## Parametric estimation of 3D tubular structures for diffuse optical tomography |

Biomedical Optics Express, Vol. 4, Issue 2, pp. 271-286 (2013)

http://dx.doi.org/10.1364/BOE.4.000271

Acrobat PDF (1447 KB)

### Abstract

We explore the use of diffuse optical tomography (DOT) for the recovery of 3D tubular shapes representing vascular structures in breast tissue. Using a parametric level set method (PaLS) our method incorporates the connectedness of vascular structures in breast tissue to reconstruct shape and absorption values from severely limited data sets. The approach is based on a decomposition of the unknown structure into a series of two dimensional slices. Using a simplified physical model that ignores 3D effects of the complete structure, we develop a novel inter-slice regularization strategy to obtain global regularity. We report on simulated and experimental reconstructions using realistic optical contrasts where our method provides a more accurate estimate compared to an unregularized approach and a pixel based reconstruction.

© 2013 OSA

## 1. Introduction

1. Y. Yang, A. Sassaroli, D. K. Chen, M. J. Homer, R. A. Graham, and S. Fantini, “Near-infrared, broad-band spectral imaging of the human breast for quantitative oximetry: applications to healthy and cancerous breasts,” J. Innov. Opt. Health Sci. **3**, 267–277 (2010). [CrossRef]

19. M. E. Kilmer, E. L. Miller, A. Barbaro, and David Boas, “Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. **42**, 3129–3144 (2003). [CrossRef] [PubMed]

1. Y. Yang, A. Sassaroli, D. K. Chen, M. J. Homer, R. A. Graham, and S. Fantini, “Near-infrared, broad-band spectral imaging of the human breast for quantitative oximetry: applications to healthy and cancerous breasts,” J. Innov. Opt. Health Sci. **3**, 267–277 (2010). [CrossRef]

6. R. A. Jesinger, G. E. Lattin, E. A. Ballard, S. M. Zelasko, and L. M. Glassman, “Vascular abnormalities of the breast: arterial and venous disorders, vascular masses, and mimic lesions with radiologic-pathologic correlation,” Radiographics **31**, E117–E136 (2011). [CrossRef] [PubMed]

1. Y. Yang, A. Sassaroli, D. K. Chen, M. J. Homer, R. A. Graham, and S. Fantini, “Near-infrared, broad-band spectral imaging of the human breast for quantitative oximetry: applications to healthy and cancerous breasts,” J. Innov. Opt. Health Sci. **3**, 267–277 (2010). [CrossRef]

3. S. van de Ven, A. Wiethoff, T. Nielsen, B. Brendel, M. van der Voort, R. Nachabe, M. Van der Mark, M. Van Beek, L. Bakker, L. Fels, S. Elias, P. Luijten, and W. Mali, “A novel fluorescent imaging agent for diffuse optical tomography of the breast: first clinical experience in patients,” Mol. Imaging Biol. **12**, 343–348, 2010. [CrossRef]

10. H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: Initial simulation, phantom, and clinical results,” Appl. Opt. **42**, 135–145 (2004). [CrossRef]

2. A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous diffuse optical tomography,” Opt. Lett. **29**, 256–258 (2004). [CrossRef] [PubMed]

4. P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express **15**, 8043–8058 (2007). [CrossRef] [PubMed]

16. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**, 2950–2961 (1999). [CrossRef]

12. F. Larusson, S. Fantini, and E. L. Miller, “Hyperspectral image reconstruction for diffuse optical tomography,” Biomed. Opt. Express **2**, 947–965 (2011). [CrossRef]

15. F. Larusson, S. Fantini, and E. L. Miller, “Parametric level set reconstruction methods for hyperspectral diffuse optical tomography,” Biomed. Opt. Express **3**, 1006–1024 (2012). [CrossRef] [PubMed]

15. F. Larusson, S. Fantini, and E. L. Miller, “Parametric level set reconstruction methods for hyperspectral diffuse optical tomography,” Biomed. Opt. Express **3**, 1006–1024 (2012). [CrossRef] [PubMed]

