T-matrix-based inverse algorithm for morphologic characterization of nonspherical particles using multispectral diffuse optical tomography

Abstract

An inverse algorithm is presented for tomographically imaging morphologic characteristics of nonspherical particles in heterogeneous turbid media. The particles are assumed to have spheroidal shapes with random orientations. The inverse algorithm is based on a relationship of the particle scattering spectra, obtained from multispectral diffuse optical tomography, and the size, concentration, and aspect ratio of spheroidal particles through the T-matrix method. The algorithm is implemented based on Tikhonov–Marquardt regularization techniques that minimize the difference between the observed and calculated scattering spectra. Different statistical models are assumed for the suspended nonspherical particles and the performance of the inverse algorithm is tested using noise-corrupted data up to 50% noise added to the observed scattering spectra.

1. Introduction

Optical imaging techniques for extracting cellular morphological information in tissue are beneficial for biomedical applications, such as tumor diagnosis and cellular understanding of tumor formation and growth [1–4]. We have previously shown [1–3] that the statistical parameters of the suspended particles embedded in heterogeneous turbid media or in vivo tissue can be noninvasively extracted using the scattering spectra obtained using multispectral diffuse optical tomography (MSDOT), an emerging modality that can provide tomographic images of tissue optical properties (i.e., absorption and scattering coefficients) using measured data collected along the tissue boundary [5]. In these early studies, the particles used were spherical, and the Mie theory was used to relate the reduced scattering spectra to the mean size and the concentration of particles under Gaussian particle size distribution. In a realistic setting, however, the particles involved in light scattering are not spherical in most cases. In fact, a slight deviation from the spherical shape may cause considerable changes in the scattering response [6].

To overcome the limitations associated with the spherical particle based Mie scattering model, a nonspherical particle scattering model needs to be considered [7,8]. For example, an equiphase-sphere approximation was proposed to calculate the light scattering from an arbitrary-shaped nonspherical particle; however, the extension of this method for calculating the light scattering from an aggregate of randomly oriented particles with different sizes is not straightforward. On the other hand, the T-matrix method, also referred as extended boundary conditions, is a powerful and accurate method for studying the light scattering from nonspherical particles [6]. The T-matrix method is directly derived from the Maxwell equations and it reduces to the Lorenz–Mie theory for spherical particles. It has been shown that as well as a single nonspherical particle, the T-matrix analysis can be used for an ensemble of independently scattering nonspherical particles [9,10]. The T-matrix method has been applied for light scattering analysis of biological tissue and nonbiological media [11–16]. In addition, the T-matrix method has been recently used to determine the size and shape of spheroidal particles based on the angle-resolved low-coherence interferometry [17–19].

The novelty of this work lies in the T-matrix based reconstruction of bulk statistical parameters of the suspended nonspherical particles embedded in heterogeneous turbid media by inverting the reduced scattering spectra obtained from MSDOT. Depending on the statistical model assumed, expressions relating the reduced scattering coefficient to the size, concentration, and aspect ratio of the particles are derived and the multiple unknown parameters are simultaneously retrieved using a Tikhonov–Marquardt regularization-based inverse algorithm. Both Gaussian and log-normal size distributions are assumed for the particles and the effect of the noise on the inverse algorithm is investigated. In this study, the T-matrix database is generated using the publicly available code developed by M. Mishchenko [10].

2. Methodology

A. Forward Model

In the framework of the T-matrix, the incident and scattering fields are expanded in outgoing (M̄_mn and N̄_mn) and regular (RgM̄_mn and RgN̄_mn) vector spherical wave functions. The relation between the expansion coefficients is established using a transition matrix or T-matrix [6,7]:

$${\overline{E}}^{\text{inc}}(\overline{R})=\sum _{n=1}^{\infty }\sum _{m=-n}^n[a_{mn}\text{Rg}{\overline{M}}_{mn}(k_1\overline{R})+b_{mn}\text{Rg}{\overline{N}}_{mn}(k_1\overline{R})],$$ (1a)

