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Biomedical Optics Express

Biomedical Optics Express

  • Editor: Joseph A. Izatt
  • Vol. 3, Iss. 9 — Sep. 1, 2012
  • pp: 2036–2049
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Singular value decomposition based regularization prior to spectral mixing improves crosstalk in dynamic imaging using spectral diffuse optical tomography

Yuxuan Zhan, Adam T. Eggebrecht, Joseph P. Culver, and Hamid Dehghani  »View Author Affiliations


Biomedical Optics Express, Vol. 3, Issue 9, pp. 2036-2049 (2012)
http://dx.doi.org/10.1364/BOE.3.002036


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Abstract

The spectrally constrained diffuse optical tomography (DOT) method relies on incorporating spectral prior information directly into the image reconstruction algorithm, thereby correlating the underlying optical properties across multiple wavelengths. Although this method has been shown to provide a solution that is stable, the use of conventional Tikhonov-type regularization techniques can lead to additional crosstalk between parameters, particularly in linear, single-step dynamic imaging applications. This is due mainly to the suboptimal regularization of the spectral Jacobian matrix, which smoothes not only the image-data space, but also the spectral mapping space. In this work a novel regularization technique based on the singular value decomposition (SVD) is presented that preserves the spectral prior information while regularizing the Jacobian matrix, leading to dramatically reduced crosstalk between the recovered parameters. Using simulated data, images of changes in oxygenated and deoxygenated hemoglobin concentrations are reconstructed via the SVD-based approach and compared with images reconstructed by using non-spectral and conventional spectral methods. In a 2D, two wavelength example, it is shown that the proposed approach provides a 98% reduction in crosstalk between recovered parameters as compared with conventional spectral reconstruction algorithms, and 60% as compared with non-spectrally constrained algorithms. Using a subject specific multilayered model of the human head, a noiseless dynamic simulation of cortical activation is performed to further demonstrate such improvement in crosstalk. However, with the addition of realistic noise in the data, both non-spectral and proposed algorithms perform similarly, indicating that the use of spectrally constrained reconstruction algorithms in dynamic DOT may be limited by the contrast of the signal as well as the noise characteristics of the system.

© 2012 OSA

1. Introduction

Diffuse optical tomography (DOT) is finding widespread application in the diagnosis and characterization of breast cancer [1

1. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50(4), R1–R43 (2005). [CrossRef] [PubMed]

5

5. Q. Fang, J. Selb, S. A. Carp, G. Boverman, E. L. Miller, D. H. Brooks, R. H. Moore, D. B. Kopans, and D. A. Boas, “Combined optical and X-ray tomosynthesis breast imaging,” Radiology 258(1), 89–97 (2011). [CrossRef] [PubMed]

], monitoring brain function [6

6. A. Bluestone, G. Abdoulaev, C. Schmitz, R. Barbour, and A. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express 9(6), 272–286 (2001). [CrossRef] [PubMed]

14

14. H. Niu, S. Khadka, F. Tian, Z. J. Lin, C. Lu, C. Zhu, and H. Liu, “Resting-state functional connectivity assessed with two diffuse optical tomographic systems,” J. Biomed. Opt. 16(4), 046006 (2011). [CrossRef] [PubMed]

], and small animal imaging for the study of disease detection, progression and treatment [15

15. H. Dehghani, B. W. Pogue, J. Shudong, B. Brooksby, and K. D. Paulsen, “Three-dimensional optical tomography: resolution in small-object imaging,” Appl. Opt. 42(16), 3117–3128 (2003). [CrossRef] [PubMed]

18

18. G. Zacharakis, H. Kambara, H. Shih, J. Ripoll, J. Grimm, Y. Saeki, R. Weissleder, and V. Ntziachristos, “Volumetric tomography of fluorescent proteins through small animals in vivo,” Proc. Natl. Acad. Sci. U.S.A. 102(51), 18252–18257 (2005). [CrossRef] [PubMed]

]. Specifically in DOT of human brain function, an imaging cap that uses an array of continuous-wave, near-infrared (NIR) light sources and detectors are placed on the scalp surface, allowing measurement of light level variance due to underlying functional changes [19

19. H. Obrig and A. Villringer, “Beyond the visible--imaging the human brain with light,” J. Cereb. Blood Flow Metab. 23(1), 1–18 (2003). [CrossRef] [PubMed]

]. A minimum of two sets of measurements are typically required to allow dynamic imaging, i.e. an activation set and a reference set, where the differences between these sets are used to recover spatial maps of hemodynamic changes within the brain, which relies on a forward model for light propagation in tissue. The image reconstruction problem in this case is single-step and linear, but remains ill-conditioned and ill-posed [20

20. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]

,21

21. H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Philos. Transact. A Math. Phys. Eng. Sci. 367(1900), 3073–3093 (2009). [CrossRef] [PubMed]

].

