June  2020, 14(3): 535-568. doi: 10.3934/ipi.2020025

Robust reconstruction of fluorescence molecular tomography with an optimized illumination pattern

1. 

Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland

2. 

Institute for Biomedical Engineering, University of Zurich and ETH Zurich, 8093 Zurich, Switzerland

3. 

Biomedical Optics Research Laboratory, University Hospital Zurich, 8091 Zurich, Switzerland

* Corresponding author: Wuwei Ren

Received  September 2019 Revised  December 2019 Published  March 2020

Fund Project: This work is supported by the Swiss National Science Foundation and Swiss Innovation Agent BRIDGE proof-of-concept fellowship (No. 178262)

Fluorescence molecular tomography (FMT) is an emerging tool for biomedical research. There are two factors that influence FMT reconstruction most effectively. The first one is regularization techniques. Traditional methods such as Tikhonov regularization suffer from low resolution and poor signal to noise ratio. Therefore, sparse regularization techniques have been introduced to improve the reconstruction quality. The second factor is the illumination pattern. A better illumination pattern ensures the quantity and quality of the information content of the data set, thus leading to better reconstructions. In this work, we take advantage of the discrete formulation of the forward problem to give a rigorous definition of an illumination pattern as well as the admissible set of patterns. We add restrictions in the admissible set as different types of regularizers to a discrepancy functional, generating another inverse problem with the illumination pattern as unknown. Both inverse problems of reconstructing the fluorescence distribution and finding the optimal illumination pattern are solved by efficient iterative algorithms. Numerical experiments have shown that with a suitable choice of regularization parameters, the two-step approach converges to an optimal illumination pattern quickly and the reconstruction result is improved significantly regardless of the initial illumination setting.

Citation: Yan Liu, Wuwei Ren, Habib Ammari. Robust reconstruction of fluorescence molecular tomography with an optimized illumination pattern. Inverse Problems & Imaging, 2020, 14 (3) : 535-568. doi: 10.3934/ipi.2020025
References:
[1]

G. S. AlbertiH. AmmariF. Romero and T. Wintz, Dynamic spike superresolution and applications to ultrafast ultrasound imaging, SIAM J. Imaging Sci., 12 (2019), 1501-1527.  doi: 10.1137/18M1174775.  Google Scholar

[2]

H. Ammari, J. Garnier, H. Kang, L. H. Nguyen and L. Seppecher, Multi-wave medical imaging: Mathematical modelling & imaging reconstruction, Modelling and Simulation in Medical Imaging, 2 (2017), 688. doi: 10.1142/q0067.  Google Scholar

[3]

S. R. ArridgeM. SchweigerM. Hiraoka and D. T. Delpy, A finite element approach for modeling photon transport in tissue, Medical Physics, 20 (1993), 299-309.  doi: 10.1118/1.597069.  Google Scholar

[4]

S. R. Arridge and J. C. Schotland, Optical tomography: Forward and inverse problems, Inverse Problems, 25 (2009), 123010, 59pp. doi: 10.1088/0266-5611/25/12/123010.  Google Scholar

[5]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[6]

M. Bergounioux, E. Bretin and Y. Privat, How to position sensors in thermo-acoustic tomography, Inverse Problems, 35 (2019), 074003, 25pp. doi: 10.1088/1361-6420/ab0e4d.  Google Scholar

[7]

E. Candes, N. Braun and M. Wakin, Sparse signal and image recovery from compressive samples, 2007 4th IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2007, 976–979. doi: 10.1109/ISBI.2007.357017.  Google Scholar

[8]

T. CorreiaM. KochA. AleV. Ntziachristos and S. Arridge, Patch-based anisotropic diffusion scheme for fluorescence diffuse optical tomography–part 2: Image reconstruction, Phys. Med. Biol., 61 (2016), 1452-1475.  doi: 10.1088/0031-9155/61/4/1452.  Google Scholar

[9]