5. Y. Bresler, J. A. Fessler, and A. Macovski, “A bayesian approach to reconstruction from incomplete projections of a multiple object 3D domain,” IEEE Trans. Pattern Anal. Mach. Intell. **2**, 840–858 (1989). [CrossRef]

5. Y. Bresler, J. A. Fessler, and A. Macovski, “A bayesian approach to reconstruction from incomplete projections of a multiple object 3D domain,” IEEE Trans. Pattern Anal. Mach. Intell. **2**, 840–858 (1989). [CrossRef]

*x*−

*y*−

*z*), if

*z*is assumed to correspond to the “vertical”, then each PaLS primitive resides in a

*x*−

*z*plane. This takes advantage of the tubular nature of vessels found in breast tissue [6

6. R. A. Jesinger, G. E. Lattin, E. A. Ballard, S. M. Zelasko, and L. M. Glassman, “Vascular abnormalities of the breast: arterial and venous disorders, vascular masses, and mimic lesions with radiologic-pathologic correlation,” Radiographics **31**, E117–E136 (2011). [CrossRef] [PubMed]

## 2. Forward problem

*x*axis, acquiring data in slices along the

*y*axis, where detectors are restricted to the same

*z*−

*x*plane as the source. The medium is considered to consist of a nominal and generally homogeneous background and a collection of localized structures such as blood vessels in our application. Taking advantage of this arrangement the forward model is simplified by employing a linearization of the forward model adapted to the problem of recovering 2D slice information. Linear approximations used in DOT have been proven to produce reasonable good images in cases where the variation in optical properties within the medium is relatively small, which is the scenario often encountered in breast imaging [15

15. F. Larusson, S. Fantini, and E. L. Miller, “Parametric level set reconstruction methods for hyperspectral diffuse optical tomography,” Biomed. Opt. Express **3**, 1006–1024 (2012). [CrossRef] [PubMed]

28. T. Tarvainen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Corrections to linear methods for diffuse optical tomography using approximation error modelling,” Biomed. Opt. Express **1**, 209–222 (2010). [CrossRef]

**r**) is the photon fluence at position

**r**due to light injected into the medium,

*v*is the electromagnetic propagation velocity in the medium,

*S*(

**r**) =

*δ*(

**r**) represents a point source. Lastly

*D*is the diffusion coefficient, given by

*D*=

*v*/(3

*μ*′

*) where*

_{s}*μ*′

*is the reduced scattering coefficient. Here we consider uniform scattering properties that mimic those of breast tissue, hence we assume that*

_{s}*D*is constant with respect to the spatial variable

**r**.

*μ*, and a spatially varying perturbation Δ

_{a}*μ*(

_{a}**r**). This entails defining the fluence as the sum of the incident field, Φ

*(*

^{i}**r**), and a scattered fluence, Φ

*(*

^{s}**r**). To obtain a linear relationship between the measurements and the absorption concentrations, we subtract Eq. (1) from the perturbed version which leaves us with an equation for the scattered fluence where

*k*

^{2}(

**r**) = (

*v*/

*D*)Δ

*μ*(

_{a}**r**). For a wide variety of unbounded and bounded geometries, the Green’s function,

*G*(

**r**,

**r**′), for the operator on the left hand side of Eq. (2) can be obtained [20

20. A. Mandelis, *Diffusion-Wave Fields: Mathematical Methods and Green Functions* (Springer, 2001), 1st ed. [CrossRef]

*k*

^{2}=

*v*/

*D*Δ

*μ*in Eq. (2),

_{a}**r**

*is the location of the detector and Φ*

_{d}*(*

_{i}**r**′,

**r**

*) denotes the incident field at position*

_{s}**r**and due to a source located at

**r**

*.*

_{s}*x*axis, yielding

*K*scans along the

*y*axis. Using our forward model we define

**c**

*∈ ℝ*

_{k}^{Np}as the vector of discretized

*μ*associated with the

_{a}*k*slice in the rectangular region, where

^{th}*N*is number of pixels in each slice image, and

_{p}*m*,

*j*)

*element of the*

^{th}**K**

*represents the*

_{k}*m*source-detector pair and

^{th}*j*pixel in the

^{th}*k*slice of the 3D medium. Assuming that for a given experiment

^{th}*N*source-detector pairs are used for all

_{sd}*K*slices then if

*N*is the number of pixels in each slice the dimensions of the whole matrix

_{p}**K**is

*N*×

_{sd}K*N*.