The two primary chromophore species of interest in cerebral hemodynamics studies are oxyhemoglobin (HbO2) and deoxyhemoglobin (HbR). Given their difference in the near-infrared absorption spectra (Fig. 1
Fig. 1 Absorption spectra for oxyhemoglobin (HbO2) and deoxyhemoglobin (HbR) in the near-infrared spectrum. The molar extinction coefficients used in this work are indicated by the four red-dots in this figure, which are 0.1193 mm−1 and 0.3236 mm−1 for HbO2 and HbR at 750 nm, and 0.2436 mm−1 and 0.1592 mm−1 for HbO2 and HbR at 850 nm.
), measurements at two wavelengths are typically sufficient to recover the change in concentration for both chromophores [22

22. D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage 23(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

]. Most existing optical neuroimaging systems are equipped with two wavelengths of light sources, including the CW5 (690 and 830 nm) [14

14. H. Niu, S. Khadka, F. Tian, Z. J. Lin, C. Lu, C. Zhu, and H. Liu, “Resting-state functional connectivity assessed with two diffuse optical tomographic systems,” J. Biomed. Opt. 16(4), 046006 (2011). [CrossRef] [PubMed]

,23

23. R. C. Mesquita, M. A. Franceschini, and D. A. Boas, “Resting state functional connectivity of the whole head with near-infrared spectroscopy,” Biomed. Opt. Express 1(1), 324–336 (2010). [CrossRef] [PubMed]

], the DYNOT (760 and 830 nm) [12

12. S. P. Koch, C. Habermehl, J. Mehnert, C. H. Schmitz, S. Holtze, A. Villringer, J. Steinbrink, and H. Obrig, “High-resolution optical functional mapping of the human somatosensory cortex,” Front Neuroenergetics 2, 12 (2010). [PubMed]

,14

14. H. Niu, S. Khadka, F. Tian, Z. J. Lin, C. Lu, C. Zhu, and H. Liu, “Resting-state functional connectivity assessed with two diffuse optical tomographic systems,” J. Biomed. Opt. 16(4), 046006 (2011). [CrossRef] [PubMed]

,24

24. B. Khan, P. Chand, and G. Alexandrakis, “Spatiotemporal relations of primary sensorimotor and secondary motor activation patterns mapped by NIR imaging,” Biomed. Opt. Express 2(12), 3367–3386 (2011). [CrossRef] [PubMed]

], and the high-density-DOT imaging system at Washington University School of Medicine (750 and 850 nm) [9

9. B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A. 104(29), 12169–12174 (2007). [CrossRef] [PubMed]

,10

10. B. R. White, A. Z. Snyder, A. L. Cohen, S. E. Petersen, M. E. Raichle, B. L. Schlaggar, and J. P. Culver, “Resting-state functional connectivity in the human brain revealed with diffuse optical tomography,” Neuroimage 47(1), 148–156 (2009). [CrossRef] [PubMed]

,13

13. B. R. White and J. P. Culver, “Phase-encoded retinotopy as an evaluation of diffuse optical neuroimaging,” Neuroimage 49(1), 568–577 (2010). [CrossRef] [PubMed]

,25

25. A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” Neuroimage 61(4), 1120–1128 (2012). [CrossRef] [PubMed]

]. For this reason our analyses in this work will be focused on the dual-wavelength setup using 750 and 850 nm.

The spectrally constrained image reconstruction method was first documented by Corlu [26

26. A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. Hillman, S. R. Arridge, and A. G. Yodh, “Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography,” Opt. Lett. 28(23), 2339–2341 (2003). [CrossRef] [PubMed]

] in non-linear DOT, wherein absolute concentrations of chromophores are solved iteratively with a minimization function [21

21. H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Philos. Transact. A Math. Phys. Eng. Sci. 367(1900), 3073–3093 (2009). [CrossRef] [PubMed]

]. When compared with non-spectral methods, whereby images of absorption and scattering coefficients are first recovered and then un-mixed to provide chromophore concentrations, spectrally constrained reconstruction techniques are found to suppress artifacts in water and scattering power images, and also reduce crosstalk between chromophores and scatter parameters in breast imaging [27

27. S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, and K. D. Paulsen, “Spectrally constrained chromophore and scattering near-infrared tomography provides quantitative and robust reconstruction,” Appl. Opt. 44(10), 1858–1869 (2005). [CrossRef] [PubMed]

29

29. B. Brooksby, S. Srinivasan, S. Jiang, H. Dehghani, B. W. Pogue, K. D. Paulsen, J. Weaver, C. Kogel, and S. P. Poplack, “Spectral priors improve near-infrared diffuse tomography more than spatial priors,” Opt. Lett. 30(15), 1968–1970 (2005). [CrossRef] [PubMed]

]. Li [30

30. A. Li, G. Boverman, Y. Zhang, D. Brooks, E. L. Miller, M. E. Kilmer, Q. Zhang, E. M. Hillman, and D. A. Boas, “Optimal linear inverse solution with multiple priors in diffuse optical tomography,” Appl. Opt. 44(10), 1948–1956 (2005). [CrossRef] [PubMed]

] further demonstrated that when considering dynamic (also known as temporal or differential) imaging, linear DOT algorithms that incorporate spectral priors directly in the image reconstruction, correlate the absorption coefficients across multiple wavelengths, thereby providing a solution that is more stable, albeit with an increase in crosstalk between chromophores. Such crosstalk is partially due to the suboptimal regularization and update of the Jacobian matrix (which relates a small change in measured data to changes in optical parameters) as pointed out by Eames [31

31. M. E. Eames and H. Dehghani, “Wavelength dependence of sensitivity in spectral diffuse optical imaging: effect of normalization on image reconstruction,” Opt. Express 16(22), 17780–17791 (2008). [CrossRef] [PubMed]

], who proposed a Jacobian normalization technique in non-linear DOT that allows more uniform regularization and parameter update in image reconstruction, hence reducing imaging crosstalk. However in DOT of brain function where the imaging problem is linear and measurements are mostly reflectance rather than transmittance, such normalization techniques would inappropriately bias the measurement sensitivities towards deeper regions within the brain and cause significant imaging error.

