Abstract

Many cancer cells over-express folate receptors, and this provides an opportunity for both folate-targeted fluorescence imaging and the development of targeted anti-cancer drugs. We present an optical imaging modality that allows for the monitoring and evaluation of drug delivery and release through disulfide bond reduction inside a tumor in vivo for the first time. A near-infrared folate-targeting fluorophore pair was synthesized and used to image a xenograft tumor grown from KB cells in a live mouse. The in vivo results are shown to be in agreement with previous in vitro studies, confirming the validity and feasibility of our method as an effective tool for preclinical studies in drug development.

© 2014 Optical Society of America

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2014

C. Darne, Y. Lu, and E. M. Sevick-Muraca, “Small animal fluorescence and bioluminescence tomography: a review of approaches, algorithms and technology update,” Phys. Med. Biol.59, R1–R64 (2014).
[CrossRef]

2013

2012

H.-R. Tsai, F. Enderli, T. Feurer, and K. J. Webb, “Optimization-based terahertz imaging,” IEEE Trans. THz Sci. Technol.2, 493–503 (2012).
[CrossRef]

2011

G. M. van Dam, G. Themelis, L. M. A. Crane, N. J. Harlaar, R. G. Pleijhuis, W. Kelder, A. Sarantopoulos, J. S. de Jong, H. J. G. Arts, A. G. J. van der Zee, J. Bart, P. S. Low, and V. Ntziachristos, “Intraoperative tumor-specific fluorescent imaging in ovarian cancer by folate receptor-α targeting: first in-human results,” Nat. Med.17, 1315–1319 (2011).
[CrossRef] [PubMed]

J. McGinty, D. W. Stuckey, V. Y. Soloviev, R. Laine, M. Wylezinska-Arridge, D. J. Wells, S. R. Arridge, P. M. W. French, J. V. Hajnal, and A. Sardini, “In vivo fluorescence lifetime tomography of a FRET probe expressed in mouse,” Biomed. Opt. Express2, 1907–1917 (2011).
[CrossRef] [PubMed]

2010

Q. T. Nguyen, E. S. Olson, T. A. Aguilera, T. Jiang, M. Scadeng, L. G. Ellies, and R. Y. Tsien, “Surgery with molecular fluorescence imaging using activatable cell-penetrating peptides decreases residual cancer and improves survival,” Proc. Natl. Acad. Sci. U.S.A.107, 4317–4322 (2010).
[CrossRef] [PubMed]

W. Xia and P. S. Low, “Folate-targeted therapies for cancer,” J. Med. Chem.53, 6811–6824 (2010).
[CrossRef] [PubMed]

S. A. Hilderbrand and R. Weissleder, “Near-infrared fluorescence: application to in vivo molecular imaging,” Curr. Opin. Chem. Biol.14, 71–79 (2010).
[CrossRef]

V. Gaind, S. Kularatne, P. S. Low, and K. J. Webb, “Deep tissue imaging of intramolecular fluorescence resonance energy transfer parameters,” Opt. Lett.35, 1314–1316 (2010).
[CrossRef] [PubMed]

2009

2008

P. S. Low, W. A. Henne, and D. D. Doorneweerd, “Discovery and development of folic-acid-based receptor targeting for imaging and therapy of cancer and inflammatory diseases,” Acc. Chem. Res.41, 120–129 (2008).
[CrossRef]

B. Alacam, B. Yazici, X. Intes, S. Nioka, and B. Chance, “Pharmacokinetic-rate images of indocyanine green for breast tumors using near-infrared optical methods,” Phys. Med. Biol.53, 837–859 (2008).
[CrossRef] [PubMed]

2006

J. Yang, H. Chen, I. R. Vlahov, J.-X. Cheng, and P. S. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. USA103, 13872–13877 (2006).
[CrossRef] [PubMed]

2005

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2, 932–940 (2005).
[CrossRef] [PubMed]

A. B. Milstein, K. J. Webb, and C. A. Bouman, “Estimation of kinetic model parameters in fluorescence optical diffusion tomography,” J. Opt. Soc. Am. A22, 1357–1368 (2005).
[CrossRef]

N. Parker, M. J. Turk, E. Westrick, J. D. Lewis, P. S. Low, and C. P. Leamon, “Folate receptor expression in carcinomas and normal tissues determined by a quantitative radioligand binding assay,” Anal. Biochem.338, 284–293 (2005).
[CrossRef] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

S. Oh, A. B. Milstein, C. A. Bouman, and K. J. Webb, “A general framework for nonlinear multigrid inversion,” IEEE Trans. Image Process.14, 125–140 (2005).
[CrossRef] [PubMed]

