August 2018, 12(4): 883-901. doi: 10.3934/ipi.2018037

Use of an optimized spatial prior in D-bar reconstructions of EIT tank data

1. 

Department of Mathematics, Gonzaga University, MSC 2615, Spokane, WA 99258, USA

2. 

Department of Mathematics and School of Biomedical Engineering, Colorado State University, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA

* Corresponding author: Melody Alsaker

Received  March 2017 Revised  December 2017 Published  June 2018

The aim of this paper is to demonstrate the feasibility of using spatial a priori information in the 2-D D-bar method to improve the spatial resolution of EIT reconstructions of experimentally collected data. The prior consists of imperfectly known information about the spatial locations of inclusions and the assumption that the conductivity is a mollified piecewise constant function. The conductivity values for the prior are constructed using a novel method in which a nonlinear constrained optimization routine is used to select the values for the piecewise constant function that give the best fit to the scattering transform computed from the measured data in a disk. The prior is then included in the high-frequency components of the scattering transform and in the computation of the solution of the D-bar equation, with weights to control the influence of the prior. In addition, a new technique is described for selecting regularization parameters to truncate the measured scattering data, in which complex scattering frequencies for which the values of the scattering transform differ greatly from those in the scattering prior are omitted. The effectiveness of the method is demonstrated on EIT data collected on saline-filled tanks with agar heart and lungs with various added inhomogeneities.

Citation: Melody Alsaker, Jennifer L. Mueller. Use of an optimized spatial prior in D-bar reconstructions of EIT tank data. Inverse Problems & Imaging, 2018, 12 (4) : 883-901. doi: 10.3934/ipi.2018037
References:
[1]

M. AlsakerS. Hamilton and A. Hauptmann, A direct D-bar method for partial boundary data electrical impedance tomography with a priori information, Inverse Problems and Imaging, 11 (2017), 427-454. doi: 10.3934/ipi.2017020.

[2]

M. Alsaker and J. Mueller, A D-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM J. Imaging Sci, 9 (2016), 1619-1654. doi: 10.1137/15M1020137.

[3]

M. AradS. ZlochiverT. DavidsonY. ShoenfeldA. Adunsky and A. Abboud, The detection of pleural effusion using a parametric eit technique, Physiol. Meas., 30 (2009), 421-428. doi: 10.1088/0967-3334/30/4/006.

[4]

N. J. Avis and D. C. Barber, Incorporating a priori information into the Sheffield filtered backprojection algorithm, Physiol. Meas., 16 (1995), A111-A122. doi: 10.1088/0967-3334/16/3A/011.

[5]

U. Baysal and B. M. Eyüboglu, Use of a priori information in estimating tissue resistivities - a simulation study, Phys. Med. and Biol., 43 (1998), 3589-3606.

[6]

B. H. Brown, Electrical impedance tomography (EIT): A review, Journal of medical engineering & technology, 27 (2003), 97-108. doi: 10.1080/0309190021000059687.

[7]

E. D. L. B. Camargo, Development of an Absolute Electrical Impedance Imaging Algorithm for Clinical Use, PhD thesis, University of São Paulo, 2013.

[8]

E. CostaC. ChavesS. GomesM. BeraldoM. VolpeM. TucciI. SchettinoS. BohmC. CarvalhoH. Tanaka and L. R.G. and M. Amato, Real-time detection of pneumothorax using electrical impedance tomography, Critical Care Medicine, 36 (2008), 1230-1238. doi: 10.1097/CCM.0b013e31816a0380.

[9]

E. Costa, R. Gonzalez Lima and M. Amato, Electrical impedance tomography, in Intensive Care Medicine (ed. J. Vincent), Springer, New York, 2009, 394–404.

[10]

H. DehghaniD. C. Barber and I. Basarab-Horwath, Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography, Physiol. Meas., 20 (1999), 87-102.