Qualitative and quantitative evaluations are conducted among non-spectral method and spectral methods (using conventional and SVD-based regularization respectively) first on a two dimensional (2D) circular model and then on a three dimensional (3D) subject specific head geometry. The 2D model provides an initial proof-of-concept analysis whereas the 3D model has more degrees of freedom, is highly ill-determined and more realistic to in vivo diffuse optical tomography of human brain studies.

2. Theory

2.1 Non-spectral method

The non-spectrally constrained image reconstruction problem consists of two steps. The goal of the first step is the recovery of a temporal change of absorption Δμλ at wavelength λ for a selection of wavelengths. This requires two sets of boundary data Φλ,ref and Φλ at each wavelength λ which are acquired before and after a change of absorption, Δμλ. Given that Δμλ is relatively small, the forward problem can be linearized by the Rytov approximation (whereby the log of the intensity measurements are considered) [20

20. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]

]

ΦλΦλ,ref=ΔΦλ=JλΔμλ
(1)

where Jλ is the wavelength dependent Jacobian matrix for wavelength λ, with a size of number of measurements by number of nodes. Since Jλis non-invertible, the Moore-Penrose generalized inverse is applied

Δμλ=JλT(JλJλT+αλ2I)1ΔΦλ
(2)

where αλ2 is the Tikhonov regularization parameter [20

20. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]

,35

35. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, Bristol, 1998), Chap. 5.

]. Assuming a dual-wavelength system and two chromophores of interest, we have

(Δμλ1Δμλ2)=Ms(ΔHbO2ΔHbR)
(3)

where Ms is the spectral composition matrix (containing spectral prior information) of the following form

Ms=[εc1,λ1εc2,λ1εc1,λ2εc2,λ2]
(4)

where εc,λ is the molar extinction coefficient of the absorbing chromophore c at wavelength λ (Fig. 1). Hence the second step inverse problem is a spectral decomposition process

(ΔHbO2ΔHbR)=Ms1(Δμλ1Δμλ2)
(5)

This step provides the mapping from images of change in wavelength dependent absorptions to images of change in chromophore concentrations.

2.2 Spectral method using conventional regularization

Instead of reconstructing changes in absorption at each wavelength (Eq. (2)) and then spectrally decomposing these into changes of chromophore concentrations (Eq. (5)), the spectral prior information Ms can be directly incorporated into the forward problem by constructing a spectral Jacobian matrix Js

Js=(Jλ1εc1,λ1Jλ1εc2,λ1Jλ2εc1,λ2Jλ2εc2,λ2)
(6)

By doing so the forward problem can be expressed as

(ΔΦλ1TΔΦλ2T)=Js(ΔHbO2TΔHbRT)
(7)

and the spectral image reconstruction using conventional Tikhonov regularization becomes

(ΔHbO2TΔHbRT)=JsT(JsJsT+αs2I)1(ΔΦλ1TΔΦλ2T)
(8)

2.3 Spectral method using SVD based regularization

Instead of constructing the spectral Jacobian matrix first and performing the regularization second, we propose an alternative regularization technique that reverses the order of these two operations. This is achieved by first applying the singular value decomposition (SVD) on each wavelength dependent Jacobian matrix, which yields a triplet of matrices

Jλ=USVT=Udiag(σλ,i)VT,i=1:rank(Jλ)
(9)

where U and V are orthonormal matrices containing the singular vectors of Jλ, and S is a diagonal matrix that contains the i number of singular values of Jλ, i.e., σλ,i. Since applying Tikhonov regularization of weight α2 changes the singular values of Jλ from σλ,i to σλ,i2+α2(Fig. 2
Fig. 2 Singular value spectra of the 750nm (blue line) and 850 nm (red line) wavelength Jacobian matrices of the 2D circular model as described in section 3.1, before (solid line) and after (dashed line) Tikhonov regularization.
), we can recompose the regularized Jλ, denoted J^λ, as

J^λ=USS+diag(αλ2)VT
(10)

We can then construct the regularized Js, denoted J^s, as

J^s=(J^λ1εc1,λ1J^λ1εc2,λ1J^λ2εc1,λ2J^λ2εc2,λ2)
(11)

The inverse problem therefore becomes

(ΔHbO2TΔHbRT)=JsT(J^sJ^sT)1(ΔΦλ1TΔΦλ2T)
(12)

The advantage here is that regularization is applied to the Jacobian matrix while (unlike all other previously proposed methods) the spectral prior information Ms is exactly preserved and therefore contains all information needed for the wavelength unmixing operation. In addition, αλ2 in the non-spectral method is now directly used in the spectral method, avoiding the operation to find an equivalent αs2 for comparative studies which can be problematic due to its subjective choice.