2004

R. M. Sandoval, M. D. Kennedy, P. S. Low, and B. A. Molitoris, “Uptake and trafficking of fluorescent conjugates of folic acid in intact kidney determined using intravital two-photon microscopy,” Am. J. Physiol.-Cell Ph.287, C517–C526 (2004).
[CrossRef]

C. M. Paulos, J. A. Reddy, C. P. Leamon, M. J. Turk, and P. S. Low, “Ligand binding and kinetics of folate receptor recycling in vivo: impact on receptor-mediated drug delivery,” Mol. Pharmacol.66, 1406–1414 (2004).
[CrossRef] [PubMed]

C. M. Paulos, M. J. Turk, G. J. Breur, and P. S. Low, “Folate receptor-mediated targeting of therapeutic and imaging agents to activated macrophages in rheumatoid arthritis,” Adv. Drug Deliv. Rev.56, 1205–1217 (2004).
[CrossRef] [PubMed]

2003

2002

V. Ntziachristos, C.-H. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med.8, 757–761 (2002).
[CrossRef] [PubMed]

A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, and R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett.27, 95–97 (2002).
[CrossRef]

C. H. Tung, Y. Lin, W. K. Moon, and R. Weissleder, “A receptor-targeted near-infrared fluorescence probe for in vivo tumor imaging,” ChemBioChem8, 784–786 (2002).
[CrossRef]

2001

J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process.10, 909–922 (2001).
[CrossRef]

2000

M. Gurfinkel, A. B. Thompson, W. B. Ralston, T. L. Troy, A. L. Moore, T. A. Moore, J. D. Gust, D. Tatman, J. S. Reynolds, B. Muggenburg, K. Nikula, R. Pandey, R. H. Mayer, D. J. Hawrysz, and E. M. Sevick-Muraca, “Pharmacokinetics of ICG and HPPH-car for the detection of normal and tumor tissue using fluorescence, near-infrared reflectance imaging: a case study,” Photochem. Photobiol.72, 94–102 (2000).
[CrossRef] [PubMed]

J. Sudimack and R. J. Lee, “Targeted drug delivery via the folate receptor,” Adv. Drug Deliv. Rev.41, 147–162 (2000).
[CrossRef] [PubMed]

1999

Y. Shin, K. A. Winans, B. J. Backes, S. B. H. Kent, J. A. Ellman, and C. R. Bertozzi, “Fmoc-based synthesis of peptide-αthioesters: application to the total chemical synthesis of a glycoprotein by native chemical ligation,” J. Am. Chem. Soc.121, 11684–11689 (1999).
[CrossRef]

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

J. C. Ye, K. J. Webb, C. A. Bouman, and R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A16, 2400–2412 (1999).
[CrossRef]

1997

1996

1994

M.-Y. Su, J.-C. Jao, and O. Nalcioglu, “Measurement of vascular volume fraction and blood-tissue permeability constants with a pharmacokinetic model: Studies in rat muscle tumors with dynamic gd-DTPA enhanced MRI,” Magn. Reson. Med.32, 714–724 (1994).
[CrossRef] [PubMed]

1993

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20, 299–309 (1993).
[CrossRef] [PubMed]

1989

1973

A. C. Riches, J. G. Sharp, D. B. Thomas, and S. V. Smith, “Blood volume determination in the mouse,” J. Physiol.228, 279–284 (1973).
[PubMed]

1948

T. Förster, “Zwischenmolekulare energiewanderung und fluoreszenze,” Ann. Physik2, 55 (1948).
[CrossRef]

Aguilera, T. A.

Q. T. Nguyen, E. S. Olson, T. A. Aguilera, T. Jiang, M. Scadeng, L. G. Ellies, and R. Y. Tsien, “Surgery with molecular fluorescence imaging using activatable cell-penetrating peptides decreases residual cancer and improves survival,” Proc. Natl. Acad. Sci. U.S.A.107, 4317–4322 (2010).
[CrossRef] [PubMed]

Alacam, B.

B. Alacam, B. Yazici, X. Intes, S. Nioka, and B. Chance, “Pharmacokinetic-rate images of indocyanine green for breast tumors using near-infrared optical methods,” Phys. Med. Biol.53, 837–859 (2008).
[CrossRef] [PubMed]

Alexandrakis, G.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

Arridge, S. R.