S. C. DavisH. DehghaniJ. WangS. JiangB. W. Pogue and K. D. Paulsen, Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization, Optics Express, 15 (2007), 4066-4082.  doi: 10.1364/OE.15.004066.  Google Scholar

[10]

E. DemidenkoA. HartovN. Soni and K. D. Paulsen, On optimal current patterns for electrical impedance tomography, IEEE Trans. on Biomedical Engineering, 52 (2005), 238-248.  doi: 10.1109/TBME.2004.840506.  Google Scholar

[11]

N. DucrosC. D'AndreaA. BassiG. Valentini and S. Arridge, A virtual source pattern method for fluorescence tomography with structured light, Phys. Med. Biol., 57 (2012), 3811-3832.  doi: 10.1088/0031-9155/57/12/3811.  Google Scholar

[12]

J. DuttaS. AhnA. Joshi and R. M. Leahy, Illumination pattern optimization for fluorescence tomography: Theory and simulation studies, Phys. Med. Biol., 55 (2010), 2961-2982.  doi: 10.1088/0031-9155/55/10/011.  Google Scholar

[13]

J. DuttaS. AhnC. LiS. R. Cherry and R. M. Leahy, Joint L1 and total variation regularization for fluorescence molecular tomography, Phys. Med. Biol., 57 (2012), 1459-1476.  doi: 10.1088/0031-9155/57/6/1459.  Google Scholar

[14]

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3$^rd$ edition, Prentice-Hall, Upper Saddle River, New Jersey, 2006. Google Scholar

[15]

R. A. J. GroenhuisH. A. Ferwerda and J. J. T. Bosch, Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory, Appl. Opt., 22 (1983), 2456-2462.  doi: 10.1364/AO.22.002456.  Google Scholar

[16] T. HastieR. Tibshirani and M. Wainwright, Statistical Learning with Sparsity: The Lasso and Generalizations, CRC Press, Boca Raton, FL, 2015.   Google Scholar
[17]

N. HyvönenA. Seppänen and S. Staboulis, Optimizing electrode positions in electrical impedance tomography, SIAM J. Appl. Math., 74 (2014), 1831-1851.  doi: 10.1137/140966174.  Google Scholar

[18]

J. KaipioA. SeppänenA. Voutilainen and H. Haario, Optimal current patterns in dynamical electrical impedance tomography imaging, Inverse Problems, 23 (2007), 1201-1214.  doi: 10.1088/0266-5611/23/3/021.  Google Scholar

[19]

J. R. Lorenzo, Principles of Diffuse Light Propagation: Light Propagation in Tissues with Applications in Biology and Medicine, World Scientific, 2012. doi: 10.1142/7609.  Google Scholar

[20]

A. LyonsF. TonoliniA. BoccoliniA. RepettiR. HendersonY. Wiaux and D. Faccio, Computational time-of-flight diffuse optical tomography, Nature Photonics, 13 (2019), 575-579.  doi: 10.1038/s41566-019-0439-x.  Google Scholar

[21]

V. NtziachristosC. H. TungC. Bremer and R. Weissleder, Fluorescence molecular tomography resolves protease activity in vivo, Nature Medicine, 8 (2002), 757-761.  doi: 10.1038/nm729.  Google Scholar

[22]

V. Ntziachristos, Going deeper than microscopy: The optical imaging frontier in biology, Nature Methods, 7 (2010), 603-614.  doi: 10.1038/nmeth.1483.  Google Scholar

[23]

V. Ntziachristos and R. Weissleder, Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation, Optics Letters, 26 (2001), 893-895.  doi: 10.1364/OL.26.000893.  Google Scholar

[24]

W. RenA. ElmerD. BuehlmannM.-A. AugathD. VatsJ. Ripoll and M. Rudin, Dynamic measurement of tumor vascular permeability and perfusion using a hybrid system for simultaneous magnetic resonance and fluorescence imaging, Mol. Imaging Biol., 18 (2016), 191-200.  doi: 10.1007/s11307-015-0884-y.  Google Scholar

