Abstract

Perturbation of normal corneal water content is a common manifestation of many eye diseases. Terahertz (THz) imaging has the potential to serve as a clinical tool for screening and diagnosing such corneal diseases. In this study, we first investigate the diffusive properties of a corneal phantom using simultaneous THz time-domain spectroscopy (THz-TDS) and gravimetric measurements. We will then utilize a variable-thickness diffusion model combined with a stratified composite-media model to simulate changes in thickness, hydration profile, and the THz-TDS signal as a function of time. The simulated THz-TDS signals show very good agreement with the reflection measurements. Results show that the THz-TDS technique can be used to understand water diffusion dynamics in corneal phantoms as a step towards future in vivo quantitative hydration sensing.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

Q. Sun, R. I. Stantchev, J. Wang, E. P. Parrott, A. Cottenden, T. W. Chiu, A. T. Ahuja, and E. Pickwell-MacPherson, “In vivo estimation of water diffusivity in occluded human skin using terahertz reflection spectroscopy,” J. Biophotonics 12(2), e201800145 (2019).
[Crossref]

S. J. Vincent, D. Alonso-Caneiro, H. Kricancic, and M. J. Collins, “Scleral contact lens thickness profiles: The relationship between average and centre lens thickness,” Cont. Lens Anterior Eye 42(1), 55–62 (2019).
[Crossref]

2018 (3)

Q. Sun, E. P. Parrott, Y. He, and E. Pickwell-MacPherson, “In vivo THz imaging of human skin: Accounting for occlusion effects,” J. Biophotonics 11(2), e201700111 (2018).
[Crossref]

S. Sung, S. Selvin, N. Bajwa, S. Chantra, B. Nowroozi, J. Garritano, J. Goell, A. Li, S. X. Deng, E. Brown, W. S. Grundfest, and Z. D. Taylor, “THz imaging system for in vivo human cornea,” IEEE Trans. Terahertz Sci. Technol. 8(1), 27–37 (2018).
[Crossref]

S. Sung, S. Dabironezare, N. Llombart, S. Selvin, N. Bajwa, S. Chantra, B. Nowroozi, J. Garritano, J. Goell, A. Li, S. X. Deng, E. Brown, W. S. Grundfest, and Z. D. Taylor, “Optical system design for noncontact, normal incidence, THz imaging of in vivo human cornea,” IEEE Trans. Terahertz Sci. Technol. 8(1), 1–12 (2018).
[Crossref]

2017 (1)

2015 (5)

Z. D. Taylor, J. Garritano, S. Sung, N. Bajwa, D. B. Bennett, B. Nowroozi, P. Tewari, J. W. Sayre, J. Hubschman, S. X. Deng, E. R. Brown, and W. S. Grundfest, “THz and mm-wave sensing of corneal tissue water content: in vivo sensing and imaging results,” IEEE Trans. Terahertz Sci. Technol. 5(2), 184–196 (2015).
[Crossref]

Z. D. Taylor, J. Garritano, S. Sung, N. Bajwa, D. B. Bennett, B. Nowroozi, P. Tewari, J. Sayre, J.-P. Hubschman, S. Deng, E. R. Brown, and W. S. Grundfest, “THz and mm-wave sensing of corneal tissue water content: electromagnetic modeling and analysis,” IEEE Trans. Terahertz Sci. Technol. 5(2), 170–183 (2015).
[Crossref]

S. Sung, S. Chantra, N. Bajwa, R. McCurdy, G. Kerezyte, J. Garritano, J.-P. Hubschman, W. Grundfest, S. X. Deng, and Z. Taylor, “Direct measurement of corneal tissue water content by reflection imaging at Terahertz Frequencies,” Invest. Ophthalmol. Visual Sci. 56, 1644 (2015).

T. Ahmed, I. V. Belova, and G. E. Murch, “Finite difference solution of the diffusion equation and calculation of the interdiffusion coefficient using the Sauer-Freise and Hall Methods in binary systems,” Procedia Eng. 105, 570–575 (2015).
[Crossref]

X. Cheng, S. J. Petsche, and P. M. Pinsky, “A structural model for the in vivo human cornea including collagen-swelling interaction,” J. R. Soc. Interface 12(109), 20150241 (2015).
[Crossref]

2012 (2)

G. P. Kniffin and L. M. Zurk, “Model-based material parameter estimation for terahertz reflection spectroscopy,” IEEE Trans. Terahertz Sci. Technol. 2(2), 231–241 (2012).
[Crossref]