$${\overline{E}}^{\text{scat}}(\overline{R})=\sum _{n=1}^{\infty }\sum _{m=-n}^n[p_{mn}{\overline{M}}_{mn}(k_1\overline{R})+q_{mn}{\overline{N}}_{mn}(k_1\overline{R})],$$ (1b)

where Ē^inc and Ē^scat represent the incident and scattered waves, and k₁ and R̄ are the wave number and the position vector in the surrounding medium, respectively. a_mn and b_mn represent the expansion coefficients of the incident wave, while p_mn and q_mn are the expansion coefficients of the scatted wave. In a matrix form, the coefficients of the scattered field are obtained from the linear combinations of those of the incident field [6]:

$$\left[\begin{array}{c}p\\ q\end{array}\right]=T\left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{cc}{T}^{11}& T^{12}\\ T^{21}& T^{22}\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right].$$ (2)

The T-matrix is independent of the incident and scattered fields, and depends only on the shape, size, and refractive index of the particle. The details for calculating the coefficients and the T-matrix elements are given in [6]. After calculating the T-matrix, the scattering cross section of a nonspherical particle is given as follows:

$$C_{\text{sca}}=\frac{1}{k_1^2{\mid E_0^{\text{inc}}\mid }^2}\sum _{n=1}^{\infty }\sum _{m=-n}^n[{\mid p_{mn}\mid }^2+{\mid q_{mn}\mid }^2],$$ (3)

where $E_0^{\text{inc}}$ is the amplitude of the incident wave. The T-matrix analysis can be extended to an ensemble of randomly oriented monodisperse particles, provided that the multiple scattering between the particles is not considered. In this case, the following expression is obtained for ensemble-averaged scattering cross section [6]:

$$\langle C_{\text{sca}}\rangle =\frac{2\pi }{k_1^2}\sum _{n=1}^{\infty }\sum _{m=-n}^n\sum _{n^{\prime }=1}^{\infty }\sum _{m^{\prime }=-n^{\prime }}^{n^{\prime }}({\mid T_{mn{m}^{\prime }{n}^{\prime }}^{11}\mid }^2+{\mid T_{mn{m}^{\prime }{n}^{\prime }}^{12}\mid }^2+{\mid T_{mn{m}^{\prime }{n}^{\prime }}^{21}\mid }^2+{\mid T_{mn{m}^{\prime }{n}^{\prime }}^{22}\mid }^2),$$ (4)

where $T_{mn{m}^{\prime }{n}^{\prime }}^{ij}$ are the elements of the T-matrix calculated in the particle reference frame.

The asymmetry parameter and consequently the reduced scattering cross section per particle are calculated as follows [6]:

$$g=\langle cos\theta \rangle =\frac{a_1^1}{3},$$ (5)

$$\langle C_{\text{sca}}^{\prime }(r)\rangle =(1-g)\langle C_{\text{sca}}(r)\rangle ,$$ (6)

where $a_1^1$ is a coefficient obtained by expansion of the phase function, F̃₁₁(θ), in generalized spherical functions [6,7]. The phase function is (1, 1) element of the normalized scattering matrix:

$${\stackrel{\sim }{F}}_{11}(\theta )=\sum _{S=0}^{S_{\text{max}}}{a}_1^s{P}_{00}^s(cos\theta ).$$ (7)

The above analysis is valid only for monodisperse nonspherical particles. If the suspended particles have varying sizes and aspect ratios, as shown in Fig. 1, the scattering cross section and the asymmetry parameter should be averaged over appropriate size and aspect ratio distributions. Assuming that the particle equivalent radius is varying in an interval of r ∈ (r_min, r_max) and the aspect ratio is varying in an interval of ε ∈ (ε_min, ε_max), the following expressions can be straightforwardly derived for the averaged scattering cross section and the asymmetry parameter:

$$\langle C_{\text{sca}}\rangle ={\int }_{r=r_{min}}^{r=r_{max}}{\int }_{\epsilon ={\epsilon }_{min}}^{\epsilon ={\epsilon }_{max}}\text{d}r\text{d}\epsilon f(r,\epsilon )\langle C_{\text{sca}}(r,\epsilon )\rangle ,$$ (9)