3. Methods and results

3.1 Two-dimensional circular model

Linear, single-step reconstructed images of hemodynamic changes using non-spectral (denoted ‘Non-Spec’), spectral with conventional regularization (‘Conv-Spec’), and spectral with SVD-based regularization (‘Svd-Spec’) are shown in Figs. 4
Fig. 4 Reconstructed images of ΔHbO2 (upper row) and ΔHbR (lower row) concentration at three different locations (a-c) using methods ‘Svd-Spec’, ‘Non-Spec’ and ‘Conv-Spec’ with 0% noise.
6
Fig. 6 Same as Fig. 4, but with 0.5% added noise.
with 0%, 0.2% and 0.5% added Gaussian random noise respectively. Within each figure, three scenarios (a-c) having the ΔHbO2 target moving towards the center of the circular model are presented, emulating a gradual decrease in signal-to-noise ratio (SNR) in the measurement; that is as the target moves deeper within the domain, the magnitude of the detected differential signal decreases. It is evident from Fig. 4 that in the case of 0% noise a region of crosstalk can be easily identified in the recovered ΔHbR images along with some imaging artifacts due to measurement sensitivity distribution for all three methods, with ‘Svd-Spec’ demonstrating the least magnitude of crosstalk and imaging artifacts while ‘Conv-Spec’ produces the most. As noise increases to 0.2%, imaging artifacts due to noise appear although at smaller magnitudes as compared to the crosstalk. When noise reaches 0.5% the artifacts begin to dominate the ΔHbR concentration images for both the ‘Svd-Spec’ and ‘Non-Spec’. When the target is at the center of the model (43 mm from boundary, Fig. 6(c)), the recovered images from ‘Svd-Spec’ and ‘Non-Spec’ become indistinguishable. In comparison, crosstalk and imaging artifacts in ‘Conv-Spec’ still dominate the ΔHbR concentration images even at 0.5% noise level.

For quantitative analysis, the crosstalk of ΔHbO2 into ΔHbR, defined as the mean value of the known region of crosstalk in the ΔHbR concentration image divided by the mean value of the known target region in the ΔHbO2 concentration image, are calculated. In Fig. 7
Fig. 7 Crosstalk versus depth of target: 13, 28, 43 mm, representing scenario (a)–(c) in Figs. 46, respectively.
the plotted crosstalk versus depth of target is shown for all cases presented in Figs. 46. Again it can be seen that crosstalk increases along with depth of target due to decreasing magnitude of the differential signal and/or SNR. On average, crosstalk is ~2×103 for ‘Svd-Spec’, representing a 60% reduction from ~5×103 for ‘Non-Spec’, and a 98% reduction from ~10−1 for ‘Conv-Spec’. At 0.5% noise the imaging artifacts dominate over the crosstalk for both ‘Svd-Spec’ and ‘Non-Spec’, particularly when the target is located at the center of the model (Fig. 6(c)), where crosstalk for both methods are approximately 5×103, indicating little difference between the two methods, however still much better as compared to conventional spectral reconstruction, ‘Conv-Spec’. These simulations were repeated with a ΔHbR target and similar results were obtained.

3.2 Three-dimensional head model

A high-density (HD) imaging array which contains 24 sources and 28 detectors as described previously [9

9. B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A. 104(29), 12169–12174 (2007). [CrossRef] [PubMed]

,38

38. B. R. White and J. P. Culver, “Quantitative evaluation of high-density diffuse optical tomography: in vivo resolution and mapping performance,” J. Biomed. Opt. 15(2), 026006 (2010). [CrossRef] [PubMed]

], was placed on the back of the head over the visual cortex, Figs. 8(c) and 8(d). Within this optode arrangement, nearest neighbor measurements can be defined based on source-detector separations, Fig. 8(e). In this study, 260 differential measurements per wavelength were used to image the hemodynamic change of human brain function, which include first, second and third nearest neighbor measurements, corresponding to a maximum source-detector separation of 40 mm.

Reconstructed chromophore concentration images of simulated brain activation using non-spectral (‘Non-Spec’) and spectral method with SVD-based regularization (‘Svd-Spec’) with noise free and noise added data are displayed on the surface rendered FEM model of the brain in Fig. 9. Similar to our finding with the 2D model, a region of crosstalk of ΔHbR into ΔHbO2 is identified when noiseless data are used for image reconstruction, with the magnitude of crosstalk from ‘Svd-Spec’ significantly smaller than ‘Non-Spec’ (Fig. 9(c)). The crosstalk (as defined in the 2D case) in Fig. 9(c) is 4.9×105 for ‘Svd-Spec’, representing a 95% reduction from 1.1×103 for ‘Non-Spec’. When measurement noise based on a current imaging system specification is added, the two methods produce images that are similar with crosstalk both at approximately 2×103 (Fig. 9(d)) showing little difference between non-spectral and spectral reconstruction techniques.