D. Bennett, Z. Taylor, P. Tewari, S. Sung, A. Maccabi, R. Singh, M. Culjat, W. Grundfest, J. P. Hubschman, and E. Brown, “Assessment of corneal hydration sensing in the terahertz band: in vivo results at 100 GHz,” J. Biomed. Opt. 17(9), 0970081 (2012).
[Crossref]

2011 (4)

D. B. Bennett, Z. D. Taylor, P. Tewari, R. S. Singh, M. O. Culjat, W. S. Grundfest, D. J. Sassoon, R. D. Johnson, J. P. Hubschman, and E. R. Brown, “Terahertz sensing in corneal tissues,” J. Biomed. Opt. 16(5), 057003 (2011).
[Crossref]

D. B. Bennett, W. Li, Z. D. Taylor, W. S. Grundfest, and E. R. Brown, “Stratified media model for terahertz reflectometry of the skin,” IEEE Sens. J. 11(5), 1253–1262 (2011).
[Crossref]

Z. D. Taylor, R. S. Singh, D. B. Bennett, P. Tewari, C. P. Kealey, N. Bajwa, M. O. Culjat, A. Stojadinovic, H. Lee, J.-P. Hubschman, E. R. Brown, and W. S. Grundfest, “THz medical imaging: in vivo hydration sensing,” IEEE Trans. Terahertz Sci. Technol. 1(1), 201–219 (2011).
[Crossref]

M. H. Arbab, T. C. Dickey, D. P. Winebrenner, A. Chen, M. B. Klein, and P. D. Mourad, “Terahertz reflectometry of burn wounds in a rat model,” Biomed. Opt. Express 2(8), 2339–2347 (2011).
[Crossref]

2008 (1)

Z. Taylor, R. Singh, M. Culjat, J. Suen, W. Grundfest, and E. Brown, “THz imaging based on water-concentration contrast - art. no. 69490D,” Proc. SPIE 6949, 69490D (2008).
[Crossref]

2007 (1)

N. Efron, P. B. Morgan, I. D. Cameron, N. A. Brennan, and M. Goodwin, “Oxygen permeability and water content of silicone hydrogel contact lens materials,” Optom. Vis. Sci. 84(4), E328–E337 (2007).
[Crossref]

2006 (2)

2004 (3)

E. Pickwell-MacPherson, B. Cole, A. Fitzgerald, V. Wallace, and M. Pepper, “Simulation of terahertz pulse propagation in biological systems,” Appl. Phys. Lett. 84(12), 2190–2192 (2004).
[Crossref]

J. Fischbarg and D. M. Maurice, “An update on corneal hydration control,” Exp. Eye Res. 78(3), 537–541 (2004).
[Crossref]

Z. Gu and P. Alexandridis, “Drying of poloxamer hydrogel films,” J. Pharm. Sci. 93(6), 1454–1470 (2004).
[Crossref]

2000 (1)

P. McConville and J. M. Pope, “A comparison of water binding and mobility in contact lens hydrogels from NMR measurements of the water self-diffusion coefficient,” Polymer 41(26), 9081–9088 (2000).
[Crossref]

1998 (3)

C. J. Budd and G. J. Collins, “An invariant moving mesh scheme for the nonlinear diffusion equation,” Appl. Numer. Math. 26(1-2), 23–39 (1998).
[Crossref]

N. J. Bauer, J. P. Wicksted, F. H. Jongsma, W. F. March, F. Hendrikse, and M. Motamedi, “Noninvasive assessment of the hydration gradient across the cornea using confocal Raman spectroscopy,” Invest. Ophthalmol. Vis. Sci. 39, 831–835 (1998).

Y. S. Rabinowitz, “Keratoconus,” Surv. Ophthalmol. 42(4), 297–319 (1998).
[Crossref]

1996 (1)

J. T. Kindt and C. A. Schmuttenmaer, “Far-Infrared dielectric properties of polar liquids probed by femtosecond terahertz pulse spectroscopy,” J. Phys. Chem. 100(24), 10373–10379 (1996).
[Crossref]

1995 (1)

L. Ion and J. M. Vergnaud, “Process of drying a polymeric paint by diffusion-evaporation and shrinkage. Determination of the concentration-dependent diffusivity,” Polym. Test. 14(5), 479–487 (1995).
[Crossref]

1993 (1)