$$g=\frac{1}{3\langle C_{\text{sca}}\rangle }{\int }_{r=r_{min}}^{r=r_{max}}{\int }_{\epsilon ={\epsilon }_{min}}^{\epsilon ={\epsilon }_{max}}\text{d}r\text{d}\epsilon f(r,\epsilon )\langle C_{\text{sca}}(r,\epsilon )\rangle {\alpha }_1^1(r,\epsilon ),$$ (10)

where 〈C_sca(r, ε)〉 represents the averaged scattering cross section of an ensemble of randomly oriented monodisperse spheroidal particles, having an equivalent radius of r and aspect ratio of ε. ${\alpha }_1^1(r,\epsilon )$ refers to the corresponding expansion coefficient [see Eq. (7)] and f (r, ε) is an appropriate two-dimensional (2D) distribution function, satisfying the following constraint:

$${\int }_{r=r_{min}}^{r=r_{max}}{\int }_{\epsilon ={\epsilon }_{min}}^{\epsilon ={\epsilon }_{max}}\text{d}r\text{d}\epsilon f(r,\epsilon )=1.$$ (11)

Fig. 1.

(Color online) Randomly oriented spheroidal particles with different sizes and aspect ratios.

If we assume that the particles have a concentration of φ (number/mm³), then the reduced scattering coefficient of the bulk suspension at a wavelength of λ, can be calculated as follows:

$${\mu }_s^{\prime }(\lambda )=\phi {\int }_{r=r_{min}}^{r=r_{max}}{\int }_{\epsilon ={\epsilon }_{min}}^{\epsilon ={\epsilon }_{max}}\text{d}r\text{d}\epsilon f(r,\epsilon )\times \left(1-\frac{{\alpha }_1^1(r,\epsilon )}{3}\right)\langle C_{\text{sca}}(r,\epsilon )\rangle .$$ (12)

B. Inverse Algorithm

Equation (12) serves as the forward for calculating the reduced scattering spectra of the spheroidal particles. Our inverse problem can be stated as the minimization of the following objective functional:

$${\chi }^2=\sum _{{\lambda }_j}{({\mu }_s^{\prime }{({\lambda }_j)}^o-{\mu }_s^{\prime }{({\lambda }_j)}^c)}^2,$$ (13)

where ${\mu }_s^{\prime }{({\lambda }_j)}^o$ represents the observed reduced scattering coefficient, and ${\mu }_s^{\prime }{({\lambda }_j)}^c$ is calculated as follows:

$${\mu }_s^{\prime }{({\lambda }_j)}^c=\phi {\int }_{r=r_{min}}^{r=r_{max}}{\int }_{\epsilon ={\epsilon }_{min}}^{\epsilon ={\epsilon }_{max}}\frac{e^{-\frac{1}{2}\left(\frac{{(r-r_m)}^2}{{\delta }_m^2}+\frac{{(\epsilon -{\epsilon }_m)}^2}{{\delta }_{\epsilon }^2}\right)}}{2\pi {\delta }_r{\delta }_{\epsilon }}\times \left(1-\frac{{\alpha }_1^1(r,\epsilon )}{3}\right)\langle C_{\text{sca}}(r,\epsilon )\rangle \text{d}r\text{d}\epsilon ,$$ (14)

where a Gaussian particle size distribution is assumed:

$$f(r,\epsilon )=\frac{1}{2\pi {\delta }_r{\delta }_{\epsilon }}{e}^{-\frac{1}{2}\left[\frac{{(r-r_m)}^2}{{\delta }_m^2}+\frac{{(\epsilon -{\epsilon }_m)}^2}{{\delta }_{\epsilon }^2}\right]},$$ (15)

where r_m and ε_m are the mean particle radius and aspect ratio, respectively. In this work, the sizes of the spheroidal particles are expressed in term of the radius of sphere having equivalent surface area. δ_r and δ_ε represent the standard deviations of the particle size and aspect ratio, respectively.