_{p}K*y*-axis. This has a significant impact on the accuracy of the forward model where “out-of-plane” effects on the photon migration from the overall tubular structure are not modeled by the forward model. Our method, detailed in Sections 3 and 4, of parameterizing the shape and regularization is able to recover accurate 3D structures in spite of this, even with limited data sets. Additionally this approach is easily expandable, by filling in the off-diagonal elements of

**K**which will be considered in future developments discussed in Section 8.

## 3. Parametric level-set method

**3**, 1006–1024 (2012). [CrossRef] [PubMed]

*χ*(

*x*,

*y*) we then represent each slice image to be reconstructed as where

*k*= 1, 2,...,

*K*. For the purpose of reconstructing absorption concentration the unknown value in this formulation is the constant concentration values of the primitive,

*μ*.

_{a}*χ*(

*x*,

*y*) is defined to be the

*τ*level set of a Lipschitz continuous object function 𝒪 : ℱ → ℝ such that 𝒪 >

*τ*in Ω(

*x*,

*y*), 𝒪 <

*τ*in ℱ\Ω and 𝒪(

*x*,

*y*) =

*τ*in

*∂*Ω. Using 𝒪(

*x*,

*y*),

*χ*(

*x*,

*y*) is written as where

*H*is the step function where in practice we use smooth approximations of the step function and its derivative, the delta function [21].

*x*,

*y*) is represented parametrically, so instead of using a dense collection of pixel or voxel values [22], we represent it by using basis functions where

*κ*’s are the weight coefficients whereas

_{i}*ψ*(

_{i}*x*,

*y*) are the functions which belong to the basis set of 𝒫 = {

*ψ*

_{1},

*ψ*

_{2},...,

*ψ*}. Possible choices for the 𝒫 basis set include polynomial or radial basis functions, where for the purpose of this paper we use compactly supported radial basis function [18

_{L}18. A. Aghasi, M. Kilmer, and E. L. Miller, “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. **4**, 618–650 (2011). [CrossRef]

27. O. Semerci and E. L. Miller, “A parametric level set approach to simultaneous object identification and background reconstruction for dual energy computed tomography,” IEEE Trans. Image Process. **21**, 2719–2734 (2012). [CrossRef] [PubMed]

*β*defines the dilation factor of the radial basis function.

_{i}*κ*, of each basis function keeping their position fixed. To ensure the adaptability of the method to different shapes we now allow each basis function to “roam” within the imaging medium. As discussed in [15

_{i}**3**, 1006–1024 (2012). [CrossRef] [PubMed]

*L*, used in a fixed grid framework directly affected the results, and sometimes resulted in reconstruction errors. By removing the fixed grid, and instead estimating the centers of the basis functions allows for greater accuracy and adaptability for the method. We represent the center point of the

*i*basis function as

^{th}**r**

*= (*

_{i}*x*,

_{i}*y*).

_{i}*K*represents the number of slices, where

*L*basis functions reside giving

*κ*= [

_{k}*κ*

_{1},...,

*κ*]

_{L}*,*

^{T}*β*= [

_{k}*β*

_{1},...,

*β*]

_{L}*and*

^{T}**r**

*= [*

_{k}**r**

*,...,*

_{i}**r**

*]*

_{L}*. Now our forward model in Eq. (4) can be expressed as This parameterization of the forward model contains far fewer unknowns than a pixel based method and allows for high adaptability with the movable basis functions. As an example, a single image with 4544 pixels is reconstructed using 21 basis functions, requiring only 85 unknowns.*

^{T}*K*, and the plane in which the

*k*primitive resides as

^{th}*y*=

*y*, where

_{k}*k*= 1,...,

*K*and the number of basis functions by

*L*in each plane where

*l*= 1,...,

*L*. For simplicity, we assume that the primitives are equally spaced, though this assumption can be easily relaxed. Thus far, our model only defines the object at

*K*points on the

*y*-axis, where in essence the primitives may be interpreted as cross section of the overall 3D object. The object description at all other points on the

*y*axis is recovered independently, and then combined to represent the 3D structure.