4. Discussion

In this study we have presented a new regularization technique for linear image reconstruction in spectral diffuse optical tomography. This technique utilizes the singular value decomposition (SVD), allowing regularization on the spectral Jacobian matrix to be applied without altering the underlying spectral prior information. Specifically, this is achieved by regularizing the wavelength dependant Jacobian matrix prior to mixing with the spectral coefficients, a completely new method where the regularization does not affect the mapping of absorption changes onto chromophore concentrations, thereby providing a much more accurate spectral inversion technique. This method is different to previously published work, whereby the Jacobian is first spectrally mixed and then regularized and inverted.

Through a series of proof-of-concept analyses using a 2D circular model, we have shown that the SVD-based regularization technique followed by spectral mixing dramatically reduces the crosstalk between chromophores. In comparison to conventional spectral method, the crosstalk is reduced by approximately 98%. When compared with non-spectral method, this technique also demonstrates 60% crosstalk reduction with noise free data and demonstrated consistently robust performance against crosstalk as noise increases, before the image is dominated by imaging artifacts due to poor SNR (Fig. 6(c)). We extended our evaluation analysis on a 3D subject specific head model with simulated focal activation on the visual cortex, with the results showing 95% reduction in crosstalk when using the proposed algorithm as compared to non-spectral method.

In the case of spectral techniques, using conventional regularization (Eq. (8)) the regularization parameter is applied to a spectral Jacobian (one that relates a small change in measurements to a change of either oxyhemoglobin or deoxyhemoglobin concentrations) and therefore any smoothing applied will also have an effect on the spectral mixing. However, using the proposed SVD-based spectral method (Eq. (12)) the spectral Jacobian is a pre-regularized matrix (which only relates a small change in measurements to change in absorption) and the spectral mixing is unaffected by this regularization, therefore implying that the latter method should provide better crosstalk in parameter recovery. In the case of noise-added data, it is possible to expand Eq. (12) by adding noise to the final term on the right-hand-side and then demonstrate that at a certain noise level, the improvement in crosstalk are gradually minimized as the effect of noise on the (un-regularized) spectral matrix becomes dominant.

Validation of the SVD-based method as presented in this paper has been limited at a specific wavelength pair, i.e. 750 and 850 nm. In this case the crosstalk between ΔHbR and ΔHbO2 is systematically small, owing to the large and opposite difference between their molar extinction coefficients at 750 and 850 nm (Fig. 1). In the concept of wavelength optimization [26

26. A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. Hillman, S. R. Arridge, and A. G. Yodh, “Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography,” Opt. Lett. 28(23), 2339–2341 (2003). [CrossRef] [PubMed]

], this means that the condition number of their spectral composition matrix Ms is relatively low as compared to other wavelength pairs. Nevertheless since the crosstalk improvement by our SVD-based method relies on no assumption on either specific spectral range or specific combinations of the molar extinction coefficients, we expect the improvement to be consistent when using other pairs of wavelengths. A full validation of this could be carried out over an extensive range of wavelength pairs as an extension of this work.

When compared with the linear, one-step DOT as presented here, non-linear DOT reconstruction has the added benefit of iteratively updating the recovered chromophore concentrations simultaneously with a minimization function until the solution finally converges to a predefined threshold. The improvement at each iteration is therefore accumulative and the final recovered imaged are shown to provide better contrast and robustness to noise. However, in the case of dynamic (or temporal or differential) imaging, as shown here, appropriate regularization is shown to be critical to minimize crosstalk while maintaining the benefits of spectral constraints. We expect that in the presence of larger magnitude differential data and lower noise levels, this proposed reconstruction algorithm to be superior to non-spectral techniques. Finally, we expect that in line with other published works in spectrally constrained non-linear imaging techniques, the addition of more wavelengths will provide better contrast and minimize crosstalk further.

5. Conclusion

While incorporating spectral prior information in spectral image reconstruction correlates the absorption coefficients across multiple wavelengths, providing a solution that is stable, the use of conventional regularization techniques can result in additional crosstalk between chromophores. A novel regularization technique that regularizes the Jacobian using the singular value decomposition (SVD) prior to spectral mixing has been presented in an attempt to reduce this crosstalk effect. Simulations have shown that using SVD-based regularization can dramatically reduce the crosstalk presented in images recovered using conventional regularization technique in linear, single-step spectral DOT. Although our analysis has shown that given the current HD-DOT imaging system specification, such improvement in crosstalk over the non-spectral method may not be substantial, it is evident that future instruments with higher SNR measurements would only yield better image quality for the spectral method over the non-spectral method. The use of SVD in matrix regularization as described in this study is also potentially applicable to other one-step linear imaging problems, offering an alternative approach to the conventional Tikhonov regularization and should also play an important part in non-linear spectral imaging techniques in DOT.

Acknowledgment

This work has been funded by the National Institutes of Health (NIH) Grant R01EB009233-2.

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A. Custo, D. A. Boas, D. Tsuzuki, I. Dan, R. Mesquita, B. Fischl, W. E. Grimson, and W. Wells 3rd, “Anatomical atlas-guided diffuse optical tomography of brain activation,” Neuroimage 49(1), 561–567 (2010). [CrossRef] [PubMed]

12.

