formalism to derive the Lippmann-Schwinger equation for the scattered photon density , the Robin Green function (), and the background photon density () for the homogeneous medium (in the absence of any anomaly) are required. The functions and are, respectively, the solutions to the boundary value problems
Notice that, by an appropriate use of the Green’s theorem and the boundary condition in (16),
where is the infinitesimal differential element on . Moreover, thanks to the boundary condition in (17),
From an application of the Green’s theorem on , one arrives at
Thanks to (16), one can see that
Using Green’s identity on the last term and by (16), one arrives at
After fairly easy manipulations, it can be seen that
(22) is the Lippmann-Schwinger integral equation that relates the scattered field to the perturbations in the optical properties. In particular, when the absorption perturbations are only present, the scattered photon flux measurement at the boundary is given by
Appendix B Bulk optical property estimation
The uniform bulk optical properties were found by fitting the experimental data to the model based data (diffusion equation in this case) using the iterative Newton-Raphson scheme (46). Under this scheme, two parameters and
were estimated through linear regression, whereis amplitude, is phase of the data (Fig. 9).
The minimization problem in (23) is the root finding problem which can be efficiently solved by Newton-Raphson method. Let . Newton-Raphson method find the roots of a real-valued function, such that
where the derivative is computed numerically.