We present a method that is capable of imaging particle size and concentration in heterogeneous turbid media using multispectral diffuse optical tomography (MSDOT). Successful images of mean particle size and particle concentration are achieved for the first time using tissue phantom data obtained from a ten-wavelength MSDOT system.
©2004 Optical Society of America
The interest of diffuse optical tomography (DOT) in breast cancer detection continues to grow rapidly due to its relatively high contrast in both tissue absorption and scattering and tissue functional information available from MSDOT. While absorption spectra have been studied extensively for functional imaging, the exploration of scattering spectra has just begun. Optical spectroscopy studies have shown that scattering spectra are correlated with tissue morphology and that both nuclei and mitochondria contribute to tissue scattering significantly [1–4]. On the other hand, it is well known in pathology that tumor cells/nuclei are considerably enlarged relative to normal ones . Thus one can expect that tissue morphology can be imaged using scattering spectra obtained from MSDOT. Thus far scattering spectra in MSDOT has been studied with a simple power law [6, 7], which was originated from spectroscopy studies with homogeneous media [8–11]. The advantages of the power law are its simplicity and reasonable accuracy under certain conditions. The disadvantage is that the constants in the power law have no clear physical meanings and that particle size and concentration cannot be directly obtained from this empirical relationship.
In this paper, we describe a method for studying scattering spectra in MSDOT based on more rigorous Mie scattering theory without the limitations associated with the power law assumption. We present for the first time images of particle size and concentration using tissue phantom experiments where a target is embedded in a scattering medium. The recovered images obtained are quantitative in terms of the target size and shape, and the particle size and concentration in both the target and background media.
2. Methods and materials
2.1 Reconstruction algorithms
Two inverse algorithms are required in order to obtain particle size distribution and concentration with MSDOT. The first is a DOT algorithm for the recovery of spectroscopic scattering images in heterogeneous turbid media. Our DOT algorithm, described in detail elsewhere [12–14], uses a regularized Newton’s method to update an initial optical property distribution iteratively in order to minimize an object function composed of a weighted sum of the squared difference between computed and measured optical data at the medium surface. The computed optical data (i.e., photon intensity) is obtained by solving the photon diffusion equation with finite element method. The second algorithm is one that extracts the particle morphological information using the scattering spectra obtained from the DOT algorithm. It casts the reconstruction as an optimization problem in which the optimization parameters are coefficients in a probability function such as Gaussian distribution function using a priori assumptions. This inverse algorithm is based on a least squares optimization, where the difference between measured and computed scattering spectra is iteratively minimized by adjusting the optimization parameters under Mie scattering theory [15–18].
Once the reduced scattering spectra, µ′s(λ), are recovered using DOT algorithm, the following relationship from Mie theory allows us to obtain the particle size distribution and concentration [15, 17]:
where Qscat is the scattering efficiency; g is the average cosine of scattering angles; x is the particle size; n is the refractive index of particles; ϕ is the particle concentration/volume fraction; f(x) is the particle size distribution. Both Qscat and g can be computed with Mie theory . In Eq. (1) we have assumed that particles act as independent scatterers without particle-particle interaction. In order to solve for f(x) and ϕ from measured scattering spectra, an inversion of Eq. (1) must be obtained. Our numerical inversion is based on a Newton-type iterative scheme through least-squares minimization of the objective functional:
where (µ′s and (µ′s are the observed and computed reduced scattering coefficients at ten wavelengths, j=λ1, λ2,…, λ10 (more wavelengths can be used, depending on the number of wavelength available from the experimental system). In the reconstruction, we have assumed a Gaussian particle size distribution in this study,
Now the particle sizing task becomes to recover three parameters a, b and ϕ. As described in detail in Refs. 15–16, we have used a combined Marquardt-Tikhonov regularization scheme to stabilize the reconstruction procedure. We have also found that the particle size inversion is not sensitive to the choice of the initial parameters which can be 100% greater than the true values.
Phantom experiments were conducted using our ten-wavelength DOT system (638, 673, 690, 733, 775, 808, 840, 915, 922 and 960nm). This newly developed imaging system and its calibration were described in detail elsewhere [20, 21]. Two sets of phantom experiments were conducted to demonstrate the overall approach for imaging particle size and concentration using MSDOT. The cylindrical background phantom had a radius of 25mm, an absorption coefficient of 0.005/mm (India ink as absorber) and a reduced scattering coefficient of 1.0/mm (Intralipid as scatterer). A thin glass tube (9mm in inner diameter, 0.4mm in thickness) containing polystyrene suspensions (Polysciences, Warrington, PA) was embedded off-center in the background solid phantom. Two different types of polystyrene spheres were used in the experiments: one had a diameter of 2.06µm and a concentration of 0.52%, and the other had a diameter 5.66µm and a concentration of 2.62%. The refractive index of the spheres and their surrounding aqueous medium are 1.59 and 1.33, respectively.