S. P. Koch, C. Habermehl, J. Mehnert, C. H. Schmitz, S. Holtze, A. Villringer, J. Steinbrink, and H. Obrig, “High-resolution optical functional mapping of the human somatosensory cortex,” Front Neuroenergetics 2, 12 (2010). [PubMed]

13.

B. R. White and J. P. Culver, “Phase-encoded retinotopy as an evaluation of diffuse optical neuroimaging,” Neuroimage 49(1), 568–577 (2010). [CrossRef] [PubMed]

14.

H. Niu, S. Khadka, F. Tian, Z. J. Lin, C. Lu, C. Zhu, and H. Liu, “Resting-state functional connectivity assessed with two diffuse optical tomographic systems,” J. Biomed. Opt. 16(4), 046006 (2011). [CrossRef] [PubMed]

15.

H. Dehghani, B. W. Pogue, J. Shudong, B. Brooksby, and K. D. Paulsen, “Three-dimensional optical tomography: resolution in small-object imaging,” Appl. Opt. 42(16), 3117–3128 (2003). [CrossRef] [PubMed]

16.

H. Xu, R. Springett, H. Dehghani, B. W. Pogue, K. D. Paulsen, and J. F. Dunn, “Magnetic-resonance-imaging-coupled broadband near-infrared tomography system for small animal brain studies,” Appl. Opt. 44(11), 2177–2188 (2005). [CrossRef] [PubMed]

17.

H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, and J. F. Dunn, “Near-infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. 8(1), 102–110 (2003). [CrossRef] [PubMed]

18.

G. Zacharakis, H. Kambara, H. Shih, J. Ripoll, J. Grimm, Y. Saeki, R. Weissleder, and V. Ntziachristos, “Volumetric tomography of fluorescent proteins through small animals in vivo,” Proc. Natl. Acad. Sci. U.S.A. 102(51), 18252–18257 (2005). [CrossRef] [PubMed]

19.

H. Obrig and A. Villringer, “Beyond the visible--imaging the human brain with light,” J. Cereb. Blood Flow Metab. 23(1), 1–18 (2003). [CrossRef] [PubMed]

20.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]

21.

H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Philos. Transact. A Math. Phys. Eng. Sci. 367(1900), 3073–3093 (2009). [CrossRef] [PubMed]

22.

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage 23(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

23.

R. C. Mesquita, M. A. Franceschini, and D. A. Boas, “Resting state functional connectivity of the whole head with near-infrared spectroscopy,” Biomed. Opt. Express 1(1), 324–336 (2010). [CrossRef] [PubMed]

24.

B. Khan, P. Chand, and G. Alexandrakis, “Spatiotemporal relations of primary sensorimotor and secondary motor activation patterns mapped by NIR imaging,” Biomed. Opt. Express 2(12), 3367–3386 (2011). [CrossRef] [PubMed]

25.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” Neuroimage 61(4), 1120–1128 (2012). [CrossRef] [PubMed]

26.

A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. Hillman, S. R. Arridge, and A. G. Yodh, “Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography,” Opt. Lett. 28(23), 2339–2341 (2003). [CrossRef] [PubMed]

27.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, and K. D. Paulsen, “Spectrally constrained chromophore and scattering near-infrared tomography provides quantitative and robust reconstruction,” Appl. Opt. 44(10), 1858–1869 (2005). [CrossRef] [PubMed]

28.

S. Srinivasan, B. W. Pogue, B. Brooksby, S. Jiang, H. Dehghani, C. Kogel, W. A. Wells, S. P. Poplack, and K. D. Paulsen, “Near-infrared characterization of breast tumors in vivo using spectrally-constrained reconstruction,” Technol. Cancer Res. Treat. 4(5), 513–526 (2005). [PubMed]

29.

B. Brooksby, S. Srinivasan, S. Jiang, H. Dehghani, B. W. Pogue, K. D. Paulsen, J. Weaver, C. Kogel, and S. P. Poplack, “Spectral priors improve near-infrared diffuse tomography more than spatial priors,” Opt. Lett. 30(15), 1968–1970 (2005). [CrossRef] [PubMed]

30.

A. Li, G. Boverman, Y. Zhang, D. Brooks, E. L. Miller, M. E. Kilmer, Q. Zhang, E. M. Hillman, and D. A. Boas, “Optimal linear inverse solution with multiple priors in diffuse optical tomography,” Appl. Opt. 44(10), 1948–1956 (2005). [CrossRef] [PubMed]

31.

M. E. Eames and H. Dehghani, “Wavelength dependence of sensitivity in spectral diffuse optical imaging: effect of normalization on image reconstruction,” Opt. Express 16(22), 17780–17791 (2008). [CrossRef] [PubMed]

32.

P. K. Yalavarthy, H. Dehghani, B. W. Pogue, and K. D. Paulsen, “Critical computational aspects of near infrared circular tomographic imaging: Analysis of measurement number, mesh resolution and reconstruction basis,” Opt. Express 14(13), 6113–6127 (2006). [CrossRef] [PubMed]

33.