Suppose that the vector Δχ is the error between the observed and calculated reduced scattering coefficient and that Δζ is the vector that updates the unknown parameters:

$$\mathrm{\Delta }\chi =\left[\begin{array}{c}{\mu }_s^{\prime }{({\lambda }_1)}^o-{\mu }_s^{\prime }{({\lambda }_1)}^c\\ {\mu }_s^{\prime }{({\lambda }_2)}^o-{\mu }_s^{\prime }{({\lambda }_2)}^c\\ ·\end{array}\right],$$ (16a)

$$\mathrm{\Delta }\zeta =\left[\begin{array}{c}\mathrm{\Delta }\phi \\ \mathrm{\Delta }{\delta }_r\\ \mathrm{\Delta }{r}_m\\ \mathrm{\Delta }{\delta }_{\epsilon }\\ \mathrm{\Delta }{\epsilon }_m\end{array}\right].$$ (16b)

Using the Newton’s method [1,5], we obtain the following expression for calculating Δζ:

$$J^T\mathrm{\Delta }\chi =J^{T}J\mathrm{\Delta }\zeta ,$$ (17)

where J is the Jacobian matrix, calculated as follows:

$$J=\left[\begin{array}{ccccc}\frac{\partial }{\partial \phi }{\mu }_s^{\prime }({\lambda }_1)& \frac{\partial }{\partial {\delta }_r}{\mu }_s^{\prime }({\lambda }_1)& \frac{\partial }{\partial r_m}{\mu }_s^{\prime }({\lambda }_1)& \frac{\partial }{\partial {\delta }_{\epsilon }}{\mu }_s^{\prime }({\lambda }_1)& \frac{\partial }{\partial {\epsilon }_m}{\mu }_s^{\prime }({\lambda }_1)\\ \frac{\partial }{\partial \phi }{\mu }_s^{\prime }({\lambda }_2)& \frac{\partial }{\partial {\delta }_r}{\mu }_s^{\prime }({\lambda }_2)& \frac{\partial }{\partial r_m}{\mu }_s^{\prime }({\lambda }_2)& \frac{\partial }{\partial {\delta }_{\epsilon }}{\mu }_s^{\prime }({\lambda }_2)& \frac{\partial }{\partial {\epsilon }_m}{\mu }_s^{\prime }({\lambda }_2)\\ ·& ·& ·& ·& ·\\ ·& ·& ·& ·& ·\\ ·& ·& ·& ·& ·\end{array}\right].$$ (18)

The elements of the Jacobian matrix are determined by direct derivation of Eq. (14) with respect to the model parameters. It is known that the matrix J^TJ in Eq. (17) often becomes ill-conditioned. Therefore, it is necessary to employ appropriate regularization schemes to stabilize the matrix. In this study, we use Tikhonov–Marquardt regularization schemes as follows [5,20]:

$$(J^{T}J+\alpha I)\mathrm{\Delta }\zeta =J^T\mathrm{\Delta }\chi ,$$ (19)

where α is an appropriately chosen regularization parameter and I is the identity matrix.

After solving Δζ from Eq. (19), the parameters of interests are subsequently updated:

$${\zeta }_{\text{new}}={\zeta }_{\text{old}}+\mathrm{\Delta }\zeta .$$ (20)

The above analysis can be easily modified for other types of particle size/aspect ratio distributions. For example, for log-normal distribution, when there is no correlation between the size and the aspect ratio, the following expression is used for ${\mu }_s^{\prime }{({\lambda }_j)}^c$:

$${\mu }_s^{\prime }{({\lambda }_j)}^c=\phi {\int }_{r=r_{min}}^{r=r_{max}}{\int }_{\epsilon ={\epsilon }_{min}}^{\epsilon ={\epsilon }_{max}}\frac{e^{-\frac{1}{2}\left[\frac{{(log(r)-r_m)}^2}{{\delta }_m^2}+\frac{{(log(\epsilon )-{\epsilon }_m)}^2}{{\delta }_{\epsilon }^2}\right]}}{2\pi r\epsilon {\delta }_r{\delta }_{\epsilon }}\times \left(1-\frac{{\alpha }_1^1(r,\epsilon )}{3}\right)\langle C_{\text{sca}}(r,\epsilon )\rangle \text{d}r\text{d}\epsilon .$$ (21)

In this case, the expressions for the elements of the Jacobian matrix should be modified accordingly. In this work, we have used both Gaussian and log-normal distributions for our study as these two distributions are typically used to characterize cellular light scattering in tissue.