A. P. Adamis, V. Filatov, B. J. Tripathi, and R. C. Tripathi, “Fuchs’ endothelial dystrophy of the cornea,” Surv. Ophthalmol. 38(2), 149–168 (1993).
[Crossref]

1987 (1)

H. P. Blandin, J. C. David, J. M. Vergnaud, J. P. Illien, and M. Malizewicz, “Modelling of drying of coatings: Effect of the thickness, temperature and concentration of solvent,” Prof. Org. Coat. 15(2), 163–172 (1987).
[Crossref]

1965 (1)

J. Ytteborg and C. H. Dohlman, “Corneal edema and intraocular pressure. II. Clinical results,” Arch Ophthalmol. 74(4), 477–484 (1965).
[Crossref]

Adamis, A. P.

A. P. Adamis, V. Filatov, B. J. Tripathi, and R. C. Tripathi, “Fuchs’ endothelial dystrophy of the cornea,” Surv. Ophthalmol. 38(2), 149–168 (1993).
[Crossref]

Ahmed, T.

T. Ahmed, I. V. Belova, and G. E. Murch, “Finite difference solution of the diffusion equation and calculation of the interdiffusion coefficient using the Sauer-Freise and Hall Methods in binary systems,” Procedia Eng. 105, 570–575 (2015).
[Crossref]

Ahuja, A. T.

Q. Sun, R. I. Stantchev, J. Wang, E. P. Parrott, A. Cottenden, T. W. Chiu, A. T. Ahuja, and E. Pickwell-MacPherson, “In vivo estimation of water diffusivity in occluded human skin using terahertz reflection spectroscopy,” J. Biophotonics 12(2), e201800145 (2019).
[Crossref]

Alexandridis, P.

Z. Gu and P. Alexandridis, “Drying of poloxamer hydrogel films,” J. Pharm. Sci. 93(6), 1454–1470 (2004).
[Crossref]

Alonso-Caneiro, D.

S. J. Vincent, D. Alonso-Caneiro, H. Kricancic, and M. J. Collins, “Scleral contact lens thickness profiles: The relationship between average and centre lens thickness,” Cont. Lens Anterior Eye 42(1), 55–62 (2019).
[Crossref]

Arbab, M. H.

Bajwa, N.

S. Sung, S. Selvin, N. Bajwa, S. Chantra, B. Nowroozi, J. Garritano, J. Goell, A. Li, S. X. Deng, E. Brown, W. S. Grundfest, and Z. D. Taylor, “THz imaging system for in vivo human cornea,” IEEE Trans. Terahertz Sci. Technol. 8(1), 27–37 (2018).
[Crossref]

S. Sung, S. Dabironezare, N. Llombart, S. Selvin, N. Bajwa, S. Chantra, B. Nowroozi, J. Garritano, J. Goell, A. Li, S. X. Deng, E. Brown, W. S. Grundfest, and Z. D. Taylor, “Optical system design for noncontact, normal incidence, THz imaging of in vivo human cornea,” IEEE Trans. Terahertz Sci. Technol. 8(1), 1–12 (2018).
[Crossref]

Z. D. Taylor, J. Garritano, S. Sung, N. Bajwa, D. B. Bennett, B. Nowroozi, P. Tewari, J. W. Sayre, J. Hubschman, S. X. Deng, E. R. Brown, and W. S. Grundfest, “THz and mm-wave sensing of corneal tissue water content: in vivo sensing and imaging results,” IEEE Trans. Terahertz Sci. Technol. 5(2), 184–196 (2015).
[Crossref]

Z. D. Taylor, J. Garritano, S. Sung, N. Bajwa, D. B. Bennett, B. Nowroozi, P. Tewari, J. Sayre, J.-P. Hubschman, S. Deng, E. R. Brown, and W. S. Grundfest, “THz and mm-wave sensing of corneal tissue water content: electromagnetic modeling and analysis,” IEEE Trans. Terahertz Sci. Technol. 5(2), 170–183 (2015).
[Crossref]

S. Sung, S. Chantra, N. Bajwa, R. McCurdy, G. Kerezyte, J. Garritano, J.-P. Hubschman, W. Grundfest, S. X. Deng, and Z. Taylor, “Direct measurement of corneal tissue water content by reflection imaging at Terahertz Frequencies,” Invest. Ophthalmol. Visual Sci. 56, 1644 (2015).