[11]

D. C. Dobson and F. Santosa, An image-enhancement technique for electrical impedance tomography, Inverse Probl., 10 (1994), 317-334. doi: 10.1088/0266-5611/10/2/008.

[12]

M. Dodd and J. Mueller, A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Probl. Imag., 8 (2014), 1013-1031. doi: 10.3934/ipi.2014.8.1013.

[13]

L. D. Faddeev, Increasing solutions of the schroedinger equation, Fifty Years of Mathematical Physics, (2016), 34-36. doi: 10.1142/9789814340960_0003.

[14]

D. FerrarioB. GrychtolA. AdlerJ. SolaS. H. Bohm and M. Bodenstein, Toward morphological thoracic EIT: Major signal sources correspond to respective organ locations in CT, IEEE T. Med. Imaging, 59 (2012), 3000-3008. doi: 10.1109/TBME.2012.2209116.

[15]

D. Flores-Tapia and S. Pistorius, Electrical impedance tomography reconstruction using a monotonicity approach based on a priori knowledge, in Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE, 2010, 4996–4999. doi: 10.1109/IEMBS.2010.5627204.

[16]

I. FrerichsS. PulletzG. ElkeF. ReifferscheidD. SchädlerJ. Scholz and N. Weiler, Assessment of changes in distribution of lung perfusion by electrical impedance tomography, Respiration, 77 (2009), 282-291. doi: 10.1159/000193994.

[17]

S. HamiltonJ. Mueller and M. Alsaker, Incorporating a spatial prior into nonlinear d-bar eit imaging for complex admittivities, IEEE T. Med. Imaging, 36 (2017), 457-466. doi: 10.1109/TMI.2016.2613511.

[18]

C. N. L. HerreraM. F. M. VallejoJ. L. Mueller and R. G. Lima, Direct 2-D reconstructions of conductivity and permittivity from EIT data on a human chest, IEEE T. Med. Imaging, 34 (2015), 267-274.

[19]

D. S. Holder, Electrical Impedance Tomography: Methods, History and Applications, CRC Press, 2004. doi: 10.1201/9781420034462.

[20]

D. IsaacsonJ. L. MuellerJ. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE T. Med. Imaging, 23 (2004), 821-828. doi: 10.1109/TMI.2004.827482.

[21]

J. P. KaipioV. KolehmainenM. Vauhkonen and E. Somersalo, Inverse problems with structural prior information, Inverse Probl., 15 (1999), 713-729. doi: 10.1088/0266-5611/15/3/306.

[22]

K. KnudsenM. LassasJ. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imag., 3 (2009), 599-624. doi: 10.3934/ipi.2009.3.599.

[23]

K. LowhagenS. Lundin and O. Stenqvist, Regional intratidal gas distribution in acute lung injury and acute respiratory distress syndrome - assessed by electric impedance tomography, Minerva Anestesiologica, 76 (2010), 1024-1035.

[24]

M. Mellenthin, J. Mueller, E. de Camargo, F. de Moura, T. Santos, R. Lima, S. Hamilton, P. Muller and M. Alsaker, The ACE1 electrical impedance tomography system for thoracic imaging, In review.

[25]

T. MudersH. Luepschen and C. Putensen, Impedance tomography as a new monitoring technique, Curr Opin Crit Care, 16 (2010), 269-275. doi: 10.1097/MCC.0b013e3283390cbf.

[26]

J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, SIAM, Philadelphia, PA, 2012. doi: 10.1137/1.9781611972344.

[27]

E. K. Murphy and J. L. Mueller, Effect of domain shape modeling and measurement errors on the 2-D D-bar method for EIT, IEEE T. Med. Imaging, 28 (2009), 1576-1584. doi: 10.1109/TMI.2009.2021611.

[28]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math., 143 (1996), 71-96. doi: 10.2307/2118653.

[29]

D. NguyenJ. C. Thiagalingam and A. A. McEwan, A review on electrical impedance tomography for pulmonary perfusion imaging, Physiol. Meas., 33 (2012), 695-706. doi: 10.1088/0967-3334/33/5/695.

