Abstract

We compare reconstructions based on the radiative transport and diffusion equations in optical tomography for media of small sizes. While it is well known that the diffusion approximation is less accurate to describe light propagation in such media, it has not yet been shown how this inaccuracy affects the images obtained with model-based iterative image reconstructions schemes. Using synthetic nondifferential data we calculate the error in the reconstructed images of optical properties as functions of source modulation frequency, noise level in measurement, and diffusion extrapolation length. We observe that the differences between diffusion and transport reconstructions are large when high modulation frequencies and noise-free data are used in the reconstructions. When the noise in data reaches a certain level, approximately 12% in our simulations, the differences between diffusion- and transport-based reconstructions become almost indistinguishable. Given that state-of-the-art optical imaging systems operate at much lower noise levels, the benefits of transport-based reconstructions of small imaging domains can be realized with most of the currently available systems. However, transport-based reconstructions are considerably slower than diffusion-based reconstructions.

© 2007 Optical Society of America

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2006

K. Ren, G. Bal, and A. H. Hielscher, "Frequency domain optical tomography based on the equation of radiative transfer," SIAM J. Sci. Comput. (USA) 28, 1463-1489 (2006).
[CrossRef]

2005

A. H. Hielscher, "Optical tomographic imaging of small animals," Curr. Opi. Biotechnol. 16, 79-88 (2005).
[CrossRef]

M. Francocur, R. Vaillon, and D. R. Rousse, "Theoretical analysis of frequency and time-domain methods for optical characterization of absorbing and scattering media," J. Quant. Spectrosc. Radiat. Transf. 93, 139-150 (2005).
[CrossRef]

H. Xu, B. W. Pogue, R. Springett, and H. Dehghani, "Spectral derivative based image reconstruction provides inherent insensitivity to coupling and geometric errors," Opt. Lett. 30, 2912-2914 (2005).
[CrossRef] [PubMed]

2004

K. Ren, G. S. Abdoulaev, G. Bal, and Andreas H. Hielscher, "Algorithm for solving the equation of radiative transfer in the frequency domain," Opt. Lett. 29, 578-580 (2004).
[CrossRef] [PubMed]

A. Y. Bluestone, M. Stewart, B. Lei, I. S. Kass, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, part 1: Hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

E. E. Graves, R. Weissleder, and V. Ntziachristos, "Fluorescence molecular imaging of small animal tumor models," Current Molecular Medicine 4, 419-430 (2004).
[CrossRef] [PubMed]

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, 1506-1508 (2004).
[CrossRef] [PubMed]

P. Taroni, G. Danesini, A. Torricelli, A. Pifferi, L. Spinelli, and R. Cubeddu, "Clinical trial of time-resolved scanning optical mammography at 4 wavelengths between 683 and 975 nm," J. Biomed. Opt. 9, 464-473 (2004).
[CrossRef] [PubMed]

A. H. Hielscher, A. D. Klose, A. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, "Sagittal laser optical tomography for imaging of rheumatoid finger joints," Phys. Med. Biol. 49, 1147-1163 (2004).
[CrossRef] [PubMed]

2003

2002

G. Bal, "Particle transport through scattering regions with clear layers and inclusions," J. Comp. Physiol. 180, 659-685 (2002).

G. Bal and Y. Maday, "Coupling of transport and diffusion models in linear transport theory," Math. Modell. Numer. Anal. 36, 69-86 (2002).
[CrossRef]

Y. Xu, N. Iftimia, H. Jiang, L. Key, and M. Bolster, "Three-dimensional diffuse optical tomography of bones and joints," J. Biomed. Opt. 7, 88-92 (2002).
[CrossRef] [PubMed]

A. D. Klose and A. H. Hielscher, "Optical tomography using the time-independent equation of radiative transfer. 2. inverse model," J. Quant. Spectrosc. Radiat. Transf. 72, 715-202 (2002).
[CrossRef]