## 4. Image reconstruction

**c**from Φ

*, is formed as a regularized optimization problem of the form where*

^{s}**W**represents the structure of the noise corrupting the data. The first term in Eq. (10) requires that the estimated value of

**c**is consistent with the observed measurement of Φ

*. The second term of Eq. (10) is a regularization term that correlates the parameter vector between slices. Considering the prior information of tubular structure anatomy of breast tissues, it encourages correlating reconstructions between slices in the cost functional. Therefore the second term Eq. (10) ensures that a reconstruction between slices will result in connected structures, which provides better approximation of the structure than a unregularized function. We structure*

_{s}**L**to penalize the difference between similar parameters on adjacent primitives. That is to say, we impose a penalty for the difference between centers,

**r**

*and*

_{i}**r**

_{i}_{+1}for

*i*= 1,...,

*K*− 1, the value of absorption,

*κ*so that

_{i}**L**is given by. Where

**I**is a diagonal matrix where number of diagonal elements are the same as number of elements in

*θ*, and

_{k}**A**⊗

**B**is the Kronecker product [13] of

**A**and

**B**and

**L**

*is written as In order to demonstrate the effectiveness of our regularization method, we evaluate a tomographic reconstruction over a range of values for the regularization parameter*

_{d}*α*. As

*α*is varied the algorithm trades off the cost associated with the regularization penalty against the cost associated with the data. To select the optimal regularization parameter we employ the commonly used L-curve method, detailed in Section 7 [7

7. M. Belge, M. E. Kilmer, and E. L. Miller, “Efficient determination of multiple regularization parameters in a generalized l-curve framework,” Inverse Probl. **18**, 1161–1183 (2002). [CrossRef]

**W**matrix reflects the structure of the noise corrupting the data [9

9. R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techinques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. **45**, 1051–1069 (2000). [CrossRef] [PubMed]

*m*elements of Φ,

^{th}**W**is constructed as a diagonal matrix with 1/

*σ*the

_{m}*m*element along the diagonal. For the experimental and simulated data the variance is calculated from where Ω(

^{th}*m*) corresponds to the photon count for each source-detector pair. The SNR for each element of Φ is then calculated from

**J**is required. The Jacobian contains derivatives of

*ε*with respect to each element in the parameter vector

*θ*The solution is then obtained by updating

*θ*at each iteration as

*θ*

^{n+1}=

*θ*+

^{n}**h**where

**h**is the solution to the following linear system where

**I**is the identity matrix,

*ρ*is the damping parameter affecting the size and direction of

**h**and found via and appropriate line search algorithm [23]. The stopping criteria used when iterating Eq. (18) is the discrepancy principle [24

24. C. R. Vogel, *Computational Methods for Inverse Problems*, 1st ed. (SIAM, 2002). [CrossRef]

## 5. Simulation analysis

*z*−

*x*and

*z*−

*y*planes. The true geometries of phantoms 1 and 2 are shown in Fig. 3(b) and (c), respectively. Each cylinder in the medium has Δ

*μ*= 0.04 cm

_{a}^{−1}where the background has

*μ*= 0.02, giving absorption contrast of 2:1, comparable to what is found in a clinical setting [25

_{a}25. B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: analysis of intersubject variability and menstrual cycles changes,” J. Biomed. Opt. **9** (2004). [CrossRef] [PubMed]

*μ*′

*= 10.1 cm*

_{s}^{−1}at 690 nm [26

26. S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. A., and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. **35**, 446–455 (2008). [CrossRef] [PubMed]

**S**

*the 0 − 1 characteristic matrix corresponding to the estimated shape, and*

^{est}**S**

*the 0 − 1 characteristic matrix corresponding to the actual object. Let*

^{act}*N*denote the number of overall voxels in the region of interest with

_{v}*i*denoting the

*i*element of

^{th}**S**

*and*

^{est}**S**

*. Symmetric Difference,*

^{act}*d*, is the fraction of entries in

_{sd}**S**

*where the corresponding entries in*

^{est}**S**

*are not equal. Mathematically, this is expressed as where 1*

^{act}_{{·}}is the indicator function. Another metric to judge shape estimation the Dice coefficient,