[30]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Verlag, 2006.

[31]

R. G. Novikov, Multidimensional inverse spectral problem for the equation —$δ$$ψ$ + (v(x) — eu(x))$ψ$ = 0, Functional Analysis and Its Applications, 22 (1988), 263-272. doi: 10.1007/BF01077418.

[32]

H. ReiniusJ. B. BorgesF. FredénL. JideusE. D. CamargoM. B. AmatoG. HedenstiernaA. Larsson and F. Lennmyr, Real-time ventilation and perfusion distributions by electrical impedance tomography during one-lung ventilation with capnothorax, Acta Anaesthesiol Scand., 59 (2015), 354-368. doi: 10.1111/aas.12455.

[33]

S. SiltanenJ. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem, Inverse Probl., 16 (2000), 681-699. doi: 10.1088/0266-5611/16/3/310.

[34]

S. Siltanen, Electrical Impedance Tomography and Faddeev Green's Functions, PhD thesis, Helsinki University of Technology, 1999.

[35]

M. Soleimani, Electrical impedance tomography imaging using a priori ultrasound data, BioMed. Eng. OnLine, 5.

[36]

M. VauhkonenD. VadaszP. A. KarjalainenE. Somersalo and J. P. Kaipio, Tikhonov regularization and prior information in electrical impedance tomography, IEEE T. Med. Imaging, 17 (1998), 285-293. doi: 10.1109/42.700740.

[37]

J. VictorinoJ. BorgesV. OkamotoG. MatosM. TucciM. CaramezH. TanakaF. SipmannD. SantosC. BarbasC. Carvalho and M. P. Amato, Imbalances in regional lung ventilation: a validation study on electrical impedance tomography, American Journal of Respiratory and Critical Care Medicine, 169 (2004), 791-800. doi: 10.1164/rccm.200301-133OC.

show all references

References:
[1]

M. AlsakerS. Hamilton and A. Hauptmann, A direct D-bar method for partial boundary data electrical impedance tomography with a priori information, Inverse Problems and Imaging, 11 (2017), 427-454. doi: 10.3934/ipi.2017020.

[2]

M. Alsaker and J. Mueller, A D-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM J. Imaging Sci, 9 (2016), 1619-1654. doi: 10.1137/15M1020137.

[3]

M. AradS. ZlochiverT. DavidsonY. ShoenfeldA. Adunsky and A. Abboud, The detection of pleural effusion using a parametric eit technique, Physiol. Meas., 30 (2009), 421-428. doi: 10.1088/0967-3334/30/4/006.

[4]

N. J. Avis and D. C. Barber, Incorporating a priori information into the Sheffield filtered backprojection algorithm, Physiol. Meas., 16 (1995), A111-A122. doi: 10.1088/0967-3334/16/3A/011.

[5]

U. Baysal and B. M. Eyüboglu, Use of a priori information in estimating tissue resistivities - a simulation study, Phys. Med. and Biol., 43 (1998), 3589-3606.

[6]

B. H. Brown, Electrical impedance tomography (EIT): A review, Journal of medical engineering & technology, 27 (2003), 97-108. doi: 10.1080/0309190021000059687.

[7]

E. D. L. B. Camargo, Development of an Absolute Electrical Impedance Imaging Algorithm for Clinical Use, PhD thesis, University of São Paulo, 2013.

[8]

E. CostaC. ChavesS. GomesM. BeraldoM. VolpeM. TucciI. SchettinoS. BohmC. CarvalhoH. Tanaka and L. R.G. and M. Amato, Real-time detection of pneumothorax using electrical impedance tomography, Critical Care Medicine, 36 (2008), 1230-1238. doi: 10.1097/CCM.0b013e31816a0380.

[9]

E. Costa, R. Gonzalez Lima and M. Amato, Electrical impedance tomography, in Intensive Care Medicine (ed. J. Vincent), Springer, New York, 2009, 394–404.