V. A. Markel and J. C. Schotland, "Inverse problem in optical diffusion tomography. 2. Role of boundary conditions," J. Opt. Soc. Am. A 19, 558-566 (2002).
[CrossRef]

R. Elaloufi, R. Carminati, and J. Greffet, "Time-dependent transport through scattering media: from radiative transfer to diffusion," J. Opt. A , Pure Appl. Opt. 4, S103-S108 (2002).
[CrossRef]

G. Bal, "Transport through diffusive and nondiffusive regions, embedded objects, and clear layers," SIAM J. Appl. Math. 62, 1677-1697 (2002).
[CrossRef]

C. H. Schmitz, M. Löcker, J. M. Lasker, A. H. Hielscher, and R. L. Barbour, "Instrumentation for fast functional optical tomography," Rev. Sci. Instrum. 73, 429-439 (2002).
[CrossRef]

2001

A. Kienle, F. K. Forster, and R. Hibst, "Influence of the phase function on determination of the optical properties of biological tissue by spatially resolved reflectance," Opt. Lett. 26, 1571-1573 (2001).
[CrossRef]

Y. Pei, H. L. Graber, and R. L. Barbour, "Influence of systematic errors in reference states on image quality and on stability of derived information for DC optical imaging," Appl. Opt. 40, 5755-5769 (2001).
[CrossRef]

V. A. Markel and J. C. Schotland, "Inverse problem in optical diffusion tomography. 1. Fourier-Laplace inverse formula," J. Opt. Soc. Am. A 18, 1336-1347 (2001).
[CrossRef]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag. 18, 57-75 (2001).
[CrossRef]

E. M. C. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmidt, D. T. Delpy, and S. R. Arridge, "Time resolved optical tomography of the human forearm," Phys. Med. Biol. 46, 1117-1130 (2001).
[CrossRef] [PubMed]

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, and K. D. Paulsen, "Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast," Radiology 218, 261-266 (2001).
[PubMed]

U. Netz, J. Beuthan, H. J. Capius, H. C. Koch, A. D. Klose, and A. H. Hielscher, "Imaging of rheumatoid arthritis in finger joints by sagittal optical tomography," Medical Laser Application 16, 306-310 (2001).
[CrossRef]

2000

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, "Noninvasive functional imaging of human brain using light," J. Cereb. Blood Flow Metab. 20, 469-477 (2000).
[CrossRef] [PubMed]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, "Optical tomography in the presence of void regions," J. Opt. Soc. Am. A 17, 1659-1670 (2000).
[CrossRef]

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, "Noninvasive in vivo characterization of breast tumors using photon migration spectroscopy," Neoplasia 2, 26-40 (2000).
[CrossRef] [PubMed]

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
[CrossRef] [PubMed]

1999

1998

A. D. Klose, A. H. Hielscher, K. M. Hanson, and J. Beuthan, "Three-dimensional optical tomography of a finger joint model for diagnostic of rheumatoid arthritis," Proc. SPIE 3566, 151-160 (1998).
[CrossRef]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissue," Phys. Med. Biol. 43, 1285-1302 (1998).
[CrossRef] [PubMed]

O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. Eng. 14, 1107-1130 (1998).
[CrossRef]

1997

V. Prapavat, W. Runge, J. Mans, A. Krause, J. Beuthan, and G. Müller, "Development of a finger joint phantom for the optical simulation of early stages of rheumatoid arthritis," Biomed. Tech. 42, 319-326 (1997).
[CrossRef]

B. W. Pogue, M. Testorf, T. O. McBride, U. L. Österberg, and K. D. Paulsen, "Instrumentation and design of frequency-domain diffuse optical imager for breast cancer detection," Opt. Express 1, 391-403 (1997).
[CrossRef] [PubMed]

1996

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, "An investigation of light transport through scattering bodies with nonscattering regions," Phys. Med. Biol. 41, 767-783 (1996).
[CrossRef] [PubMed]

1995

A. G. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, "Initial assessment of a simple system for frequency domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995).
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Österberg, B. W. Pogue, and M. S. Patterson, "Simultaneous reconstruction of optical absorption and scattering maps turbid media from near-infrared frequency-domain data," Opt. Lett. 20, 2128-2130 (1995).
[CrossRef] [PubMed]

1994

1993

M. Schweiger, S. Arridge, and D. Delpy, "Application of the finite-element method for the forward and inverse models in optical tomography," J. Math. Imaging Vision 3, 263-283 (1993).
[CrossRef]

1941

L. G. Henvey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 90, 70-83 (1941).
[CrossRef]

Appl. Opt.