*d*, measures the similarity between

_{di}**S**

*and*

^{est}**S**

*judge how well the concentrations are located by computing where a perfectly reconstructed image gives*

^{act}*d*= 1 and failure gives

_{di}*d*=0. Finally the Mean Square Error,

_{di}*MSE*, measures how well the reconstructed structure

**ĉ**recovers the true absorption values and shape

**c**, computed by The symmetric distance is an important measure of the quality of reconstruction because it measures the overall quality of shape reconstruction, by penalizing errors in detecting object voxels as background and similarly background voxels as object voxels. Symmetric difference assigns an equal penalty to an erroneous voxel, irrespective of whether it is detected as background or object. An important limitation of the symmetric difference measure is that it does not reflect well on how close the estimated absorption concentration value in the reconstructed image is to the true value. The mean square error fills this gap by providing a measure on the quantitative accuracy for each slices that measures both how well the shape and value of the Δ

*μ*is recovered.

_{a}**L**=

**I**. Tikhonov regularization is widely used for image reconstruction for multiple imaging modalities and provides a suitable comparison for our method [4

4. P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express **15**, 8043–8058 (2007). [CrossRef] [PubMed]

16. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**, 2950–2961 (1999). [CrossRef]

## 6. Experimental analysis

*μ*= 0.16 cm

_{a}^{−1}and

*μ*′

*= 10.1 cm*

_{s}^{−1}at 690 nm. Embedded in the slab are two absorbing cuboids, each with a height and width of 1 cm and length of 4.5 cm, separated by 1.6 cm. The geometry of the inclusions embedded in the phantom is shown in Fig. 4. In this experimental setup we utilize analytical Green’s functions for the forward model in Eq. (4). The cuboid inclusions have the same

*μ*′

*as the slab and their*

_{s}*μ*is ten and three times higher than the background, respectively. The 10× absorption results in a highly absorbing rod, where we define Δ

_{a}*μ*= 1.28 cm

_{a}^{−1}for ground truth comparison, where the 3× cuboid has Δ

*μ*= 0.33 cm

_{a}^{−1}. The 10× absorbing rod has high absorption, mimicking the strong absorption of blood where the 3× rod is close to realistic values for breast cancer. This experimental setup allows us to test our algorithm to recover realistic tubular structures accurately even when highly absorbing areas, exceeding the Born approximation limit, are present in the medium [8

8. S. Fantini and A. Sassaroli, “Near-infrared optical mammography for breast cancer detection with intrinsic contrast,” Ann. Biomed. Eng. **40**, 398–407(2011). [CrossRef] [PubMed]

25. B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: analysis of intersubject variability and menstrual cycles changes,” J. Biomed. Opt. **9** (2004). [CrossRef] [PubMed]

*φ*is defined as the angle between the direction of the cuboids and the scanning direction, as shown in Fig. 1. The first set is obtained where the inclusions are exactly perpendicular to the scanning direction,

*φ*= 90°, and a second set where

*φ*= 30°. These two setups are demonstrated in Fig. 4. The instrument performs a two dimensional planar scan, with an illumination and detection fiber operating in transmission geometry. For three different detector positions at

*x*= {±1, 0} cm a light source is placed on the opposite side of the phantom using a 4 mm diameter fiber. For each scan 32 light sources are considered with 0.2 increments resulting in 96 source-detector pairs for each slice, where slices are spaced 0.2 cm along the

*y*-axis. Using a Xenon arc lamp light source emitting a power of 13 mW, optical data is then found by spatially sampling 25 points/cm

^{2}at wavelengths from 650–900 nm. The light is collected by an optical fiber that delivers light to a spectrograph (Model No. SP-150, Acton Research Corp., Acton, MA) with a 2 mm wide slit entrance. The wavelengths are resolved by a cooled CCD Camera (Model No. DU420A-BR-DD, Andor Technology, South Windsor, CT) giving a spectral sampling rate of 0.5 nm

^{−1}. Reconstructions are performed at a wavelength of 690 nm. Considering the geometry of the phantom we use the Green’s functions for slab geometry [14], computed by where

## 7. Results

### 7.1. Simulations

*d*and

_{di}*d*the middle rod is recovered as a separate structure when the regularization is present. Reconstruction results for the branching structure of phantom 2 is shown in Fig. 9 with corresponding error metrics are shown in Table 1.