3. Results and Discussion

A. Reconstruction of Two Hexagonal Targets

In the first example, the inverse algorithm is tested for retrieving the mean size, concentration, and aspect ratio of spheoroidal particles suspended in two hexagonal-shaped inclusions embedded in a circular background. The diameter of each target is 10 mm. The background has a diameter of 56 mm and contains spherical particles with a mean radius of r_m = 0.5 μm and concentration of φ = 1.2 × 10⁶/mm³. A Gaussian distribution is assumed for the suspended particles in this case. The first target is made of monodisperse oblate particles and its center is located at (10 mm, 10 mm). It contains particles with a mean size of r_m = 1.5 μm, a concentration of φ = 0.7 × 10⁶/mm³, and a mean aspect ratio of ε = 1.8. The second target is made of mono-disperse prolate particles and is centrally located at (−10 mm, −10 mm). The second target has r_m = 1.0 μm, φ = 0.7 × 10⁶/mm³, and ε = 0.6. The standard deviations are assumed to be 10% of the corresponding mean parameter as δ_r = 0.1r_m and δ_ε = 0.1ε_m. The reduced scattering coefficients are calculated using Eq. (14) at the following nine wavelengths: 638 nm, 673 nm, 690 nm, 733 nm, 775 nm, 808 nm, 840 nm, 915 nm, 922 nm, and 965 nm. The refractive indices of the targets and the background are 1.59 and 1.33, respectively. A finite element mesh with 700 nodes and 1334 triangle elements was used for reconstructions. The initial guesses for the particle size, concentration, and the aspect ratio, at each node, are randomly chosen within ±20% of the actual values.

The observed reduced scattering coefficients at the wavelengths of interest are given in Table 1. The values of the model parameters used for generating the observed reduced scattering coefficients are close to the ones for breast tissue. Figure 2(a) presents the averaged error (over all nodes) between the observed and computed scattering spectra versus the iteration number, while Fig. 2(b) gives all the nodes of the finite element mesh used to average out the reconstruction error. As can be seen, the reconstruction error, given by Eq. (13), drops from 2 × 10⁻² to 6 × 10⁻⁶ as the iteration number increases.

Table 1.

Reduced Scattering Coefficient at Nine Wavelengths of Interest (mm⁻¹)

Wavelength/Target	638 nm	673 nm	690 nm	733 nm	775 nm	808 nm	840 nm	915 nm	922 nm
Target 1	1.36	1.348	1.345	1.329	1.31	1.295	1.284	1.25	1.25
Target 2	0.605	0.594	0.589	0.577	0.565	0.556	0.547	0.529	0.527
Background	0.204	0.198	0.196	0.19	0.184	0.18	0.176	0.167	0.167

Fig. 2.

(a) Averaged reconstruction error versus the iteration number. (b) The mesh used in calculation of averaged reconstruction error.

The recovered images of particles size, concentration, and aspect ratio are shown in Figs. 3(a), 3(c), and 3(e), respectively. We see that the shape and parameters of the targets and the background medium are quantitatively reconstructed. However, we do note some artifacts in the background of the aspect ratio image. In Figs. 3(b), 3(d), and 3(f), the values of retrieved parameters are shown along a line passing through centers of the two targets (x = y line). For comparison, the true model parameters are shown in dotted line. We can observe that the retrieved parameters agree very well with the true ones.

Fig. 3.

Reconstructed two-target images of (a, b) particle size, (c, d) concentration, and (e, f) aspect ratio without noise added.

