Abstract

We have examined the possibility of determining the optical properties of a two-layer medium by using a diffusion approximation radiation transport model [Appl. Opt. 37, 779 (1998)]. Continuous-wave and frequency-domain (FD) low-noise Monte Carlo (MC) data were fitted to the model. Marquardt–Levenberg and a simulated annealing algorithm were used and compared as optimization techniques. Our particular choice of optical properties for the two-layer model was consistent with skin and underlying fat in the presence of an exogenous chromophore [Appl. Opt. 37, 1958 (1998)]. The results are therefore specific to this set of optical properties. It was found that the cw diffusion solution could never be used to estimate all optical properties reliably. The combined cw and FD solutions could not be used to estimate some of the top-layer optical properties to an accuracy of better than 10%, although the absorption and the transport scattering coefficients of the bottom layer could be estimated to within 7% and 0.5%, respectively. No improvement was found from simultaneously fitting MC data at three different modulation frequencies. These results point to the need for a more accurate radiation transfer model at small source–detector separations.

© 1998 Optical Society of America

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References

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  1. B. Chance, Q. Luo, S. Nioka, D. C. Alsop, J. A. Detre, “Optical investigation of physiology: a study of biomedical contrast: intrinsic and extrinsic,” Philos. Trans. R. Soc. London B 352, 707–716 (1997).
    [CrossRef]
  2. R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of noninvasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
    [CrossRef] [PubMed]
  3. J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Corn, T. J. Johnson, T. Shimada, “Spectroscopic diagnosis of bladder cancer with elastic light scattering,” Lasers Surg. Med. 17, 350–357 (1995).
    [CrossRef] [PubMed]
  4. J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, M. Essenpreis, G. Schmelzeiser-Redeker, D. Bocker, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. 22, 190–192 (1997).
    [CrossRef] [PubMed]
  5. T. J. Farrell, M. S. Patterson, M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry,” Appl. Opt. 37, 1958–1972 (1998).
    [CrossRef]
  6. P. L. Williams, R. Warwick, “The integument,” in Gray’s Anatomy, 36th edition (Churchill Livingstone, Edinburgh, UK, 1986), pp. 1216–1222.
  7. A. Kienle, M. S. Patterson, N. Utke, R. Bays, G. Wagnieres, H. Van Den Bergh, “Determination of the optical properties of two-layer turbid media,” Appl. Opt. 37, 779–791 (1998).
    [CrossRef]
  8. I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium. A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
    [CrossRef]
  9. A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
    [CrossRef] [PubMed]
  10. B. W. Pogue, M. S. Patterson, “Error assessment of a wavelength tunable frequency domain system for noninvasive tissue spectroscopy,” J. Biomed. Opt. 1, 311–323 (1996).
    [CrossRef] [PubMed]
  11. T. J. Farrell, M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
    [CrossRef] [PubMed]
  12. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  13. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, London, 1990).
  14. W. C. Goffe, G. D. Ferrier, J. Rogers, “Global optimization of statistical functions with simulated annealing,” J. Economet. 60, 65–100 (1994).
    [CrossRef]
  15. L. Wang, S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746–1752 (1993).
    [CrossRef]
  16. S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue, Proc. SPIE908, 20–28 (1988).
    [CrossRef]

1998 (2)

1997 (3)

J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, M. Essenpreis, G. Schmelzeiser-Redeker, D. Bocker, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. 22, 190–192 (1997).
[CrossRef] [PubMed]

B. Chance, Q. Luo, S. Nioka, D. C. Alsop, J. A. Detre, “Optical investigation of physiology: a study of biomedical contrast: intrinsic and extrinsic,” Philos. Trans. R. Soc. London B 352, 707–716 (1997).
[CrossRef]

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of noninvasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

1996 (2)

1995 (1)

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Corn, T. J. Johnson, T. Shimada, “Spectroscopic diagnosis of bladder cancer with elastic light scattering,” Lasers Surg. Med. 17, 350–357 (1995).
[CrossRef] [PubMed]

1994 (2)

W. C. Goffe, G. D. Ferrier, J. Rogers, “Global optimization of statistical functions with simulated annealing,” J. Economet. 60, 65–100 (1994).
[CrossRef]

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

1993 (1)

1992 (2)

T. J. Farrell, M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium. A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Alsop, D. C.