_{sd}*α*we implement the L-curve method. In this method, we generate a plot of log(‖

**L**

*θ*‖

^{2}) against log(‖

**W**(

**K**(

*θ*) − Φ)‖

^{2}) as

*α*is varied. Fig. 6(a) depicts the reconstructed object when

*α*= 0, where no regularization is being applied. From visual observation as well as examining error metrics defined in Section 5, it is clear that some degree of regularization is beneficial. The L-curve plot for the reconstruction of the simulated phantom is shown in Fig. 3(a). Note the encircled point on the curve denotes the “best” reconstruction given the data. The parameter

*α*was obtained in a similar fashion in all our experiments. However, here in order to save space, we have only demonstrated our results for a single case.

*μ*. As expected using the Born Approximation, the absorption values are underestimated, but these results are encouraging, considering the limited data sets being employed, and how each slice reconstruction is not modeled to incorporate effects from the total 3D structure.

_{a}### 7.2. Experimental validation

*d*,

_{di}*d*and MSE shown in Table 2 it is clear that this method allows for recovery of tubular structures in a realistic breast phantom. It is notable that the 10× absorbing inclusion is recovered as a larger structure, whereas the 3× cuboid is recovered close to its true shape with more accurate absorption value. This is demonstrated in an example slice image for the

_{sd}*φ*= 90° case in Fig. 12. This is expected due to the aforementioned limitations of the Born Approximation [17

17. D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express **1**, 404–413 (1997). [CrossRef] [PubMed]

*φ*= 30° case, although shape is recovered much better when regularization is introduced. For the 30° case improvements in both absorption values and shape are evident and examining Fig. 11 shows clearly that we are able to recover structures even though they are angled close to the scanning direction. The correlation term in Eq. (10) is shown to be as important for experimental reconstructions as in simulations, both in error metrics in Table 2 and visually, in Fig. 11(a). For both experimental sets, the primitive 3D PaLS method resolves the location and the shape of the inclusion more accurately, which is verified by all metrics. It should be noted in Table 2 that

*d*is computed strictly for regions where the inclusions are present. This is due to the Dice coefficient not being a useful metric to judge reconstructions when the ground truth is an empty set image.

_{di}## 8. Conclusion

*n*order hold functions, sinc functions, or spines could be used to interpolate the primitives to represent the 3D structures. Future work will also expand the algorithm to recover images of multiple chromophore concentrations, as well as scattering amplitude. This can be achieved with minor changes to our method, and has been previously demonstrated with the PaLS method [15

^{th}**3**, 1006–1024 (2012). [CrossRef] [PubMed]

*y*-axis. This of course significantly affects the mismatch between the model and the true scenario, but our method demonstrated that correlating the slice images and parameterizing the reconstruction allows for accurate recovery of the vessel like structures. Future efforts will examine the effect of computing the off diagonal elements of Eq. (4) where it would be see how results would change if a certain segment along the

*y*-axis would be modeled in 3D. This would physically represent stacking 3D slices with a certain thickness to recover a larger 3D structure and examining reconstruction accuracy versus computational intensity is a natural progression of our research.