[10]

H. DehghaniD. C. Barber and I. Basarab-Horwath, Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography, Physiol. Meas., 20 (1999), 87-102.

[11]

D. C. Dobson and F. Santosa, An image-enhancement technique for electrical impedance tomography, Inverse Probl., 10 (1994), 317-334. doi: 10.1088/0266-5611/10/2/008.

[12]

M. Dodd and J. Mueller, A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Probl. Imag., 8 (2014), 1013-1031. doi: 10.3934/ipi.2014.8.1013.

[13]

L. D. Faddeev, Increasing solutions of the schroedinger equation, Fifty Years of Mathematical Physics, (2016), 34-36. doi: 10.1142/9789814340960_0003.

[14]

D. FerrarioB. GrychtolA. AdlerJ. SolaS. H. Bohm and M. Bodenstein, Toward morphological thoracic EIT: Major signal sources correspond to respective organ locations in CT, IEEE T. Med. Imaging, 59 (2012), 3000-3008. doi: 10.1109/TBME.2012.2209116.

[15]

D. Flores-Tapia and S. Pistorius, Electrical impedance tomography reconstruction using a monotonicity approach based on a priori knowledge, in Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE, 2010, 4996–4999. doi: 10.1109/IEMBS.2010.5627204.

[16]

I. FrerichsS. PulletzG. ElkeF. ReifferscheidD. SchädlerJ. Scholz and N. Weiler, Assessment of changes in distribution of lung perfusion by electrical impedance tomography, Respiration, 77 (2009), 282-291. doi: 10.1159/000193994.

[17]

S. HamiltonJ. Mueller and M. Alsaker, Incorporating a spatial prior into nonlinear d-bar eit imaging for complex admittivities, IEEE T. Med. Imaging, 36 (2017), 457-466. doi: 10.1109/TMI.2016.2613511.

[18]

C. N. L. HerreraM. F. M. VallejoJ. L. Mueller and R. G. Lima, Direct 2-D reconstructions of conductivity and permittivity from EIT data on a human chest, IEEE T. Med. Imaging, 34 (2015), 267-274.

[19]

D. S. Holder, Electrical Impedance Tomography: Methods, History and Applications, CRC Press, 2004. doi: 10.1201/9781420034462.

[20]

D. IsaacsonJ. L. MuellerJ. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE T. Med. Imaging, 23 (2004), 821-828. doi: 10.1109/TMI.2004.827482.

[21]

J. P. KaipioV. KolehmainenM. Vauhkonen and E. Somersalo, Inverse problems with structural prior information, Inverse Probl., 15 (1999), 713-729. doi: 10.1088/0266-5611/15/3/306.

[22]

K. KnudsenM. LassasJ. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imag., 3 (2009), 599-624. doi: 10.3934/ipi.2009.3.599.

[23]

K. LowhagenS. Lundin and O. Stenqvist, Regional intratidal gas distribution in acute lung injury and acute respiratory distress syndrome - assessed by electric impedance tomography, Minerva Anestesiologica, 76 (2010), 1024-1035.

[24]

M. Mellenthin, J. Mueller, E. de Camargo, F. de Moura, T. Santos, R. Lima, S. Hamilton, P. Muller and M. Alsaker, The ACE1 electrical impedance tomography system for thoracic imaging, In review.

[25]

T. MudersH. Luepschen and C. Putensen, Impedance tomography as a new monitoring technique, Curr Opin Crit Care, 16 (2010), 269-275. doi: 10.1097/MCC.0b013e3283390cbf.

[26]

J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, SIAM, Philadelphia, PA, 2012. doi: 10.1137/1.9781611972344.

[27]

E. K. Murphy and J. L. Mueller, Effect of domain shape modeling and measurement errors on the 2-D D-bar method for EIT, IEEE T. Med. Imaging, 28 (2009), 1576-1584. doi: 10.1109/TMI.2009.2021611.