B. Chance, Q. Luo, S. Nioka, D. C. Alsop, J. A. Detre, “Optical investigation of physiology: a study of biomedical contrast: intrinsic and extrinsic,” Philos. Trans. R. Soc. London B 352, 707–716 (1997).
[CrossRef]

Bays, R.

Berger, M.

Bigio, I. J.

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Corn, T. J. Johnson, T. Shimada, “Spectroscopic diagnosis of bladder cancer with elastic light scattering,” Lasers Surg. Med. 17, 350–357 (1995).
[CrossRef] [PubMed]

Bocker, D.

Boyer, J.

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Corn, T. J. Johnson, T. Shimada, “Spectroscopic diagnosis of bladder cancer with elastic light scattering,” Lasers Surg. Med. 17, 350–357 (1995).
[CrossRef] [PubMed]

Bruulsema, J. T.

Chance, B.

B. Chance, Q. Luo, S. Nioka, D. C. Alsop, J. A. Detre, “Optical investigation of physiology: a study of biomedical contrast: intrinsic and extrinsic,” Philos. Trans. R. Soc. London B 352, 707–716 (1997).
[CrossRef]

Corn, R. L.

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Corn, T. J. Johnson, T. Shimada, “Spectroscopic diagnosis of bladder cancer with elastic light scattering,” Lasers Surg. Med. 17, 350–357 (1995).
[CrossRef] [PubMed]

Dayan, I.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium. A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Detre, J. A.

B. Chance, Q. Luo, S. Nioka, D. C. Alsop, J. A. Detre, “Optical investigation of physiology: a study of biomedical contrast: intrinsic and extrinsic,” Philos. Trans. R. Soc. London B 352, 707–716 (1997).
[CrossRef]

Diamond, K. R.

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of noninvasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

Essenpreis, M.

Farrell, T. J.

Feng, T. C.

Ferrier, G. D.

W. C. Goffe, G. D. Ferrier, J. Rogers, “Global optimization of statistical functions with simulated annealing,” J. Economet. 60, 65–100 (1994).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, London, 1990).

Flock, S. T.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue, Proc. SPIE908, 20–28 (1988).
[CrossRef]

Goffe, W. C.

W. C. Goffe, G. D. Ferrier, J. Rogers, “Global optimization of statistical functions with simulated annealing,” J. Economet. 60, 65–100 (1994).
[CrossRef]

Haskell, R. C.

Havlin, S.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium. A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Hayward, J. E.

Heinemann, L.

Hibst, R.

Jacques, S. L.

Johnson, T. J.

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Corn, T. J. Johnson, T. Shimada, “Spectroscopic diagnosis of bladder cancer with elastic light scattering,” Lasers Surg. Med. 17, 350–357 (1995).
[CrossRef] [PubMed]

Kienle, A.

Koschinsky, T.

Lilge, L.

Luo, Q.

B. Chance, Q. Luo, S. Nioka, D. C. Alsop, J. A. Detre, “Optical investigation of physiology: a study of biomedical contrast: intrinsic and extrinsic,” Philos. Trans. R. Soc. London B 352, 707–716 (1997).
[CrossRef]

McAdams, M. S.

Mourant, J. R.

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Corn, T. J. Johnson, T. Shimada, “Spectroscopic diagnosis of bladder cancer with elastic light scattering,” Lasers Surg. Med. 17, 350–357 (1995).
[CrossRef] [PubMed]

Nioka, S.

B. Chance, Q. Luo, S. Nioka, D. C. Alsop, J. A. Detre, “Optical investigation of physiology: a study of biomedical contrast: intrinsic and extrinsic,” Philos. Trans. R. Soc. London B 352, 707–716 (1997).
[CrossRef]

Orskov, H.

Patterson, M. S.

A. Kienle, M. S. Patterson, N. Utke, R. Bays, G. Wagnieres, H. Van Den Bergh, “Determination of the optical properties of two-layer turbid media,” Appl. Opt. 37, 779–791 (1998).
[CrossRef]

T. J. Farrell, M. S. Patterson, M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry,” Appl. Opt. 37, 1958–1972 (1998).
[CrossRef]

J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, M. Essenpreis, G. Schmelzeiser-Redeker, D. Bocker, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. 22, 190–192 (1997).
[CrossRef] [PubMed]

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of noninvasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

B. W. Pogue, M. S. Patterson, “Error assessment of a wavelength tunable frequency domain system for noninvasive tissue spectroscopy,” J. Biomed. Opt. 1, 311–323 (1996).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue, Proc. SPIE908, 20–28 (1988).
[CrossRef]

Pogue, B. W.