## Acknowledgments

## References and links

1. | Y. Yang, A. Sassaroli, D. K. Chen, M. J. Homer, R. A. Graham, and S. Fantini, “Near-infrared, broad-band spectral imaging of the human breast for quantitative oximetry: applications to healthy and cancerous breasts,” J. Innov. Opt. Health Sci. |

2. | A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous diffuse optical tomography,” Opt. Lett. |

3. | S. van de Ven, A. Wiethoff, T. Nielsen, B. Brendel, M. van der Voort, R. Nachabe, M. Van der Mark, M. Van Beek, L. Bakker, L. Fels, S. Elias, P. Luijten, and W. Mali, “A novel fluorescent imaging agent for diffuse optical tomography of the breast: first clinical experience in patients,” Mol. Imaging Biol. |

4. | P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express |

5. | Y. Bresler, J. A. Fessler, and A. Macovski, “A bayesian approach to reconstruction from incomplete projections of a multiple object 3D domain,” IEEE Trans. Pattern Anal. Mach. Intell. |

6. | R. A. Jesinger, G. E. Lattin, E. A. Ballard, S. M. Zelasko, and L. M. Glassman, “Vascular abnormalities of the breast: arterial and venous disorders, vascular masses, and mimic lesions with radiologic-pathologic correlation,” Radiographics |

7. | M. Belge, M. E. Kilmer, and E. L. Miller, “Efficient determination of multiple regularization parameters in a generalized l-curve framework,” Inverse Probl. |

8. | S. Fantini and A. Sassaroli, “Near-infrared optical mammography for breast cancer detection with intrinsic contrast,” Ann. Biomed. Eng. |

9. | R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techinques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. |

10. | H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: Initial simulation, phantom, and clinical results,” Appl. Opt. |

11. | J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett. |

12. | F. Larusson, S. Fantini, and E. L. Miller, “Hyperspectral image reconstruction for diffuse optical tomography,” Biomed. Opt. Express |

13. | A. J. Laub, |

14. | F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, |

15. | F. Larusson, S. Fantini, and E. L. Miller, “Parametric level set reconstruction methods for hyperspectral diffuse optical tomography,” Biomed. Opt. Express |

16. | B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. |

17. | D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express |

18. | A. Aghasi, M. Kilmer, and E. L. Miller, “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. |

19. | M. E. Kilmer, E. L. Miller, A. Barbaro, and David Boas, “Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. |

20. | A. Mandelis, |

21. | T. Chan and L. Vese, “Active contours without edges,” Inverse Probl. |

22. | S. Osher and R. Fedkiw, |

23. | K. Madsen, H. B. Nielsen, and O. Tingleff, “Methods for non-linear least squares problems”, Informatics and Mathematical Modelling, Technical University of Denmark, DTU,Nielsen Lecture Notes (2004). |

24. | C. R. Vogel, |

25. | B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: analysis of intersubject variability and menstrual cycles changes,” J. Biomed. Opt. |

26. | S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. A., and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. |

27. | O. Semerci and E. L. Miller, “A parametric level set approach to simultaneous object identification and background reconstruction for dual energy computed tomography,” IEEE Trans. Image Process. |

28. | T. Tarvainen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Corrections to linear methods for diffuse optical tomography using approximation error modelling,” Biomed. Opt. Express |

29. | P. C. Hansen, |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(170.3830) Medical optics and biotechnology : Mammography

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(170.5280) Medical optics and biotechnology : Photon migration

(170.6960) Medical optics and biotechnology : Tomography

(290.1990) Scattering : Diffusion

(290.7050) Scattering : Turbid media

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: October 26, 2012

Revised Manuscript: December 21, 2012

Manuscript Accepted: December 22, 2012

Published: January 17, 2013

**Citation**

Fridrik Larusson, Pamela G. Anderson, Elizabeth Rosenberg, Misha E. Kilmer, Angelo Sassaroli, Sergio Fantini, and Eric L. Miller, "Parametric estimation of 3D tubular structures for diffuse optical tomography," Biomed. Opt. Express **4**, 271-286 (2013)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-4-2-271