3. Results and discussion
We first performed simulations to evaluate the sensitivity of particle sizing on the number of wavelengths used. “Measured” µ′s spectra were generated using Eq. (4) with a=2.86 µ m, b=0.145 µ m and ϕ=1.02% for 10, 20 and 50 wavelengths between 600 and 1000nm, respectively. When 5% noise was added to each set of “measured” µ′s spectra, we found that the relative errors of recovering the particle parameters were within 14% using the 10-wavelength spectra, while such errors were as low as 4% when 50-wavelength spectra were used. The Mie theory fittings using the extracted parameters at 10, 20 and 50 wavelengths are shown in Fig. 1(a) where the exact spectra are also presented for comparison. We see that the 10-wavelength spectra are able to provide quantitatively accurate reconstruction. We also performed simulations to test the noise sensitivity when 1, 5, or 10% random noise was added to the 10-wavelength spectra. The relative errors of the recovered parameters (a and ϕ) were calculated to be 3, 14, and 17% for parameter a and 3, 11 and 14% for the parameter ϕ. However, the recovery of the standard deviation was sensitive to noise, which had a relative error of 93% when 10% noise was added.
A finite element mesh with 634 nodes was used for the DOT reconstructions. To show the accuracy of the DOT reconstruction, Fig. 1(b) depicts the recovered µ′s spectra at a typical node location in the target area for the 2.06 µm polystyrene case, in comparison with the corresponding Mie theory fitting using the extracted particle parameters. And the DOT reconstructed absorption and reduced scattering images for the 2.06 µm polystyrene case for all ten wavelengths are shown in the Fig. 2. The recovered mean size and concentration at each node are used for imaging display. Figure 3 presents the reconstructed images of particle size and concentration for the 2.06 and 5.66µ m polystyrene cases. We immediately note that the particle size and concentration of both the target and background are quantitatively imaged. The reconstructed mean particle size and concentration in the background were found Fig. 1. (a) The µ′s spectra generated by Eq. (1) and the Mie fittings using recovered particle parameters from simulated data (5% noise) with 10, 20, and 50 wavelengths, respectively. (b) Experimental µ′s spectra DOT reconstructed at a typical node in the target area and the corresponding Mie fitting using recovered particle parameters for the 2.06 µm polystyrene case. to be within 154.7~155.1nm and 0.92~1.21% for the 2.06 µm polystyrene case, and 155.3~155.5nm and 1.17~1.95% for the 5.66 µm polystyrene case, compared to 150nm and 1%, the equivalent mean particle size and concentration of the actual Intralipid/Ink background . In the target region, the recovered mean particle sizes are in the range of 1.31 to 2.25 µm with average value of 1.72µm for the 2.06µm polystyrene case and 3.48 to 5.97 µm with average value of 4.62 µm for the 5.66 µm polystyrene case, while the reconstructed concentrations are in the range of 0.48 to 0.87% with average value of 0.65% for the 2.06 µm polystyrene case and 1.76 to 2.70% with average value of 2.25% for the 5.66 µm polystyrene case. Similar to the simulations, we see that the standard deviation recovery is sensitive to the noise. The maximum relative errors of the extracted standard deviation are up to 67.9% and 84.9% for the 2.06 µm and 5.66 µm polystyrene case respectively.
It should be noted that while only the recovered µ′s spectra were needed for particle sizing, the absorption images were quantitatively reconstructed for the experimental two cases studied [see Figs. 2(a1)-2(a10); note that the absorption contrast between the target and background was extremely low in both cases, resulting in strong artifacts along the boundary]. Others and we have recently shown repeated experimental evidence that the cross-talk between µa and µ′s images can be minimized using CW based DOT reconstructions [7, 12, 14, 22], suggesting that future work involving phantom studies with different levels of absorption in the target are feasible and worthy.
In our particle sizing, the refractive indices of the polystyrene suspensions and the surrounding medium are important parameters and have been assumed known as a priori. In a clinical situation, we can obtain these information empirically from the literature, or we can ultimately recover the refractive indices of scatterers as we reconstruct the PSD and concentration. In response to the possible perturbation of the glass tube used in the experiments, we have previously shown that such perturbation was insignificant in the image reconstruction . In this work, we assumed that the scatterers are spherical. But in tissues, while larger scatterers such as nuclei are spherical, the smaller scatterers such as mitochondria are ellipsoidal. A possible solution to this is to consider a modified Wentzel-Kramers-Brillouin model (WKB) theory for non-spherical particles as described in . Compared with the diffuse reflectance spectroscopy (DRS), our method can image the particle size and concentration in heterogeneous media whereas DRS can only deal with homogeneous media. In addition, we believe our method can provide more accurate particle sizes and concentrations than DRS, because DRS often is based on analytical solutions to the diffusion equation with the assumption of infinite or semi-infinite media.
In conclusions, we have demonstrated quantitative imaging of particle size and concentration of heterogeneous turbid media using MSDOT. The choices of the 2.06 and 5.66 µ m polystyrene particles were intended to simulate typical mitochondria and nuclei, respectively. The phantom results presented suggest that the method described in this paper may be applied for in vivo imaging of tissue morphology, adding more parameters for clinical decision-making.
This research was supported in part by a grant from the National Institutes of Health (NIH) (R01 CA90533).
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