Astrophys. J.

L. G. Henvey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 90, 70-83 (1941).
[CrossRef]

Biomed. Tech.

V. Prapavat, W. Runge, J. Mans, A. Krause, J. Beuthan, and G. Müller, "Development of a finger joint phantom for the optical simulation of early stages of rheumatoid arthritis," Biomed. Tech. 42, 319-326 (1997).
[CrossRef]

Curr. Opi. Biotechnol.

A. H. Hielscher, "Optical tomographic imaging of small animals," Curr. Opi. Biotechnol. 16, 79-88 (2005).
[CrossRef]

Current Molecular Medicine

E. E. Graves, R. Weissleder, and V. Ntziachristos, "Fluorescence molecular imaging of small animal tumor models," Current Molecular Medicine 4, 419-430 (2004).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron.

W. Cai, M. Xu, and R. R. Alfano, "Three-dimensional radiative transfer tomography for turbid media," IEEE J. Sel. Top. Quantum Electron. 9, 189-198 (2003).
[CrossRef]

IEEE Signal Process Mag.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag. 18, 57-75 (2001).
[CrossRef]

Inverse Probl.

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

Inverse Probl. Eng.

O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. Eng. 14, 1107-1130 (1998).
[CrossRef]

J. Biomed. Opt.

A. Y. Bluestone, M. Stewart, B. Lei, I. S. Kass, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, part 1: Hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Y. Xu, N. Iftimia, H. Jiang, L. Key, and M. Bolster, "Three-dimensional diffuse optical tomography of bones and joints," J. Biomed. Opt. 7, 88-92 (2002).
[CrossRef] [PubMed]

P. Taroni, G. Danesini, A. Torricelli, A. Pifferi, L. Spinelli, and R. Cubeddu, "Clinical trial of time-resolved scanning optical mammography at 4 wavelengths between 683 and 975 nm," J. Biomed. Opt. 9, 464-473 (2004).
[CrossRef] [PubMed]

J. Cereb. Blood Flow Metab.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, "Noninvasive functional imaging of human brain using light," J. Cereb. Blood Flow Metab. 20, 469-477 (2000).
[CrossRef] [PubMed]

J. Comp. Physiol.

G. Bal, "Particle transport through scattering regions with clear layers and inclusions," J. Comp. Physiol. 180, 659-685 (2002).

J. Electron. Imaging

G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomography with the equation of radiative transfer," J. Electron. Imaging 12, 594-601 (2003).
[CrossRef]

J. Math. Imaging Vision

M. Schweiger, S. Arridge, and D. Delpy, "Application of the finite-element method for the forward and inverse models in optical tomography," J. Math. Imaging Vision 3, 263-283 (1993).
[CrossRef]

J. Opt. A

R. Elaloufi, R. Carminati, and J. Greffet, "Time-dependent transport through scattering media: from radiative transfer to diffusion," J. Opt. A , Pure Appl. Opt. 4, S103-S108 (2002).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

A. D. Klose and A. H. Hielscher, "Optical tomography using the time-independent equation of radiative transfer. 2. inverse model," J. Quant. Spectrosc. Radiat. Transf. 72, 715-202 (2002).
[CrossRef]

M. Francocur, R. Vaillon, and D. R. Rousse, "Theoretical analysis of frequency and time-domain methods for optical characterization of absorbing and scattering media," J. Quant. Spectrosc. Radiat. Transf. 93, 139-150 (2005).
[CrossRef]

Math. Modell. Numer. Anal.