Sort: Year | Journal | Reset

### References

- Y. Yang, A. Sassaroli, D. K. Chen, M. J. Homer, R. A. Graham, and S. Fantini, “Near-infrared, broad-band spectral imaging of the human breast for quantitative oximetry: applications to healthy and cancerous breasts,” J. Innov. Opt. Health Sci.3, 267–277 (2010). [CrossRef]
- A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous diffuse optical tomography,” Opt. Lett.29, 256–258 (2004). [CrossRef] [PubMed]
- S. van de Ven, A. Wiethoff, T. Nielsen, B. Brendel, M. van der Voort, R. Nachabe, M. Van der Mark, M. Van Beek, L. Bakker, L. Fels, S. Elias, P. Luijten, and W. Mali, “A novel fluorescent imaging agent for diffuse optical tomography of the breast: first clinical experience in patients,” Mol. Imaging Biol.12, 343–348, 2010. [CrossRef]
- P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express15, 8043–8058 (2007). [CrossRef] [PubMed]
- Y. Bresler, J. A. Fessler, and A. Macovski, “A bayesian approach to reconstruction from incomplete projections of a multiple object 3D domain,” IEEE Trans. Pattern Anal. Mach. Intell.2, 840–858 (1989). [CrossRef]
- R. A. Jesinger, G. E. Lattin, E. A. Ballard, S. M. Zelasko, and L. M. Glassman, “Vascular abnormalities of the breast: arterial and venous disorders, vascular masses, and mimic lesions with radiologic-pathologic correlation,” Radiographics31, E117–E136 (2011). [CrossRef] [PubMed]
- M. Belge, M. E. Kilmer, and E. L. Miller, “Efficient determination of multiple regularization parameters in a generalized l-curve framework,” Inverse Probl.18, 1161–1183 (2002). [CrossRef]
- S. Fantini and A. Sassaroli, “Near-infrared optical mammography for breast cancer detection with intrinsic contrast,” Ann. Biomed. Eng.40, 398–407(2011). [CrossRef] [PubMed]
- R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techinques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol.45, 1051–1069 (2000). [CrossRef] [PubMed]
- H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: Initial simulation, phantom, and clinical results,” Appl. Opt.42, 135–145 (2004). [CrossRef]
- J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett.26, 701–703 (2001). [CrossRef]
- F. Larusson, S. Fantini, and E. L. Miller, “Hyperspectral image reconstruction for diffuse optical tomography,” Biomed. Opt. Express2, 947–965 (2011). [CrossRef]
- A. J. Laub, Matrix Analysis for Scientists and Engineers, 1st ed. (Society for Industrial and Applied Mathematics, 2004),
- F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media, 1st ed. (SPIE, 2009).
- F. Larusson, S. Fantini, and E. L. Miller, “Parametric level set reconstruction methods for hyperspectral diffuse optical tomography,” Biomed. Opt. Express3, 1006–1024 (2012). [CrossRef] [PubMed]
- B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt.38, 2950–2961 (1999). [CrossRef]
- D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express1, 404–413 (1997). [CrossRef] [PubMed]
- A. Aghasi, M. Kilmer, and E. L. Miller, “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci.4, 618–650 (2011). [CrossRef]
- M. E. Kilmer, E. L. Miller, A. Barbaro, and David Boas, “Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt.42, 3129–3144 (2003). [CrossRef] [PubMed]
- A. Mandelis, Diffusion-Wave Fields: Mathematical Methods and Green Functions (Springer, 2001), 1st ed. [CrossRef]
- T. Chan and L. Vese, “Active contours without edges,” Inverse Probl.10(2), 266–277 (2001).
- S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2002)
- K. Madsen, H. B. Nielsen, and O. Tingleff, “Methods for non-linear least squares problems”, Informatics and Mathematical Modelling, Technical University of Denmark, DTU,Nielsen Lecture Notes (2004).
- C. R. Vogel, Computational Methods for Inverse Problems, 1st ed. (SIAM, 2002). [CrossRef]
- B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: analysis of intersubject variability and menstrual cycles changes,” J. Biomed. Opt.9 (2004). [CrossRef] [PubMed]
- S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. A., and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys.35, 446–455 (2008). [CrossRef] [PubMed]
- O. Semerci and E. L. Miller, “A parametric level set approach to simultaneous object identification and background reconstruction for dual energy computed tomography,” IEEE Trans. Image Process.21, 2719–2734 (2012). [CrossRef] [PubMed]
- T. Tarvainen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Corrections to linear methods for diffuse optical tomography using approximation error modelling,” Biomed. Opt. Express1, 209–222 (2010). [CrossRef]
- P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, 1st ed. (SIAM,1997).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.