Z. D. Taylor, R. S. Singh, D. B. Bennett, P. Tewari, C. P. Kealey, N. Bajwa, M. O. Culjat, A. Stojadinovic, H. Lee, J.-P. Hubschman, E. R. Brown, and W. S. Grundfest, “THz medical imaging: in vivo hydration sensing,” IEEE Trans. Terahertz Sci. Technol. 1(1), 201–219 (2011).
[Crossref]

Bauer, N. J.

N. J. Bauer, J. P. Wicksted, F. H. Jongsma, W. F. March, F. Hendrikse, and M. Motamedi, “Noninvasive assessment of the hydration gradient across the cornea using confocal Raman spectroscopy,” Invest. Ophthalmol. Vis. Sci. 39, 831–835 (1998).

Belova, I. V.

T. Ahmed, I. V. Belova, and G. E. Murch, “Finite difference solution of the diffusion equation and calculation of the interdiffusion coefficient using the Sauer-Freise and Hall Methods in binary systems,” Procedia Eng. 105, 570–575 (2015).
[Crossref]

Bennett, D.

D. Bennett, Z. Taylor, P. Tewari, S. Sung, A. Maccabi, R. Singh, M. Culjat, W. Grundfest, J. P. Hubschman, and E. Brown, “Assessment of corneal hydration sensing in the terahertz band: in vivo results at 100 GHz,” J. Biomed. Opt. 17(9), 0970081 (2012).
[Crossref]

Bennett, D. B.

Z. D. Taylor, J. Garritano, S. Sung, N. Bajwa, D. B. Bennett, B. Nowroozi, P. Tewari, J. Sayre, J.-P. Hubschman, S. Deng, E. R. Brown, and W. S. Grundfest, “THz and mm-wave sensing of corneal tissue water content: electromagnetic modeling and analysis,” IEEE Trans. Terahertz Sci. Technol. 5(2), 170–183 (2015).
[Crossref]

Z. D. Taylor, J. Garritano, S. Sung, N. Bajwa, D. B. Bennett, B. Nowroozi, P. Tewari, J. W. Sayre, J. Hubschman, S. X. Deng, E. R. Brown, and W. S. Grundfest, “THz and mm-wave sensing of corneal tissue water content: in vivo sensing and imaging results,” IEEE Trans. Terahertz Sci. Technol. 5(2), 184–196 (2015).
[Crossref]

Z. D. Taylor, R. S. Singh, D. B. Bennett, P. Tewari, C. P. Kealey, N. Bajwa, M. O. Culjat, A. Stojadinovic, H. Lee, J.-P. Hubschman, E. R. Brown, and W. S. Grundfest, “THz medical imaging: in vivo hydration sensing,” IEEE Trans. Terahertz Sci. Technol. 1(1), 201–219 (2011).
[Crossref]

D. B. Bennett, Z. D. Taylor, P. Tewari, R. S. Singh, M. O. Culjat, W. S. Grundfest, D. J. Sassoon, R. D. Johnson, J. P. Hubschman, and E. R. Brown, “Terahertz sensing in corneal tissues,” J. Biomed. Opt. 16(5), 057003 (2011).
[Crossref]

D. B. Bennett, W. Li, Z. D. Taylor, W. S. Grundfest, and E. R. Brown, “Stratified media model for terahertz reflectometry of the skin,” IEEE Sens. J. 11(5), 1253–1262 (2011).
[Crossref]

R. S. Singh, P. Tewari, J. L. Bourges, J. P. Hubschman, D. B. Bennett, Z. D. Taylor, H. Lee, E. R. Brown, W. S. Grundfest, and M. O. Culjat, “Terahertz sensing of corneal hydration,” Conf. Proc. IEEE Eng. Med. Biol. Soc. 2010, 3021–3024 (2010).

Blandin, H. P.

H. P. Blandin, J. C. David, J. M. Vergnaud, J. P. Illien, and M. Malizewicz, “Modelling of drying of coatings: Effect of the thickness, temperature and concentration of solvent,” Prof. Org. Coat. 15(2), 163–172 (1987).
[Crossref]

Bourges, J. L.

R. S. Singh, P. Tewari, J. L. Bourges, J. P. Hubschman, D. B. Bennett, Z. D. Taylor, H. Lee, E. R. Brown, W. S. Grundfest, and M. O. Culjat, “Terahertz sensing of corneal hydration,” Conf. Proc. IEEE Eng. Med. Biol. Soc. 2010, 3021–3024 (2010).

Brennan, N. A.