[25]

W. RenH. IslerM. WolfJ. Ripoll and M. Rudin, Smart toolkit for fluorescence tomography: Simulation, reconstruction, and validation, IEEE Trans. on Biomedical Engineering, 67 (2020), 16-26.  doi: 10.1109/TBME.2019.2907460.  Google Scholar

[26]

J. Ripoll, R. B. Schulz and V. Ntziachristos, Free-space propagation of diffuse light: Theory and experiments, Phys. Rev. Lett., 91 (2003), 103901. doi: 10.1103/PhysRevLett.91.103901.  Google Scholar

[27] M. Rudin, Molecular Imaging: Basic Principles and Applications in Biomedical Research, Imperial College Press, London, 2013.   Google Scholar
[28]

M. Schweiger and S. R. Arridge, The finite-element method for the propagation of light in scattering media: Frequency domain case, Medical Physics, 24 (1997), 895-902.  doi: 10.1118/1.598008.  Google Scholar

[29]

M. Schweiger and S. R. Arridge, The toast++ software suite for forward and inverse modeling in optical tomography, J. Biomed. Opt., 19 (2014), 040801. doi: 10.1117/1.JBO.19.4.040801.  Google Scholar

[30]

D. ZhuY. ZhaoR. BaikejiangZ. Yuan and C. Li, Comparison of regularization methods in fluorescence molecular tomography, Photonics, 1 (2014), 95-109.  doi: 10.3390/photonics1020095.  Google Scholar

[31]

Y. ZhuA. K. JhaD. F. Wong and A. Rahmim, Image reconstruction in fluorescence molecular tomography with sparsity-initialized maximum-likelihood expectation maximization, Biomed. Opt. Express, 9 (2018), 3106-3121.  doi: 10.1364/BOE.9.003106.  Google Scholar

show all references

References:
[1]

G. S. AlbertiH. AmmariF. Romero and T. Wintz, Dynamic spike superresolution and applications to ultrafast ultrasound imaging, SIAM J. Imaging Sci., 12 (2019), 1501-1527.  doi: 10.1137/18M1174775.  Google Scholar

[2]

H. Ammari, J. Garnier, H. Kang, L. H. Nguyen and L. Seppecher, Multi-wave medical imaging: Mathematical modelling & imaging reconstruction, Modelling and Simulation in Medical Imaging, 2 (2017), 688. doi: 10.1142/q0067.  Google Scholar

[3]

S. R. ArridgeM. SchweigerM. Hiraoka and D. T. Delpy, A finite element approach for modeling photon transport in tissue, Medical Physics, 20 (1993), 299-309.  doi: 10.1118/1.597069.  Google Scholar

[4]

S. R. Arridge and J. C. Schotland, Optical tomography: Forward and inverse problems, Inverse Problems, 25 (2009), 123010, 59pp. doi: 10.1088/0266-5611/25/12/123010.  Google Scholar

[5]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[6]

M. Bergounioux, E. Bretin and Y. Privat, How to position sensors in thermo-acoustic tomography, Inverse Problems, 35 (2019), 074003, 25pp. doi: 10.1088/1361-6420/ab0e4d.  Google Scholar

[7]

E. Candes, N. Braun and M. Wakin, Sparse signal and image recovery from compressive samples, 2007 4th IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2007, 976–979. doi: 10.1109/ISBI.2007.357017.  Google Scholar

[8]

T. CorreiaM. KochA. AleV. Ntziachristos and S. Arridge, Patch-based anisotropic diffusion scheme for fluorescence diffuse optical tomography–part 2: Image reconstruction, Phys. Med. Biol., 61 (2016), 1452-1475.  doi: 10.1088/0031-9155/61/4/1452.  Google Scholar

[9]

S. C. DavisH. DehghaniJ. WangS. JiangB. W. Pogue and K. D. Paulsen, Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization, Optics Express, 15 (2007), 4066-4082.  doi: 10.1364/OE.15.004066.  Google Scholar