B. W. Pogue, M. S. Patterson, “Error assessment of a wavelength tunable frequency domain system for noninvasive tissue spectroscopy,” J. Biomed. Opt. 1, 311–323 (1996).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, London, 1990).

Rogers, J.

W. C. Goffe, G. D. Ferrier, J. Rogers, “Global optimization of statistical functions with simulated annealing,” J. Economet. 60, 65–100 (1994).
[CrossRef]

Sandahl-Christiansen, J.

Schmelzeiser-Redeker, G.

Shimada, T.

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Corn, T. J. Johnson, T. Shimada, “Spectroscopic diagnosis of bladder cancer with elastic light scattering,” Lasers Surg. Med. 17, 350–357 (1995).
[CrossRef] [PubMed]

Steiner, R.

Svaasand, L. O.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, London, 1990).

Tromberg, B. J.

Tsay, T. T.

Utke, N.

Van Den Bergh, H.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, London, 1990).

Wagnieres, G.

Wang, L.

Warwick, R.

P. L. Williams, R. Warwick, “The integument,” in Gray’s Anatomy, 36th edition (Churchill Livingstone, Edinburgh, UK, 1986), pp. 1216–1222.

Weersink, R. A.

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of noninvasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

Weiss, G. H.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium. A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Williams, P. L.

P. L. Williams, R. Warwick, “The integument,” in Gray’s Anatomy, 36th edition (Churchill Livingstone, Edinburgh, UK, 1986), pp. 1216–1222.

Wilson, B.

Wilson, B. C.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue, Proc. SPIE908, 20–28 (1988).
[CrossRef]

Appl. Opt. (3)

J. Biomed. Opt. (1)

B. W. Pogue, M. S. Patterson, “Error assessment of a wavelength tunable frequency domain system for noninvasive tissue spectroscopy,” J. Biomed. Opt. 1, 311–323 (1996).
[CrossRef] [PubMed]

J. Economet. (1)

W. C. Goffe, G. D. Ferrier, J. Rogers, “Global optimization of statistical functions with simulated annealing,” J. Economet. 60, 65–100 (1994).
[CrossRef]

J. Mod. Opt. (1)

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium. A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

J. Opt. Soc. Am. A (2)

Lasers Surg. Med. (1)

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Corn, T. J. Johnson, T. Shimada, “Spectroscopic diagnosis of bladder cancer with elastic light scattering,” Lasers Surg. Med. 17, 350–357 (1995).
[CrossRef] [PubMed]

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Opt. Lett. (1)

Philos. Trans. R. Soc. London B (1)

B. Chance, Q. Luo, S. Nioka, D. C. Alsop, J. A. Detre, “Optical investigation of physiology: a study of biomedical contrast: intrinsic and extrinsic,” Philos. Trans. R. Soc. London B 352, 707–716 (1997).
[CrossRef]

Photochem. Photobiol. (1)

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of noninvasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

Other (3)

P. L. Williams, R. Warwick, “The integument,” in Gray’s Anatomy, 36th edition (Churchill Livingstone, Edinburgh, UK, 1986), pp. 1216–1222.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, London, 1990).

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue, Proc. SPIE908, 20–28 (1988).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Two-layer medium cw diffusion solution of Eqs. (1) and (2) (solid curve) showing a good comparison with the corresponding MC results (open circles) for ρ > 6 mm. The optical properties used for both are shown in Table 1, True column. Similarly, the semi-infinite medium solution for the top-layer optical properties (dashed curve) agrees well with the corresponding MC results (filled circles) for ρ > 4 mm. The semi-infinite medium diffusion solution for the bottom-layer optical properties (dash–dot–dot curve) is also shown.

Fig. 2
Fig. 2

As in Fig. 1 but for the ac reflectance (ACR) of an intensity-modulated incident beam. Note the steeper drop in the ac reflectance with the radial distance compared with the cw case in Fig. 1.