J. McGinty, D. W. Stuckey, V. Y. Soloviev, R. Laine, M. Wylezinska-Arridge, D. J. Wells, S. R. Arridge, P. M. W. French, J. V. Hajnal, and A. Sardini, “In vivo fluorescence lifetime tomography of a FRET probe expressed in mouse,” Biomed. Opt. Express2, 1907–1917 (2011).
[CrossRef] [PubMed]

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

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20, 299–309 (1993).
[CrossRef] [PubMed]

Arts, H. J. G.

G. M. van Dam, G. Themelis, L. M. A. Crane, N. J. Harlaar, R. G. Pleijhuis, W. Kelder, A. Sarantopoulos, J. S. de Jong, H. J. G. Arts, A. G. J. van der Zee, J. Bart, P. S. Low, and V. Ntziachristos, “Intraoperative tumor-specific fluorescent imaging in ovarian cancer by folate receptor-α targeting: first in-human results,” Nat. Med.17, 1315–1319 (2011).
[CrossRef] [PubMed]

Backes, B. J.

Y. Shin, K. A. Winans, B. J. Backes, S. B. H. Kent, J. A. Ellman, and C. R. Bertozzi, “Fmoc-based synthesis of peptide-αthioesters: application to the total chemical synthesis of a glycoprotein by native chemical ligation,” J. Am. Chem. Soc.121, 11684–11689 (1999).
[CrossRef]

Bart, J.

G. M. van Dam, G. Themelis, L. M. A. Crane, N. J. Harlaar, R. G. Pleijhuis, W. Kelder, A. Sarantopoulos, J. S. de Jong, H. J. G. Arts, A. G. J. van der Zee, J. Bart, P. S. Low, and V. Ntziachristos, “Intraoperative tumor-specific fluorescent imaging in ovarian cancer by folate receptor-α targeting: first in-human results,” Nat. Med.17, 1315–1319 (2011).
[CrossRef] [PubMed]

Bertozzi, C. R.

Y. Shin, K. A. Winans, B. J. Backes, S. B. H. Kent, J. A. Ellman, and C. R. Bertozzi, “Fmoc-based synthesis of peptide-αthioesters: application to the total chemical synthesis of a glycoprotein by native chemical ligation,” J. Am. Chem. Soc.121, 11684–11689 (1999).
[CrossRef]

Bevilacqua, F.

Boas, D. A.

Bouman, C. A.

Bremer, C.

V. Ntziachristos, C.-H. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med.8, 757–761 (2002).
[CrossRef] [PubMed]

Breur, G. J.

C. M. Paulos, M. J. Turk, G. J. Breur, and P. S. Low, “Folate receptor-mediated targeting of therapeutic and imaging agents to activated macrophages in rheumatoid arthritis,” Adv. Drug Deliv. Rev.56, 1205–1217 (2004).
[CrossRef] [PubMed]

Chance, B.

Chatziioannou, A. F.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

Chelvam, V.

Chen, H.

J. Yang, H. Chen, I. R. Vlahov, J.-X. Cheng, and P. S. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. USA103, 13872–13877 (2006).
[CrossRef] [PubMed]

Cheng, J.-X.

J. Yang, H. Chen, I. R. Vlahov, J.-X. Cheng, and P. S. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. USA103, 13872–13877 (2006).
[CrossRef] [PubMed]

Crane, L. M. A.

G. M. van Dam, G. Themelis, L. M. A. Crane, N. J. Harlaar, R. G. Pleijhuis, W. Kelder, A. Sarantopoulos, J. S. de Jong, H. J. G. Arts, A. G. J. van der Zee, J. Bart, P. S. Low, and V. Ntziachristos, “Intraoperative tumor-specific fluorescent imaging in ovarian cancer by folate receptor-α targeting: first in-human results,” Nat. Med.17, 1315–1319 (2011).
[CrossRef] [PubMed]

Cuccia, D. J.

Darne, C.

C. Darne, Y. Lu, and E. M. Sevick-Muraca, “Small animal fluorescence and bioluminescence tomography: a review of approaches, algorithms and technology update,” Phys. Med. Biol.59, R1–R64 (2014).
[CrossRef]

de Jong, J. S.

G. M. van Dam, G. Themelis, L. M. A. Crane, N. J. Harlaar, R. G. Pleijhuis, W. Kelder, A. Sarantopoulos, J. S. de Jong, H. J. G. Arts, A. G. J. van der Zee, J. Bart, P. S. Low, and V. Ntziachristos, “Intraoperative tumor-specific fluorescent imaging in ovarian cancer by folate receptor-α targeting: first in-human results,” Nat. Med.17, 1315–1319 (2011).
[CrossRef] [PubMed]

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20, 299–309 (1993).
[CrossRef] [PubMed]

Denk, W.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2, 932–940 (2005).
[CrossRef] [PubMed]

Doorneweerd, D. D.

