Abstract

The scattering and absorption of light by randomly oriented, discretely scattering, red blood cells imbedded in a homogeneous plasma medium can be described by the P1 approximation to the one-speed transport equation, where the cells have the dual role of anisotropic sources for first scattering events and of scattering and absorption sites for subsequent scattering events. Equations for diffuse reflectance defined for a finite size receiver in the plane of a normally incident cylindrical photon beam are derived and compared with experimental data to fundamentally justify the basic sending–receiving charactreristics of a fiber optic catheter model. A model of the fiber optic catheter used for the spectrophotometric measurement of oxygen content in blood is developed from the theory and compared with experimental results to further substantiate the theoretical approach.

© 1976 Optical Society of America

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  1. R. Longini, R. Zdrojkowski, IEEE Trans. Biomed. Eng. BME-15, 4 (January1968).
    [CrossRef]
  2. R. Zdrojkowski, N. Pisharoty, IEEE Trans. Biomed. Eng. BME-17, 122 (April1970).
    [CrossRef]
  3. C. Johnson, IEEE Trans. Biomed. Eng. BME-17, 129 (April1970).
    [CrossRef]
  4. F. Janssen, Med. Biol. Eng. 10, 231 (1972).
    [CrossRef] [PubMed]
  5. L. Reynolds, “Three Dimensional Reflection and Transmission Equations for Optical Diffusion in Blood,” M.S. Thesis, Electrical Engineering Department, University of Washington (1970).
  6. L. Reynolds, “Optical Diffuse Reflectance and Transmittance from an Anisotropically Finite Blood Medium,” Ph.D. Thesis, Electrical Engineering Department, University of Washington (1975).
  7. A. Shuster, Astrophysics 21, 1 (January1905).
    [CrossRef]
  8. P. Kubelka, J. Opt. Soc. of Am. 38, 448 (May1948).
    [CrossRef]
  9. V. Twersky, Opt. Soc. Am. 60, 1084 (1970).
    [CrossRef]
  10. M. Moaveni, “A Multiple Scattering Field Theory Applied to Whole Blood,” Ph.D. Thesis, Electrical Engineering Department, University of Washington (1970).
  11. A. Cohen, R. Longini, Med. Biol. Eng. 9, 61 (January1971).
    [CrossRef] [PubMed]
  12. C. Johnson, R. Palm, D. Stewart, Assoc. Adv. Med. Instrum. 5, No. 2 (March–April 1971).
  13. M. Polanyi, R. Hehir, Rev. Sci. Instrum. 33, 1050 (1962).
    [CrossRef]
  14. K. Case, P. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8.
  15. V. Twersky, J. Math. Phys. 3, No. 4, 724 (1962).
    [CrossRef]
  16. R. Pierce, “An Experimental Determination of the Average Scattering and Absorption Cross Sections of Human Red Blood Cells for Near Infrared Light,” M.S. Thesis, Electrical Engineering Department, University of Washington (1972).
  17. L. Reynolds, J. Molcho, C. Johnson, A. Ishimaru, “Optical Cross Sections of Human Erythrocytes,” Proc. of the 27th Annual Conf. on Eng. in Med. and Biol. (1974), Vol. 16.
  18. J. Dave, “Stokes Parameters of the Radiation Scattered by a Homogeneous Sphere,” IBM Prog. 360D-17.4.002, IBM Scientific Center, Palo Alto, Calif. (1968).
  19. G. Petersen, N. McCormick, L. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. (in press).
  20. N. Anderson, P. Sekelj, Phys. Med. Biol. 12, 185 (1967).
    [CrossRef] [PubMed]
  21. F. Janssen, Med. Biol. Eng. 10, 9 (1972).
    [CrossRef] [PubMed]
  22. J. Cole, W. Martin, P. Cheung, C. Johnson, Am. J. Cardiol. 29, 383 (1972).
    [CrossRef] [PubMed]
  23. E. Woodroof, S. Koorajian, J. Assoc. Adv. Med. Instrum. 7, No. 5, 287 (December1973).
  24. W. Blom, V. H. deVilleneuve, Med. Biol. Eng. 537 (July1974).
  25. L. Reynolds, C. Johnson, “Three-Dimensional Optical Diffusion Theory for Spectrophotometric Instrumentation Design,” Proc. of the 24th Annual Conf. on Eng. in Med. and Biol. (1971), Vol. 13.

1974 (1)

W. Blom, V. H. deVilleneuve, Med. Biol. Eng. 537 (July1974).

1973 (1)

E. Woodroof, S. Koorajian, J. Assoc. Adv. Med. Instrum. 7, No. 5, 287 (December1973).

1972 (3)

F. Janssen, Med. Biol. Eng. 10, 9 (1972).
[CrossRef] [PubMed]

J. Cole, W. Martin, P. Cheung, C. Johnson, Am. J. Cardiol. 29, 383 (1972).
[CrossRef] [PubMed]

F. Janssen, Med. Biol. Eng. 10, 231 (1972).
[CrossRef] [PubMed]

1971 (2)

A. Cohen, R. Longini, Med. Biol. Eng. 9, 61 (January1971).
[CrossRef] [PubMed]

C. Johnson, R. Palm, D. Stewart, Assoc. Adv. Med. Instrum. 5, No. 2 (March–April 1971).

1970 (3)

R. Zdrojkowski, N. Pisharoty, IEEE Trans. Biomed. Eng. BME-17, 122 (April1970).
[CrossRef]

C. Johnson, IEEE Trans. Biomed. Eng. BME-17, 129 (April1970).
[CrossRef]

V. Twersky, Opt. Soc. Am. 60, 1084 (1970).
[CrossRef]

1968 (1)