G. Bal and Y. Maday, "Coupling of transport and diffusion models in linear transport theory," Math. Modell. Numer. Anal. 36, 69-86 (2002).
[CrossRef]

Med. Phys.

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
[CrossRef] [PubMed]

Medical Laser Application

U. Netz, J. Beuthan, H. J. Capius, H. C. Koch, A. D. Klose, and A. H. Hielscher, "Imaging of rheumatoid arthritis in finger joints by sagittal optical tomography," Medical Laser Application 16, 306-310 (2001).
[CrossRef]

Neoplasia

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, "Noninvasive in vivo characterization of breast tumors using photon migration spectroscopy," Neoplasia 2, 26-40 (2000).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissue," Phys. Med. Biol. 43, 1285-1302 (1998).
[CrossRef] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, "An investigation of light transport through scattering bodies with nonscattering regions," Phys. Med. Biol. 41, 767-783 (1996).
[CrossRef] [PubMed]

B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, "Initial assessment of a simple system for frequency domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995).
[CrossRef] [PubMed]

E. M. C. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmidt, D. T. Delpy, and S. R. Arridge, "Time resolved optical tomography of the human forearm," Phys. Med. Biol. 46, 1117-1130 (2001).
[CrossRef] [PubMed]

A. H. Hielscher, A. D. Klose, A. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, "Sagittal laser optical tomography for imaging of rheumatoid finger joints," Phys. Med. Biol. 49, 1147-1163 (2004).
[CrossRef] [PubMed]

Phys. Today

A. G. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

Proc. SPIE

A. D. Klose, A. H. Hielscher, K. M. Hanson, and J. Beuthan, "Three-dimensional optical tomography of a finger joint model for diagnostic of rheumatoid arthritis," Proc. SPIE 3566, 151-160 (1998).
[CrossRef]

Radiology

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, and K. D. Paulsen, "Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast," Radiology 218, 261-266 (2001).
[PubMed]

Rev. Sci. Instrum.

C. H. Schmitz, M. Löcker, J. M. Lasker, A. H. Hielscher, and R. L. Barbour, "Instrumentation for fast functional optical tomography," Rev. Sci. Instrum. 73, 429-439 (2002).
[CrossRef]

SIAM J. Appl. Math.

G. Bal, "Transport through diffusive and nondiffusive regions, embedded objects, and clear layers," SIAM J. Appl. Math. 62, 1677-1697 (2002).
[CrossRef]

SIAM J. Sci. Comput.

K. Ren, G. Bal, and A. H. Hielscher, "Frequency domain optical tomography based on the equation of radiative transfer," SIAM J. Sci. Comput. (USA) 28, 1463-1489 (2006).
[CrossRef]

Other

B. Chance, R. R. Alfano, B. J. Tromberg, and A. Katzir, eds., in Optical Tomography and Spectroscopy of Tissue (SPIE, 2003), Vol. V.

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R. Eymard, T. Gallouet, and R. Herbin, "Finite volume methods," in Handbook of Numerical Analysis VII, P. G. Ciarlet and J. L. Lions, ed. (North-Holland, 2000).
[CrossRef]

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, 1993).

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1998).

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology (Springer-Verlag, 1993), Vol. 6.
[CrossRef]

A. J. Welch and M. J. C. Van-Gemert, Optical-Thermal Response of Laser Irradiated Tissue (Plenum, 1995).

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Figures (10)

Fig. 1
Fig. 1

XZ ( a t y = 0 ) and XY ( a t z = 1 ) cross sections of the computational domain.

Fig. 2
Fig. 2

Cross sections of the reconstructed absorption coefficients in domain of small size. Top row: XZ cross section at y = 0 for transport reconstruction (left), diffusion reconstruction (middle), and their difference (right). Bottom row: corresponding XY cross sections at z = 1 . Reconstructions are done with noise-free data.