[10]

E. DemidenkoA. HartovN. Soni and K. D. Paulsen, On optimal current patterns for electrical impedance tomography, IEEE Trans. on Biomedical Engineering, 52 (2005), 238-248.  doi: 10.1109/TBME.2004.840506.  Google Scholar

[11]

N. DucrosC. D'AndreaA. BassiG. Valentini and S. Arridge, A virtual source pattern method for fluorescence tomography with structured light, Phys. Med. Biol., 57 (2012), 3811-3832.  doi: 10.1088/0031-9155/57/12/3811.  Google Scholar

[12]

J. DuttaS. AhnA. Joshi and R. M. Leahy, Illumination pattern optimization for fluorescence tomography: Theory and simulation studies, Phys. Med. Biol., 55 (2010), 2961-2982.  doi: 10.1088/0031-9155/55/10/011.  Google Scholar

[13]

J. DuttaS. AhnC. LiS. R. Cherry and R. M. Leahy, Joint L1 and total variation regularization for fluorescence molecular tomography, Phys. Med. Biol., 57 (2012), 1459-1476.  doi: 10.1088/0031-9155/57/6/1459.  Google Scholar

[14]

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3$^rd$ edition, Prentice-Hall, Upper Saddle River, New Jersey, 2006. Google Scholar

[15]

R. A. J. GroenhuisH. A. Ferwerda and J. J. T. Bosch, Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory, Appl. Opt., 22 (1983), 2456-2462.  doi: 10.1364/AO.22.002456.  Google Scholar

[16] T. HastieR. Tibshirani and M. Wainwright, Statistical Learning with Sparsity: The Lasso and Generalizations, CRC Press, Boca Raton, FL, 2015.   Google Scholar
[17]

N. HyvönenA. Seppänen and S. Staboulis, Optimizing electrode positions in electrical impedance tomography, SIAM J. Appl. Math., 74 (2014), 1831-1851.  doi: 10.1137/140966174.  Google Scholar

[18]

J. KaipioA. SeppänenA. Voutilainen and H. Haario, Optimal current patterns in dynamical electrical impedance tomography imaging, Inverse Problems, 23 (2007), 1201-1214.  doi: 10.1088/0266-5611/23/3/021.  Google Scholar

[19]

J. R. Lorenzo, Principles of Diffuse Light Propagation: Light Propagation in Tissues with Applications in Biology and Medicine, World Scientific, 2012. doi: 10.1142/7609.  Google Scholar

[20]

A. LyonsF. TonoliniA. BoccoliniA. RepettiR. HendersonY. Wiaux and D. Faccio, Computational time-of-flight diffuse optical tomography, Nature Photonics, 13 (2019), 575-579.  doi: 10.1038/s41566-019-0439-x.  Google Scholar

[21]

V. NtziachristosC. H. TungC. Bremer and R. Weissleder, Fluorescence molecular tomography resolves protease activity in vivo, Nature Medicine, 8 (2002), 757-761.  doi: 10.1038/nm729.  Google Scholar

[22]

V. Ntziachristos, Going deeper than microscopy: The optical imaging frontier in biology, Nature Methods, 7 (2010), 603-614.  doi: 10.1038/nmeth.1483.  Google Scholar

[23]

V. Ntziachristos and R. Weissleder, Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation, Optics Letters, 26 (2001), 893-895.  doi: 10.1364/OL.26.000893.  Google Scholar

[24]

W. RenA. ElmerD. BuehlmannM.-A. AugathD. VatsJ. Ripoll and M. Rudin, Dynamic measurement of tumor vascular permeability and perfusion using a hybrid system for simultaneous magnetic resonance and fluorescence imaging, Mol. Imaging Biol., 18 (2016), 191-200.  doi: 10.1007/s11307-015-0884-y.  Google Scholar

[25]