Fig. 3
Fig. 3

Two-layer diffusion solution of Eqs. (1) and (2) for the phase of the ac reflectance (solid curve) is close to the MC calculated phase (open circles) with increasing discrepancies at small radial distances (ρ < 2 mm). For larger ρ the two-layer phase is greater than the phase calculated from the individual semi-infinite solutions for the top- (dashed curve) and bottom- (dash–dot–dot curve) layer optical properties.

Tables (10)

Tables Icon

Table 1 True Set of Optical Properties with which all MC Runs were Performed and a Selected False Initial Guess of the Parameters Used as Input to All Fitting Routinesa

Tables Icon

Table 2 Estimates of Optical Properties Obtained by Fitting MC Data for Relative cw Reflectance Measurements by Marquardt–Levenberg and a True Initial Guess (Mrqrt-t), a False Initial Guess (Mrqrt-f), and Simulated Annealing with a False Initial Guess (SA-f)a

Tables Icon

Table 3 Estimates of Optical Properties Obtained by Fitting MC Data for Absolute cw Reflectance Measurements (qr = 1.0) by Mrqrt-t and SA-f where χ2(t) = 23.3a

Tables Icon

Table 4 Estimates of Optical Properties Obtained by Fitting MC Data for Relative Phase and cw Reflectance Measurements by Mrqrt-t, SA-t, and SA-fa

Tables Icon

Table 5 Estimates of Optical Properties Obtained by Fitting MC Data for Absolute Phase and cw Reflectance Measurements (qr = 1.0, qph = 0.0) by Mrqrt-t, SA-t, and SA-fa

Tables Icon

Table 6 Estimates of Optical Properties Obtained by Fitting MC Data for Absolute cw, ac Reflectance, and Phase Measurements (qr = 1.0, qph = 0.0) at a Modulation Frequency f = 195 MHza

Tables Icon

Table 7 Estimates of Optical Properties Obtained by Fitting MC Data for Absolute cw, ac Reflectance, and Phase Measurements (qr = 1.0, qph = 0.0) at a Modulation Frequency f = 195 MHza

Tables Icon

Table 8 As in Table 6 but for a Modulation Frequency f = 300 MHz [χ2(t) = 80.3]

Tables Icon

Table 9 As in Table 6 but for a Modulation Frequency f = 1 GHz and SA-t Shown Instead of SA-f [χ2(t) = 1010]

Tables Icon

Table 10 Continuous-Wave as Well as FD MC Data at Three Different Modulation Frequencies (195 MHz, 300 MHz, 1 GHz) Simultaneously Fitted as Absolute cw, ac Reflectance, and Phase Measurements (qr = 1.0, qph = 0.0)a

Equations (16)

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D 1 2 Φ 1 r ,   ω - μ a 1 + j   ω c m Φ 1 r ,   ω = - δ x ,   y ,   z - z 0 exp j   ω c m   r ,     0 z l ,
D 2 2 Φ 2 r ,   ω - μ a 2 + j   ω c m Φ 2 r ,   ω = 0 ,     l z ,
Φ 1 z = - z b ,   r = 0 .
Φ 2 r     = 0 .
Φ 1 l = Φ 2 l ,
D 1 Φ 1 z z z = l - = D 2 Φ 2 z z z = l + .
z b = 2 D 1 .
Φ 1 ρ ,   z = 0 = 1 2 π 0   ϕ 1 z = 0 sI 0 s ρ d s ,
ϕ 1 z = 0 = sinh α 1 z b + z 0 D 1 α 1 × D 1 α 1   cosh α 1 l + D 2 α 2   sinh α 1 l D 1 α 1   cosh α 1 l + z b + D 2 α 2   sinh α 1 l + z b - sinh α 1 z 0 D 1 α 1 ,
R ρ = 1 4   Φ 1 ρ ,   z = 0 + 1 2   D 1 Φ 1 ρ ,   z z z = 0 .
θ = tan - 1 Im R ρ ,   ω Re R ρ ,   ω
ACR   =   ( Im R ρ ,   ω 2 + Re R ρ ,   ω 2 ) 1 / 2 .
χ min R 2 = i = 1 N y ρ i - q r yc ρ i σ i 2 min ,
q r = i = 1 N y ρ i yc ρ i σ i 2 i = 1 N yc ρ i σ i 2 .
χ min ph 2 = i = 1 N yp ρ i - ypc ρ i + q ph σ i 2 min ,
q ph = i = 1 N yp ρ i - ypc ρ i σ i 2 i = 1 N 1 σ i 2 .

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