P. S. Low, W. A. Henne, and D. D. Doorneweerd, “Discovery and development of folic-acid-based receptor targeting for imaging and therapy of cancer and inflammatory diseases,” Acc. Chem. Res.41, 120–129 (2008).
[CrossRef]

Durduran, T.

Durkin, A. J.

Ellies, L. G.

Q. T. Nguyen, E. S. Olson, T. A. Aguilera, T. Jiang, M. Scadeng, L. G. Ellies, and R. Y. Tsien, “Surgery with molecular fluorescence imaging using activatable cell-penetrating peptides decreases residual cancer and improves survival,” Proc. Natl. Acad. Sci. U.S.A.107, 4317–4322 (2010).
[CrossRef] [PubMed]

Ellman, J. A.

Y. Shin, K. A. Winans, B. J. Backes, S. B. H. Kent, J. A. Ellman, and C. R. Bertozzi, “Fmoc-based synthesis of peptide-αthioesters: application to the total chemical synthesis of a glycoprotein by native chemical ligation,” J. Am. Chem. Soc.121, 11684–11689 (1999).
[CrossRef]

Enderli, F.

H.-R. Tsai, F. Enderli, T. Feurer, and K. J. Webb, “Optimization-based terahertz imaging,” IEEE Trans. THz Sci. Technol.2, 493–503 (2012).
[CrossRef]

Feurer, T.

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Figures (9)

Fig. 1
Fig. 1

The folate-DA conjugate (folic acid + linker + DA), bound to the folate receptor, is internalized by the cell. An enzyme cleaves the disulfide (S-S) bond, releasing the acceptor [17].

Fig. 2
Fig. 2

Mouse compartment model for drug delivery kinetics. The folate conjugate is injected into the plasma and is internalized and unloaded by the tumor at rates k1 and k2, respectively. The folate conjugate is eliminated (cleared) from the plasma (predominantly by the kidneys) at rate k3. Finally, the acceptor fluorophore (in lieu of a drug) is released from the folate conjugate due to disulfide bond reduction inside the tumor at rate k4, which is of particular interest for targeted drug development and is determined in vivo in this work.

Fig. 3
Fig. 3

The chemical structure of our near-infrared (NIR) folate-DA conjugate with Dy-light680B as the donor and Promofluor750 as the acceptor. The green region shows the disulfide (S-S) bond.

Fig. 4
Fig. 4

Uptake of 100 nM Folate-Dylight680B-S,S-Promofluor750 in KB cells at different points in time using confocal microscopy. (a) Donor fluorescence after 1 h. (b) The cell culture at 1 h. (c) Donor fluorescence after 8 h. (d) The cell culture after 8 h. Notice the strong increase in donor fluorescence after 8 h, indicative of efficient donor-acceptor coupling before acceptor release.

Fig. 5
Fig. 5

(a) Picture of experimental setup for in vivo mouse imaging. (b) Schematic of the imaging experiment setup, including a 633 nm pulsed laser, a cooled time-gated image-intensified CCD camera, and a bandpass filter for fluorescence imaging. This setup allows for time-domain pulse measurements.

Fig. 6
Fig. 6

(a) A photo of the live mouse during the experiment and its corresponding 3-D surface profile. The black dotted circle shows the general location of the sources and detectors. (b) The dissected mouse, showing the tumor size and location.

Fig. 7
Fig. 7

(a)–(b) Isosurfaces of the reconstructed absorption (μa) at 0.05 mm−1 (showing the semi-transparent contour of the mouse) and 0.07 mm−1 (showing the reconstructed tumor), with different viewing angles in (a) and (b). (c) Cross-section of the reconstructed absorption.

Fig. 8
Fig. 8

(a)–(b) Isosurfaces of the reconstructed fluorescence magnitude (γ3) at 0.005 mm−1. (c) Cross-section of the reconstructed γ3. (d)–(e) Isosurfaces of the reconstructed fluorescence rate (γ4) at 0.1 h−1. (f) Cross-section of the reconstructed γ4.

Fig. 9
Fig. 9

(a) Fluorescence (η̃), defined in (30), with reconstructed k4 = 0.168 h−1 and k2 = 0.010 h−1. (b) Comparison of fluorescence in (a) with 5 hourly intensity measurement (based on 2-D CCD image intensity, not 3-D reconstruction). The advantage of 3-D reconstruction is that it offers more information, including the depth and size of the tumor, and thus the possibility for quantitative molecular imaging.

Tables (1)

Tables Icon

Table 1 Parameter definitions used in the kinetics model.