[28]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math., 143 (1996), 71-96. doi: 10.2307/2118653.

[29]

D. NguyenJ. C. Thiagalingam and A. A. McEwan, A review on electrical impedance tomography for pulmonary perfusion imaging, Physiol. Meas., 33 (2012), 695-706. doi: 10.1088/0967-3334/33/5/695.

[30]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Verlag, 2006.

[31]

R. G. Novikov, Multidimensional inverse spectral problem for the equation —$δ$$ψ$ + (v(x) — eu(x))$ψ$ = 0, Functional Analysis and Its Applications, 22 (1988), 263-272. doi: 10.1007/BF01077418.

[32]

H. ReiniusJ. B. BorgesF. FredénL. JideusE. D. CamargoM. B. AmatoG. HedenstiernaA. Larsson and F. Lennmyr, Real-time ventilation and perfusion distributions by electrical impedance tomography during one-lung ventilation with capnothorax, Acta Anaesthesiol Scand., 59 (2015), 354-368. doi: 10.1111/aas.12455.

[33]

S. SiltanenJ. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem, Inverse Probl., 16 (2000), 681-699. doi: 10.1088/0266-5611/16/3/310.

[34]

S. Siltanen, Electrical Impedance Tomography and Faddeev Green's Functions, PhD thesis, Helsinki University of Technology, 1999.

[35]

M. Soleimani, Electrical impedance tomography imaging using a priori ultrasound data, BioMed. Eng. OnLine, 5.

[36]

M. VauhkonenD. VadaszP. A. KarjalainenE. Somersalo and J. P. Kaipio, Tikhonov regularization and prior information in electrical impedance tomography, IEEE T. Med. Imaging, 17 (1998), 285-293. doi: 10.1109/42.700740.

[37]

J. VictorinoJ. BorgesV. OkamotoG. MatosM. TucciM. CaramezH. TanakaF. SipmannD. SantosC. BarbasC. Carvalho and M. P. Amato, Imbalances in regional lung ventilation: a validation study on electrical impedance tomography, American Journal of Respiratory and Critical Care Medicine, 169 (2004), 791-800. doi: 10.1164/rccm.200301-133OC.