W. RenH. IslerM. WolfJ. Ripoll and M. Rudin, Smart toolkit for fluorescence tomography: Simulation, reconstruction, and validation, IEEE Trans. on Biomedical Engineering, 67 (2020), 16-26.  doi: 10.1109/TBME.2019.2907460.  Google Scholar

[26]

J. Ripoll, R. B. Schulz and V. Ntziachristos, Free-space propagation of diffuse light: Theory and experiments, Phys. Rev. Lett., 91 (2003), 103901. doi: 10.1103/PhysRevLett.91.103901.  Google Scholar

[27] M. Rudin, Molecular Imaging: Basic Principles and Applications in Biomedical Research, Imperial College Press, London, 2013.   Google Scholar
[28]

M. Schweiger and S. R. Arridge, The finite-element method for the propagation of light in scattering media: Frequency domain case, Medical Physics, 24 (1997), 895-902.  doi: 10.1118/1.598008.  Google Scholar

[29]

M. Schweiger and S. R. Arridge, The toast++ software suite for forward and inverse modeling in optical tomography, J. Biomed. Opt., 19 (2014), 040801. doi: 10.1117/1.JBO.19.4.040801.  Google Scholar

[30]

D. ZhuY. ZhaoR. BaikejiangZ. Yuan and C. Li, Comparison of regularization methods in fluorescence molecular tomography, Photonics, 1 (2014), 95-109.  doi: 10.3390/photonics1020095.  Google Scholar

[31]

Y. ZhuA. K. JhaD. F. Wong and A. Rahmim, Image reconstruction in fluorescence molecular tomography with sparsity-initialized maximum-likelihood expectation maximization, Biomed. Opt. Express, 9 (2018), 3106-3121.  doi: 10.1364/BOE.9.003106.  Google Scholar