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(10), 701–703 (2001). [CrossRef] [PubMed]

34.

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 techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45(4), 1051–1070 (2000). [CrossRef] [PubMed]

35.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, Bristol, 1998), Chap. 5.

36.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009). [CrossRef] [PubMed]

37.

Materialise, “Mimics,” http://www.materialise.com/mimics.

38.

B. R. White and J. P. Culver, “Quantitative evaluation of high-density diffuse optical tomography: in vivo resolution and mapping performance,” J. Biomed. Opt. 15(2), 026006 (2010). [CrossRef] [PubMed]

39.

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Appl. Opt. 48(10), D137–D143 (2009). [CrossRef] [PubMed]

40.

Y. Zhan, A. T. Eggebrecht, J. P. Culver, and H. Dehghani, “Image quality analysis of high-density diffuse optical tomography incorporating a subject-specific head model,” Front Neuroenergetics 4, 6 (2012). [CrossRef] [PubMed]

41.

J. P. Culver, A. M. Siegel, J. J. Stott, and D. A. Boas, “Volumetric diffuse optical tomography of brain activity,” Opt. Lett. 28(21), 2061–2063 (2003). [CrossRef] [PubMed]

OCIS Codes
(110.6960) Imaging systems : Tomography
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.2655) Medical optics and biotechnology : Functional monitoring and imaging

ToC Category:
Image Reconstruction and Inverse Problems

History
Original Manuscript: June 18, 2012
Revised Manuscript: August 2, 2012
Manuscript Accepted: August 4, 2012
Published: August 9, 2012

Virtual Issues
BIOMED 2012 (2012) Biomedical Optics Express

Citation
Yuxuan Zhan, Adam T. Eggebrecht, Joseph P. Culver, and Hamid Dehghani, "Singular value decomposition based regularization prior to spectral mixing improves crosstalk in dynamic imaging using spectral diffuse optical tomography," Biomed. Opt. Express 3, 2036-2049 (2012)
http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-3-9-2036