Equations (33)

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[ D x ( r ) ϕ x ( r , ω ) ] [ μ a x ( r ) + j ω / c ] ϕ x ( r , ω ) = S x ( r ; ω )
[ D m ( r ) ϕ m ( r , ω ) ] [ μ a m ( r ) + j ω / c ] ϕ m ( r , ω ) = ϕ x ( r , ω ) S f ( r ; ω ) ,
d X p DA ( t ) / d t = ( k 1 + k 3 ) X p DA ( t ) .
X p DA ( t ) = X 0 e ( k 1 + k 3 ) t ,
d X t DA ( t ) / d t = k 1 X p DA ( t ) ( k 2 + k 4 ) X t DA ( t ) ,
d X t D ( t ) / d t = k 4 X t DA ( t ) k 2 X t D ( t ) .
X t DA ( t ) = [ X 0 k 1 k 2 + k 4 ( k 1 + k 3 ) ] [ e ( k 1 + k 3 ) t e ( k 2 + k 4 ) t ] ,
X t D ( t ) = [ X 0 k 1 k 2 + k 4 ( k 1 + k 3 ) ] [ k 4 k 2 ( k 1 + k 3 ) e ( k 1 + k 3 ) t + e ( k 2 + k 4 ) t ] [ X 0 k 1 k 2 ( k 1 + k 3 ) ] e k 2 t .
X DA ( s , t ) = v p ( s ) X p DA ( t ) + v t ( s ) X t DA ( t ) ,
X D ( s , t ) = v t ( s ) X t D ( t ) .
η ˜ DA ( s , t ) = α DA X DA ( s , t )
= α DA v p ( s ) X p DA ( t ) + α DA v t ( s ) X t DA ( t )
= w p DA ( s ) X p DA ( t ) + w t DA ( s ) X t DA ( t )
η ˜ D ( s , t ) = α D X D ( s , t )
= α D v t ( s ) X t D ( t )
= w t D ( s ) X t D ( t ) ,
S f = ( w p DA X p DA + w t DA X t DA ) 1 1 + j ω τ DA + w t D X t D 1 1 + j ω τ D .
η ˜ ( s , t ) = w p DA ( s ) X p DA ( t ) + w t DA ( s ) X t DA ( t ) + w t D ( s ) X t D ( t ) ,
η ˜ ( s , t ) = γ 1 ( s ) e γ 2 t γ 3 ( s ) e γ 4 t + γ 5 ( s ) e γ 6 t ,
γ 1 ( s ) = X 0 [ w p DA ( s ) w t DA ( s ) k 1 ( k 1 + k 3 ) ( k 2 + k 4 ) + w t D ( s ) k 1 k 4 [ ( k 1 + k 3 ) ( k 2 + k 4 ) ] ( k 1 + k 3 k 2 ) ]
γ 2 = k 1 + k 3
γ 3 ( s ) = X 0 [ k 1 w t D ( s ) k 1 w t DA ( s ) ( k 1 + k 3 ) ( k 2 + k 4 ) ]
γ 4 = k 2 + k 4
γ 5 ( s ) = X 0 [ w t D ( s ) k 1 ( k 1 + k 3 ) k 2 ]
γ 6 = k 2 .
s = 1 N γ ^ 5 ( s ) ( γ ^ 2 γ ^ 6 ) = X 0 k 1 α D [ s = 1 N v t ( s ) ] = X 0 k 1 α D ,
s = 1 N γ ^ 3 ( s ) ( γ ^ 2 γ ^ 4 ) = X 0 k 1 ( α D α DA ) [ s = 1 N v t ( s ) ] = X 0 k 1 ( α D α DA ) .
R γ 3 γ 5 = γ 5 ( s ) γ 3 ( s ) = [ ( k 1 + k 3 ) ( k 2 + k 4 ) ( k 1 + k 3 ) k 2 ] [ α D α D α DA ] .
η ˜ ( s , t ) = γ 1 ( s ) e γ 2 t + γ 3 ( s ) ( R γ 3 γ 5 e γ 6 t e γ 4 t ) .
η ˜ ( s , t ) γ 3 ( s ) ( R γ 3 γ 5 e γ 6 t e γ 4 t ) .
x ^ i = arg min x ˜ i [ y f ( x ˜ i ) Λ 2 + 1 ρ σ ρ j 𝒩 i b i j | x ˜ i x j | ρ ] ,
Γ ^ = arg min Γ [ y raw Γ f ( x ) Λ 2 ] ,
X t folate ( t ) = X 0 k 1 ( k 1 + k 3 ) k 2 [ e k 2 t e ( k 1 + k 3 ) t ] ,

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