Figure 1.  A snapshot of the latest stand-alone FMT hardware system in the animal imaging center of ETH Zurich. The main components of the hardware system include a CMOS camera (detector), the animal support (object), the laser source and the MEMS scanner. Photo credit: Andrin G. C. Rickenbacher
Figure 2.  Left: 15$ \times $15$ \times $15 $ mm^3 $ phantom with absorption coefficient $ \mu_a = 0.01 $ $ mm^{-1} $ and scattering coefficient $ \mu_s = 1 $ $ mm^{-1} $. Blue region represents the virtual detector plane that is made up of a 60 by 30 detector array. Each detector has size 1$ \times $1 $ mm^2 $ and an efficiency of 100%. Red points represent the 10 by 10 laser array. Each laser point has power 1 $ W/mm^2 $ and wavelength 670 $ nm $. The setup of the phantom is performed by the TOAST++ software [29]. The noise level of the detector is set to 0. Right: birdview of the laser and detector array from the top surface of the phantom. Laser array is placed at the center of the detector plane
Figure 3.  Fluorescence ground truth (GT) setting sliced at reference point $ (13, 9, 6) $ viewed from three directions: top, left and front of the phantom (from left to right). The fluorescence ground truth is composed of two identical bars of size 1$ \times $1$ \times $10 $ mm^3 $ and fluorescence intensity 100 $ a.u. $. The two bars are both embedded at a depth of 3 $ mm $ from the top surface of the phantom and $ 6 mm $ away from each other
Figure 4.  Reconstruction results using ART, $ l_2 $, $ l_1 $ and $ l_1 $ joint $ l_2 $ (Elastic Net, EN) regularization methods sliced at the same reference point $ (13, 9, 6) $ are compared with the ground truth (GT) in the first column. Each column contains the result of the chosen reconstruction method viewed from the top (a), left (b) and front (c) of the phantom
Figure 5.  Mean square error, Dice similarity, volume ratio and signal to noise ratio of the reconstruction results using nonsparse methods ART and $ l_2 $ regularization (red bars) and sparse methods $ l_1 $ and $ l_1 $ joint $ l_2 $ regularization (blue bars)
Figure 6.  Updated spatial distribution of lasers after round 1, 2, 3, 4, 5 and 9. Red points represent the lasers. Laser positions are attached to nodes of the finite element mesh on the top surface of the phantom. The number of lasers (illumination points) are indicated above each figure
Figure 8.  Reconstruction results sliced at the same reference point $ (13, 9, 6) $ as the ground truth in Figure 3. The first row (Illumination 0) contains reconstruction results of the initial laser setting viewed from the top (a), left (b) and front (c) of the phantom. Row 2 to row 7 are reconstruction results from the 1st, 2nd, 3rd, 4th, 5th and 9th updated illumination pattern
Figure 9.  Mean square error, Dice similarity, volume ratio and signal to noise ratio of the reconstruction results at round 1-6 and round 10. The error of the reconstruction result based on the initial laser setting, i.e., the first round of iteration is indicated by the red bars. The blue bars represent the error of following rounds of iterations that are based on updated illumination patterns
Figure 10.  A different initial illumination pattern viewed from the top surface of the same phantom. Except for the initial laser intensities and the location of the laser array, everything else remains the same as the previous experiment. The detector array is the same 60 by 30 array represented by the blue area; the laser array is the 10 by 10 point array represented by the red points
Figure 11.  Updated spatial layout of illumination points after round 1, 2, 3, 4, 5 and 9 of iteration starting from the new initial illumination pattern in Figure 10. Red points represent the lasers. The number of lasers (illumination points) are indicated above each figure
Figure 12.  Laser intensity profiles of each updated illumination pattern starting from the new initial laser setting. First column from top to bottom: laser profile for 1st, 3rd, and 5th updated illumination patterns. Second column from top to bottom: laser profile for 2nd, 4th, and 9th updated illumination pattern
Figure 14.  Reconstruction results obtained from different rounds of iteration under the new initial laser setting. Reconstruction results are sliced at the same reference point $ (13, 9, 6) $ as the ground truth in Figure 3. The first row (Illumination 0) contains reconstruction results of the initial laser setting viewed from the top (a), left (b) and front (c) of the phantom. Row 2 to row 7 are reconstruction results from the 1st, 2nd, 3rd, 4th, 5th and 9th updated illumination pattern
Figure 7.  Laser intensity profiles of each updated illumination pattern. First column from top to bottom: laser profile for 1st, 3rd, and 5th updated illumination patterns. Second column from top to bottom: laser profile for 2nd, 4rd, and 9th updated illumination pattern. On each panel the $ x- $axis indicates the node number of lasers and the $ y- $axis indicates the intensity ($ W/mm^2 $) of each laser
Figure 13.  Mean square error, Dice similarity, volume ratio and signal to noise ratio of the reconstruction results at round 1-6 and round 10. The error of the reconstruction result based on the initial laser setting, i.e. the first round of iteration is indicated by the red bars. The blue bars represent the error of following rounds of iterations that are based on updated illumination patterns
Table 1.  Comparison of quantitative error under the four metrics between the first reconstruction result based on the initial laser setting and the best reconstruction result of the first and the second experiments
experiment round MSE Dice VR SNR
$ 1^{st} $ initial 0.018436 0.47761 0.52273 1.5303
$ 1^{st} $ $ 3^{rd} $ 0.01489 0.53631 1.0341 3.6664
$ 2^{nd} $ initial 0.020026 0.35200 0.42045 0.70309
$ 2^{nd} $ $ 4^{th} $ 0.015951 0.53125 1.1818 2.9783
experiment round MSE Dice VR SNR
$ 1^{st} $ initial 0.018436 0.47761 0.52273 1.5303
$ 1^{st} $ $ 3^{rd} $ 0.01489 0.53631 1.0341 3.6664
$ 2^{nd} $ initial 0.020026 0.35200 0.42045 0.70309
$ 2^{nd} $ $ 4^{th} $ 0.015951 0.53125 1.1818 2.9783
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