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References

  1. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50(4), R1–R43 (2005). [CrossRef] [PubMed]
  2. S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A.100(21), 12349–12354 (2003). [CrossRef] [PubMed]
  3. 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(1), 135–145 (2003). [CrossRef] [PubMed]
  4. Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers,” Neoplasia7(3), 263–270 (2005). [CrossRef] [PubMed]
  5. Q. Fang, J. Selb, S. A. Carp, G. Boverman, E. L. Miller, D. H. Brooks, R. H. Moore, D. B. Kopans, and D. A. Boas, “Combined optical and X-ray tomosynthesis breast imaging,” Radiology258(1), 89–97 (2011). [CrossRef] [PubMed]
  6. A. Bluestone, G. Abdoulaev, C. Schmitz, R. Barbour, and A. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express9(6), 272–286 (2001). [CrossRef] [PubMed]
  7. D. A. Boas, K. Chen, D. Grebert, and M. A. Franceschini, “Improving the diffuse optical imaging spatial resolution of the cerebral hemodynamic response to brain activation in humans,” Opt. Lett.29(13), 1506–1508 (2004). [CrossRef] [PubMed]
  8. A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate,” Neuroimage30(2), 521–528 (2006). [CrossRef] [PubMed]
  9. B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A.104(29), 12169–12174 (2007). [CrossRef] [PubMed]
  10. B. R. White, A. Z. Snyder, A. L. Cohen, S. E. Petersen, M. E. Raichle, B. L. Schlaggar, and J. P. Culver, “Resting-state functional connectivity in the human brain revealed with diffuse optical tomography,” Neuroimage47(1), 148–156 (2009). [CrossRef] [PubMed]
  11. A. Custo, D. A. Boas, D. Tsuzuki, I. Dan, R. Mesquita, B. Fischl, W. E. Grimson, and W. Wells, “Anatomical atlas-guided diffuse optical tomography of brain activation,” Neuroimage49(1), 561–567 (2010). [CrossRef] [PubMed]
  12. S. P. Koch, C. Habermehl, J. Mehnert, C. H. Schmitz, S. Holtze, A. Villringer, J. Steinbrink, and H. Obrig, “High-resolution optical functional mapping of the human somatosensory cortex,” Front Neuroenergetics2, 12 (2010). [PubMed]
  13. B. R. White and J. P. Culver, “Phase-encoded retinotopy as an evaluation of diffuse optical neuroimaging,” Neuroimage49(1), 568–577 (2010). [CrossRef] [PubMed]
  14. H. Niu, S. Khadka, F. Tian, Z. J. Lin, C. Lu, C. Zhu, and H. Liu, “Resting-state functional connectivity assessed with two diffuse optical tomographic systems,” J. Biomed. Opt.16(4), 046006 (2011). [CrossRef] [PubMed]
  15. H. Dehghani, B. W. Pogue, J. Shudong, B. Brooksby, and K. D. Paulsen, “Three-dimensional optical tomography: resolution in small-object imaging,” Appl. Opt.42(16), 3117–3128 (2003). [CrossRef] [PubMed]
  16. H. Xu, R. Springett, H. Dehghani, B. W. Pogue, K. D. Paulsen, and J. F. Dunn, “Magnetic-resonance-imaging-coupled broadband near-infrared tomography system for small animal brain studies,” Appl. Opt.44(11), 2177–2188 (2005). [CrossRef] [PubMed]
  17. H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, and J. F. Dunn, “Near-infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt.8(1), 102–110 (2003). [CrossRef] [PubMed]
  18. G. Zacharakis, H. Kambara, H. Shih, J. Ripoll, J. Grimm, Y. Saeki, R. Weissleder, and V. Ntziachristos, “Volumetric tomography of fluorescent proteins through small animals in vivo,” Proc. Natl. Acad. Sci. U.S.A.102(51), 18252–18257 (2005). [CrossRef] [PubMed]
  19. H. Obrig and A. Villringer, “Beyond the visible--imaging the human brain with light,” J. Cereb. Blood Flow Metab.23(1), 1–18 (2003). [CrossRef] [PubMed]
  20. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl.15(2), R41–R93 (1999). [CrossRef]
  21. H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Philos. Transact. A Math. Phys. Eng. Sci.367(1900), 3073–3093 (2009). [CrossRef] [PubMed]
  22. D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage23(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]
  23. R. C. Mesquita, M. A. Franceschini, and D. A. Boas, “Resting state functional connectivity of the whole head with near-infrared spectroscopy,” Biomed. Opt. Express1(1), 324–336 (2010). [CrossRef] [PubMed]
  24. B. Khan, P. Chand, and G. Alexandrakis, “Spatiotemporal relations of primary sensorimotor and secondary motor activation patterns mapped by NIR imaging,” Biomed. Opt. Express2(12), 3367–3386 (2011). [CrossRef] [PubMed]
  25. A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” Neuroimage61(4), 1120–1128 (2012). [CrossRef] [PubMed]
  26. A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. Hillman, S. R. Arridge, and A. G. Yodh, “Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography,” Opt. Lett.28(23), 2339–2341 (2003). [CrossRef] [PubMed]
  27. S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, and K. D. Paulsen, “Spectrally constrained chromophore and scattering near-infrared tomography provides quantitative and robust reconstruction,” Appl. Opt.44(10), 1858–1869 (2005). [CrossRef] [PubMed]
  28. S. Srinivasan, B. W. Pogue, B. Brooksby, S. Jiang, H. Dehghani, C. Kogel, W. A. Wells, S. P. Poplack, and K. D. Paulsen, “Near-infrared characterization of breast tumors in vivo using spectrally-constrained reconstruction,” Technol. Cancer Res. Treat.4(5), 513–526 (2005). [PubMed]
  29. B. Brooksby, S. Srinivasan, S. Jiang, H. Dehghani, B. W. Pogue, K. D. Paulsen, J. Weaver, C. Kogel, and S. P. Poplack, “Spectral priors improve near-infrared diffuse tomography more than spatial priors,” Opt. Lett.30(15), 1968–1970 (2005). [CrossRef] [PubMed]
  30. A. Li, G. Boverman, Y. Zhang, D. Brooks, E. L. Miller, M. E. Kilmer, Q. Zhang, E. M. Hillman, and D. A. Boas, “Optimal linear inverse solution with multiple priors in diffuse optical tomography,” Appl. Opt.44(10), 1948–1956 (2005). [CrossRef] [PubMed]
  31. M. E. Eames and H. Dehghani, “Wavelength dependence of sensitivity in spectral diffuse optical imaging: effect of normalization on image reconstruction,” Opt. Express16(22), 17780–17791 (2008). [CrossRef] [PubMed]
  32. P. K. Yalavarthy, H. Dehghani, B. W. Pogue, and K. D. Paulsen, “Critical computational aspects of near infrared circular tomographic imaging: Analysis of measurement number, mesh resolution and reconstruction basis,” Opt. Express14(13), 6113–6127 (2006). [CrossRef] [PubMed]
  33. 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(10), 701–703 (2001). [CrossRef] [PubMed]
  34. 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 techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol.45(4), 1051–1070 (2000). [CrossRef] [PubMed]
  35. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, Bristol, 1998), Chap. 5.
  36. H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng.25(6), 711–732 (2009). [CrossRef] [PubMed]
  37. Materialise, “Mimics,” http://www.materialise.com/mimics .
  38. B. R. White and J. P. Culver, “Quantitative evaluation of high-density diffuse optical tomography: in vivo resolution and mapping performance,” J. Biomed. Opt.15(2), 026006 (2010). [CrossRef] [PubMed]
  39. H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Appl. Opt.48(10), D137–D143 (2009). [CrossRef] [PubMed]
  40. Y. Zhan, A. T. Eggebrecht, J. P. Culver, and H. Dehghani, “Image quality analysis of high-density diffuse optical tomography incorporating a subject-specific head model,” Front Neuroenergetics4, 6 (2012). [CrossRef] [PubMed]
  41. J. P. Culver, A. M. Siegel, J. J. Stott, and D. A. Boas, “Volumetric diffuse optical tomography of brain activity,” Opt. Lett.28(21), 2061–2063 (2003). [CrossRef] [PubMed]

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