Figure 1.  The five experimental data sets considered in this paper
Figure 2.  Illustration of artifacts and distortions in reconstructions of experimental case (iv) when ${\bf{t}}_{\text{pr}}$ is a poor match for ${\bf{t}}$. These images are reconstructions using the a priori method described here, with the scattering prior ${\bf{t}}_{\text{pr}}$ computed from output ${{\bf{c}}}^j$ of six iterative steps of the optimization routine. The reconstruction with the initial guess ${\bf{c}}^0$ is depicted in (a) and the reconstruction using the final result of the optimization routine is depicted in (f). The value of the objective function $J({\bf{c}})$ is given below each reconstruction. Smaller $J({\bf{c}})$ indicates increased goodness of fit between ${\bf{t}}_{\text{pr}}$ and ${\bf{t}}$
Figure 6.  Data Collection 2, case (ⅳ) agar heart and lungs. The influence of the prior increases from left to right in each row
Figure 3.  Data Collection 1, case (ⅰ) agar heart and lungs. The influence of the prior increases from left to right in each row
Figure 4.  Data Collection 1, case (ⅱ) agar heart and lungs with a conductive copper pipe added to right lung. The influence of the prior increases from left to right in each row
Figure 5.  Data Collection 1, case (ⅲ) agar heart and lungs with a resistive PVC pipe added to right lung. The influence of the prior increases from left to right in each row
Figure 7.  Data Collection 2, case (ⅴ) agar heart and lungs with top half of left lung removed. The influence of the prior increases from left to right in each row
Table 1.  Results from experiments with the optimization routine, in which values in the initial guess vector ${{\bf{c}}}^0$ were varied. We include statistical summary values for the initial objective function $J({{\bf{c}}}^0)$, the number of iterations required (NumIter), the resulting output vector ${{\bf{c}}}$, the optimized value of the objective function $J({{\bf{c}}})$, and the quantity $D({{\bf{c}}}): = \| {\bf{t}}_{\text{pr}}^{\text{vec}}({{\bf{c}}}) - {\bf{t}}_{\text{pr}}^{\text{vec}}({{\bf{c}}}_*) \|_{\infty}$ where ${{\bf{c}}}_*$ corresponds to the "control experiment" in our tests, which is a measure of variation between test cases of the resulting scattering data. Statistical values presented include the mean, max, min, range, standard deviation, and coefficient of variation of each quantity
Initial Guess Vector ${{\bf{c}}}^0$ Output Vector ${{\bf{c}}}$
$c^0_b$ $c^0_h$ $c^0_r$ $c^0_l$ $J({{\bf{c}}}^0)$ NumIter $c_b$ $c_h$ $c_r$ $c_l$ $J({{\bf{c}}})$ $D({{\bf{c}}})$
Mean 0.9978 1.1116 0.9312 0.9312 62.19 14.140 1.0475 1.1689 0.9083 0.9515 9.8517867 3.33E-06
Max 1.1769 1.1769 1.1769 1.1769 145.41 19 1.0547 1.1769 0.9145 0.9580 9.8517867 7.15E-06
Min 0.8658 0.8658 0.8658 0.8658 11.78 9 1.0288 1.1480 0.8921 0.9344 9.8517867 8.31E-07
Range 0.3112 0.3112 0.3112 0.3112 133.63 10 0.0259 0.0289 0.0225 0.0235 1.470E-09 6.32E-06
StdDev 0.0753 0.0904 0.0904 0.0904 46.17 2.406 0.0078 0.0087 0.0067 0.0071 3.636E-10 1.51E-06
CoeffVar 7.55% 8.13% 9.70% 9.70% 74.24% 17.02% 0.74% 0.74% 0.74% 0.74% 3.69E-09% 45.39%
Initial Guess Vector ${{\bf{c}}}^0$ Output Vector ${{\bf{c}}}$
$c^0_b$ $c^0_h$ $c^0_r$ $c^0_l$ $J({{\bf{c}}}^0)$ NumIter $c_b$ $c_h$ $c_r$ $c_l$ $J({{\bf{c}}})$ $D({{\bf{c}}})$
Mean 0.9978 1.1116 0.9312 0.9312 62.19 14.140 1.0475 1.1689 0.9083 0.9515 9.8517867 3.33E-06
Max 1.1769 1.1769 1.1769 1.1769 145.41 19 1.0547 1.1769 0.9145 0.9580 9.8517867 7.15E-06
Min 0.8658 0.8658 0.8658 0.8658 11.78 9 1.0288 1.1480 0.8921 0.9344 9.8517867 8.31E-07
Range 0.3112 0.3112 0.3112 0.3112 133.63 10 0.0259 0.0289 0.0225 0.0235 1.470E-09 6.32E-06
StdDev 0.0753 0.0904 0.0904 0.0904 46.17 2.406 0.0078 0.0087 0.0067 0.0071 3.636E-10 1.51E-06
CoeffVar 7.55% 8.13% 9.70% 9.70% 74.24% 17.02% 0.74% 0.74% 0.74% 0.74% 3.69E-09% 45.39%
[1]

Melody Alsaker, Sarah Jane Hamilton, Andreas Hauptmann. A direct D-bar method for partial boundary data electrical impedance tomography with a priori information. Inverse Problems & Imaging, 2017, 11 (3) : 427-454. doi: 10.3934/ipi.2017020

[2]

Melody Dodd, Jennifer L. Mueller. A real-time D-bar algorithm for 2-D electrical impedance tomography data. Inverse Problems & Imaging, 2014, 8 (4) : 1013-1031. doi: 10.3934/ipi.2014.8.1013

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