To test the sensitivity of our inverse algorithm to noise effect, ±20% and ±50% noise is, respectively, added to the reduced scattering coefficient at each node in the computational domain. The initial guesses, chosen for the unknown parameters, are within ±20% of the actual values. The reconstructed images for these two cases are shown in Figs. 4 and 5, respectively. From Figs. 4 and 5, we see that the targets can be recovered reasonably well when 20% or 50% noise is added to the scattering spectra. We note that the mean size and aspect ratio images [Figs. 4(a), 4(e), 5(a), and 5(e)] are clearly better reconstructed than the concentration images [Figs. 4(c) and 5(c)] for both cases. This is due to the fact that the contrast in concentration of particles between the target and background is lower compared with that in particle size and aspect ratio. To compare the retrieved parameters more quantitatively, the mean particle size, concentration, and aspect ratio along x = y line, when 20% noise is added, are shown in Figs. 4(b), 4(d), and 4(f), respectively. The corresponding results, when 50% noise is added, are shown in Figs. 5(b), 5(d), and 5(f). In both cases, the mean particle sizes show good agreement with the true ones.

Fig. 4.

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 20% noise added.

Fig. 5.

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 50% noise added.

B. Reconstruction of One Hexagonal Target Using Initial Search Algorithm

For the cases studied so far, the initial guesses are chosen within ±20% of the actual values. In a practical situation a priori information about the particle characteristics may not be available. In such cases, an initial search should be conducted to get the estimates of particle parameters in each region (i.e., target and background). The goal of the initial search algorithm is to find the initial parameters that give minimized error in Eq. (13) over a range of possible values. Here we use an example to illustrate this method. In this case, a hexagonal-shaped target, off-centered at (10 mm, 0), contains particles having a mean size, concentration, and aspect ratio of r_m = 1.5 μm, φ = 0.7 × 10⁶/mm³, and ε = 1.8. The reconstructed images with 20% and 50% noise added to the observed scattering spectra are shown in Figs. 6 and 7, respectively. We can see that the images are well reconstructed. The values of the retrieved parameters are close to the actual values, especially for the case of 20% noise. Similar to the previous examples, the retrieved parameters along y = 0 line, which passes through the center of the target, are shown in Figs. 6(b), 6(d), and 6(f) when 20% noise is added and in Figs. 7(b), 7(d), and 7(f) when 50% noise is added. It is observed that the retrieved values of particle sizes and aspect ratios are closer to the true ones compared with the concentration images.

Fig. 6.

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 20% noise added.

Fig. 7.

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 50% noise added.

C. Reconstruction with Log-Normal Distribution

Here log-normal particle size distribution is used to study the effect of different size distribution on the performance of the inverse algorithm. For this purpose, Eq. (21) is used to calculate the reduced scattering coefficients as discussed earlier. For this case, the search algorithm is used to get the estimates of initial unknown parameters. Figures 8 and 9 present the reconstructed images when 20% and 50% noise is added to the scattering spectra, respectively. We note that the image of the aspect ratio is well recovered for both cases, while relatively strong artifacts are seen in the images of mean size and concentration. The overall image quality for a log-normal distribution is notably worse than that for a Gaussian distribution [Figs. 6 and 7]. This is most likely due to the logarithmic terms in the Jacobian matrix, which may exacerbate the numerical errors, especially when the input data are already noisy.

Fig. 8.

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 20% noise added for a log-normal distribution.

Fig. 9.

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 50% noise added for a log-normal distribution.

Multiple scattering effects on a particle are not considered in this work. This is a reasonable assumption for this study. It is known that when the distance between particle centers is as small as four times their radius, such multiple scattering effects are negligible [6]. In our current study, the concentrations of particles used in the target area and the background are less than 1% by volume. According to [21,22], particle concentration of 1% or lower falls into the single scattering criteria. Nonetheless, particle scattering models for highly dense suspension of particles are under research in our lab.

4. Conclusions

We have presented a T-matrix based inverse algorithm that is capable of simultaneously retrieving mean size, concentration, and aspect ratios of nonspherical particles in heterogeneous media. The simulation results obtained demonstrate the relatively robust performance against noisy data. The experimental validation of our inverse algorithm using various nonspherical particles is under way in our laboratory, and we plan to report the experimental studies elsewhere in the near future.