Fig. 3
Fig. 3

Relative errors in transport and diffusion reconstructions using data with different noise levels (in percentage). Left: reconstructions with scattering coefficient μ s = 100 cm 1 ; Right: reconstructions with scattering coefficient μ s = 150 cm 1 .

Fig. 4
Fig. 4

Cross sections of reconstructed absorption coefficients with source of high modulation frequency. Top row: XZ cross section at y = 0 for transport reconstruction (left), diffusion reconstruction (middle), and their difference (right). Bottom row: corresponding XY cross sections at z = 1 . The modulation frequency for the sources is ω = 0.8   GHz .

Fig. 5
Fig. 5

Relative errors in reconstructions as functions of modulation frequencies (in gigahertz). Left; reconstructions with noise-free data; right: reconstructions with 12% noise in the data.

Fig. 6
Fig. 6

Cross sections of reconstructed absorption coefficients with zero extrapolation length. Top row: XZ cross section at y = 0 for transport reconstruction (left), diffusion reconstruction (middle), and their difference (right). Bottom row: corresponding XY cross sections at z = 1 . Reconstructions are done with noise-free data.

Fig. 7
Fig. 7

Relative errors in reconstructions as functions of extrapolation length. Left: reconstructions with noise-free data; Right: reconstructions with 12% noise in the data. Transport reconstructions are shown here just as a reference.

Fig. 8
Fig. 8

X Z ( y = 0 ) and X Y ( z = 1 ) cross sections of the computational domain with a void inclusion.

Fig. 9
Fig. 9

Cross sections of reconstructed absorption coefficients in media with void regions. Top row: XZ cross section at y = 0 for transport reconstruction (left), diffusion reconstruction (middle), and their difference (right). Bottom row: corresponding XY cross sections at z = 1 . A void region is embedded in the domain.

Fig. 10
Fig. 10

Relative errors in transport and diffusion reconstructions using data with different noise levels in the presence of a void. Left: reconstructions with scattering coefficient μ s = 10 cm 1 ; Right: reconstructions with scattering coefficient μ s = 15 cm 1 . Anisotropy factor g = 0 in both cases.

Equations (17)

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( i ω ν + θ · + μ t ( x ) ) u ( x , θ ) μ s ( x ) ×   S 2 k ( θ · θ ) u ( x , θ ) d θ = 0 , in   X , u ( x , θ ) = f ( x , θ ) , on   Γ _ ,
Γ ± = { ( x , θ ) Ω × S 2   such   that ± θ · v ( x ) > 0 } ,
k ( θ · θ ) = C 1 g 2 ( 1 + g 2 2 g   cos   ϕ ) 3 / 2 k ( ϕ ) ,
i ω ν U ( x ) · D U + μ a ( x ) U ( x ) = 0 , i n Ω ,
U + 3 ϵ L 3 v ( x ) · D U = Λ ( f ) ( x ) , o n δ Ω .
J T ( x ) = S + 2 θ · v ( x ) u ( x , θ ) d θ ,
J D ( x ) = v ( x ) · D U .
min μ a ( x ) [ μ a min , μ a max ] β  : =  ( μ a ) + β 2 Ω μ a · μ a d x ,
( μ a ( x ) ) 1 2 q = 1 N q J q ( x ) z q ( x ) L 2 ( Ω ) 2 .
C n C · D U d γ ( x ) + ( μ a C i ω ν ) V C U C = 0 ,
C n C ( x ) · D U d γ ( x ) = i S C , i n C ( x ) · D U d γ ( x ) .
F C , i   : = S C , i n C ( x ) · D U d γ ( x ) = D n n + D i n n 2 | S C , i | ( U C U C i ) / ,
F C ,i   : = S C , i n C ( x ) · D U d γ ( x ) = | S C , i | n L n ( U C f ) .
i F C , i + ( μ a C i ω ν ) V C U C = 0.
A U = G ,
Ω c = { ( x , z ) :   | x ( 0.5 , 0 ) | < 0.2 , 0.2 < z < 1.8 } ,
l 2 = M r M e l 2 M e l 2 .

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