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D. B. Bennett, Z. D. Taylor, P. Tewari, R. S. Singh, M. O. Culjat, W. S. Grundfest, D. J. Sassoon, R. D. Johnson, J. P. Hubschman, and E. R. Brown, “Terahertz sensing in corneal tissues,” J. Biomed. Opt. 16(5), 057003 (2011).
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S. Sung, S. Dabironezare, N. Llombart, S. Selvin, N. Bajwa, S. Chantra, B. Nowroozi, J. Garritano, J. Goell, A. Li, S. X. Deng, E. Brown, W. S. Grundfest, and Z. D. Taylor, “Optical system design for noncontact, normal incidence, THz imaging of in vivo human cornea,” IEEE Trans. Terahertz Sci. Technol. 8(1), 1–12 (2018).
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S. Sung, S. Selvin, N. Bajwa, S. Chantra, B. Nowroozi, J. Garritano, J. Goell, A. Li, S. X. Deng, E. Brown, W. S. Grundfest, and Z. D. Taylor, “THz imaging system for in vivo human cornea,” IEEE Trans. Terahertz Sci. Technol. 8(1), 27–37 (2018).
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Z. D. Taylor, J. Garritano, S. Sung, N. Bajwa, D. B. Bennett, B. Nowroozi, P. Tewari, J. Sayre, J.-P. Hubschman, S. Deng, E. R. Brown, and W. S. Grundfest, “THz and mm-wave sensing of corneal tissue water content: electromagnetic modeling and analysis,” IEEE Trans. Terahertz Sci. Technol. 5(2), 170–183 (2015).
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S. Sung, S. Dabironezare, N. Llombart, S. Selvin, N. Bajwa, S. Chantra, B. Nowroozi, J. Garritano, J. Goell, A. Li, S. X. Deng, E. Brown, W. S. Grundfest, and Z. D. Taylor, “Optical system design for noncontact, normal incidence, THz imaging of in vivo human cornea,” IEEE Trans. Terahertz Sci. Technol. 8(1), 1–12 (2018).
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D. Bennett, Z. Taylor, P. Tewari, S. Sung, A. Maccabi, R. Singh, M. Culjat, W. Grundfest, J. P. Hubschman, and E. Brown, “Assessment of corneal hydration sensing in the terahertz band: in vivo results at 100 GHz,” J. Biomed. Opt. 17(9), 0970081 (2012).
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Z. D. Taylor, J. Garritano, S. Sung, N. Bajwa, D. B. Bennett, B. Nowroozi, P. Tewari, J. Sayre, J.-P. Hubschman, S. Deng, E. R. Brown, and W. S. Grundfest, “THz and mm-wave sensing of corneal tissue water content: electromagnetic modeling and analysis,” IEEE Trans. Terahertz Sci. Technol. 5(2), 170–183 (2015).
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D. Bennett, Z. Taylor, P. Tewari, S. Sung, A. Maccabi, R. Singh, M. Culjat, W. Grundfest, J. P. Hubschman, and E. Brown, “Assessment of corneal hydration sensing in the terahertz band: in vivo results at 100 GHz,” J. Biomed. Opt. 17(9), 0970081 (2012).
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Z. D. Taylor, J. Garritano, S. Sung, N. Bajwa, D. B. Bennett, B. Nowroozi, P. Tewari, J. Sayre, J.-P. Hubschman, S. Deng, E. R. Brown, and W. S. Grundfest, “THz and mm-wave sensing of corneal tissue water content: electromagnetic modeling and analysis,” IEEE Trans. Terahertz Sci. Technol. 5(2), 170–183 (2015).
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Z. D. Taylor, R. S. Singh, D. B. Bennett, P. Tewari, C. P. Kealey, N. Bajwa, M. O. Culjat, A. Stojadinovic, H. Lee, J.-P. Hubschman, E. R. Brown, and W. S. Grundfest, “THz medical imaging: in vivo hydration sensing,” IEEE Trans. Terahertz Sci. Technol. 1(1), 201–219 (2011).
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Figures (9)

Fig. 1.
Fig. 1. (a) The experimental setup for taking simultaneous weight and THz reflection measurements from the corneal phantom is illustrated. The THz beam was focused at the apex of the stainless-steel sphere. The reference THz-TDS signal from the apex of the metallic sphere is shown in (b) time domain and (c) frequency domain. (d) The water loss of the contact lens during the dehydration experiment is shown as a function of time
Fig. 2.
Fig. 2. The flowchart for the two models used in this study is shown along with the parameters extracted from each step. This flowchart provides an overview of the models before describing them in detail in later sections.
Fig. 3.
Fig. 3. (a) The geometry of the lens is shown with a color bar to represent the thickness profile. (b) The cross-section of the contact lens is shown with discretized sections.
Fig. 4.
Fig. 4. (a) The diffusion of water (represented by blue arrow) is illustrated outwards from the sample and into the environment. The sample is assumed to have an infinite thickness and $x$ is defined as the axial direction. (b) Eq. (5) was fitted to the linear region of the gravimetric data to extract $\alpha = 3.16 \times {10^{-7}}\,\textrm{m/s}$ and ${D_0} = 1.99 \times {10^{-10}}\,{\textrm{m}^{2}}/\textrm{s}$.
Fig. 5.
Fig. 5. (a) The diffusion of water (represented by blue arrow) is illustrated outwards from the sample and into the environment. The changing thickness is represented by the black arrows. It is important to note that $x = 0$ is now defined at the bottom of the sample rather than the surface, highlighted by a red arrow. (b) Eq. (13) is fitted to the gravimetric data using $\alpha$ from the semi-infinite model. The resulting ${D_{eff}}$ and $k$ values are $3.772 \times {10^{-10}}\,{\textrm{m}^{2}}/\textrm{s}$ and 0.18 respectively.
Fig. 6.
Fig. 6. The discrete dielectric slabs are shown with the darker slabs representing increasing water content. The complex dielectric constant for each slab can be determined using Bruggeman effective media theory, Eq. (15), where each slab is a binary composite medium of water and hydrogel material. An illustration of the multiple THz reflections is shown by the red arrows and corresponding electric fields, ${E_{r_i}}$. N is the number of layers, here set to 11.
Fig. 7.
Fig. 7. The THz time-domain traces are shown over the course of the experiment. Each trace represents a single time point and corresponds to a water content shown in the color bar.
Fig. 8.
Fig. 8. The hydration profile is shown as a function of axial location in the hydrogel for the (a) thinnest and (b) thickest sections within our 2.7 mm diameter beam spot at the focus. Time is represented by the color bar.
Fig. 9.
Fig. 9. The simulated THz time-domain traces (black) and the corresponding measured THz signal (red) is shown at multiple time points.

Equations (20)

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C = G r a m s o f W a t e r V o l u m e o f H y d r o g e l = W ρ H y d r o g e l ,
W = m a s s ( t ) m a s s d r y m a s s h y d r a t e d ,
C t = x ( D 0 C ( x , t ) x ) ,
D 0 C ( x , t ) x = α ( C e q C s ) at x = 0 .
M s i ( t ) = ( C e q C 0 p ) { exp ( p 2 D 0 t ) erfc ( p D 0 t ) 1 + 2 π p D 0 t } ,
C t = x ( D ( C ) C x ) .
C ( x , t ) x = 0 at x = 0.
D ( C ) C ( x , t ) x C h ( t ) t = α ( C s C e q ) ,
D ( C ) = D e f f exp ( k C ) ,
h ( t ) = l ( 1 S + ( S 1 S ) C ¯ ( t ) C 0 )
C n t + 1 C n t Δ t = D ( C ) n t C n + 1 t 2 C n t + C n 1 t ( Δ x ) 2 + D ( C ) C | n t ( C n + 1 t C n 1 t 2 Δ x ) 2 .
M v t ( t ) = l C 0 0 h ( t ) C ( x , t ) d x .
M t o t a l ( t ) = i = 1 N M v t i ( t ) A i ,
ε ~ w ( ω ) = ε + ( ε s ε 2 ) 1 + i ω τ 1 + ( ε 2 ε ) 1 + i ω τ 2 ,
ρ w , i ( ε ~ i ε ~ w ε ~ w + 2 ε ~ i ) + ( 1 ρ w , i ) ( ε ~ i ε ~ h ε ~ h + 2 ε ~ i ) = 0 i [ i , N ] ,
r i = n ~ i 1 n ~ i n ~ i 1 + n ~ i i [ 1 , N + 1 ]
δ i = 2 π λ d n ~ i i [ 1 , N ] ,
Γ i = r i + Γ i + 1 e 2 j δ i 1 + r i Γ i + 1 e 2 j δ i , Γ N + 1 = r N + 1 i [ 1 , N + 1 ] .
E m o d e l ( t ) = F 1 { Γ 1 E r e f ( f ) } .
c o r r e c t i o n f a c t o r = 2 d m o d e l n a i r c ,