R. Longini, R. Zdrojkowski, IEEE Trans. Biomed. Eng. BME-15, 4 (January1968).
[CrossRef]

1967 (1)

N. Anderson, P. Sekelj, Phys. Med. Biol. 12, 185 (1967).
[CrossRef] [PubMed]

1962 (2)

M. Polanyi, R. Hehir, Rev. Sci. Instrum. 33, 1050 (1962).
[CrossRef]

V. Twersky, J. Math. Phys. 3, No. 4, 724 (1962).
[CrossRef]

1948 (1)

P. Kubelka, J. Opt. Soc. of Am. 38, 448 (May1948).
[CrossRef]

1905 (1)

A. Shuster, Astrophysics 21, 1 (January1905).
[CrossRef]

Anderson, N.

N. Anderson, P. Sekelj, Phys. Med. Biol. 12, 185 (1967).
[CrossRef] [PubMed]

Blom, W.

W. Blom, V. H. deVilleneuve, Med. Biol. Eng. 537 (July1974).

Case, K.

K. Case, P. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8.

Cheung, P.

J. Cole, W. Martin, P. Cheung, C. Johnson, Am. J. Cardiol. 29, 383 (1972).
[CrossRef] [PubMed]

Cohen, A.

A. Cohen, R. Longini, Med. Biol. Eng. 9, 61 (January1971).
[CrossRef] [PubMed]

Cole, J.

J. Cole, W. Martin, P. Cheung, C. Johnson, Am. J. Cardiol. 29, 383 (1972).
[CrossRef] [PubMed]

Dave, J.

J. Dave, “Stokes Parameters of the Radiation Scattered by a Homogeneous Sphere,” IBM Prog. 360D-17.4.002, IBM Scientific Center, Palo Alto, Calif. (1968).

deVilleneuve, V. H.

W. Blom, V. H. deVilleneuve, Med. Biol. Eng. 537 (July1974).

Hehir, R.

M. Polanyi, R. Hehir, Rev. Sci. Instrum. 33, 1050 (1962).
[CrossRef]

Ishimaru, A.

L. Reynolds, J. Molcho, C. Johnson, A. Ishimaru, “Optical Cross Sections of Human Erythrocytes,” Proc. of the 27th Annual Conf. on Eng. in Med. and Biol. (1974), Vol. 16.

Janssen, F.

F. Janssen, Med. Biol. Eng. 10, 9 (1972).
[CrossRef] [PubMed]

F. Janssen, Med. Biol. Eng. 10, 231 (1972).
[CrossRef] [PubMed]

Johnson, C.

J. Cole, W. Martin, P. Cheung, C. Johnson, Am. J. Cardiol. 29, 383 (1972).
[CrossRef] [PubMed]

C. Johnson, R. Palm, D. Stewart, Assoc. Adv. Med. Instrum. 5, No. 2 (March–April 1971).

C. Johnson, IEEE Trans. Biomed. Eng. BME-17, 129 (April1970).
[CrossRef]

L. Reynolds, J. Molcho, C. Johnson, A. Ishimaru, “Optical Cross Sections of Human Erythrocytes,” Proc. of the 27th Annual Conf. on Eng. in Med. and Biol. (1974), Vol. 16.

L. Reynolds, C. Johnson, “Three-Dimensional Optical Diffusion Theory for Spectrophotometric Instrumentation Design,” Proc. of the 24th Annual Conf. on Eng. in Med. and Biol. (1971), Vol. 13.

Koorajian, S.

E. Woodroof, S. Koorajian, J. Assoc. Adv. Med. Instrum. 7, No. 5, 287 (December1973).

Kubelka, P.

P. Kubelka, J. Opt. Soc. of Am. 38, 448 (May1948).
[CrossRef]

Longini, R.

A. Cohen, R. Longini, Med. Biol. Eng. 9, 61 (January1971).
[CrossRef] [PubMed]

R. Longini, R. Zdrojkowski, IEEE Trans. Biomed. Eng. BME-15, 4 (January1968).
[CrossRef]

Martin, W.

J. Cole, W. Martin, P. Cheung, C. Johnson, Am. J. Cardiol. 29, 383 (1972).
[CrossRef] [PubMed]

McCormick, N.

G. Petersen, N. McCormick, L. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. (in press).

Moaveni, M.

M. Moaveni, “A Multiple Scattering Field Theory Applied to Whole Blood,” Ph.D. Thesis, Electrical Engineering Department, University of Washington (1970).

Molcho, J.

L. Reynolds, J. Molcho, C. Johnson, A. Ishimaru, “Optical Cross Sections of Human Erythrocytes,” Proc. of the 27th Annual Conf. on Eng. in Med. and Biol. (1974), Vol. 16.

Palm, R.

C. Johnson, R. Palm, D. Stewart, Assoc. Adv. Med. Instrum. 5, No. 2 (March–April 1971).

Petersen, G.

G. Petersen, N. McCormick, L. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. (in press).

Pierce, R.

R. Pierce, “An Experimental Determination of the Average Scattering and Absorption Cross Sections of Human Red Blood Cells for Near Infrared Light,” M.S. Thesis, Electrical Engineering Department, University of Washington (1972).

Pisharoty, N.

R. Zdrojkowski, N. Pisharoty, IEEE Trans. Biomed. Eng. BME-17, 122 (April1970).
[CrossRef]

Polanyi, M.

M. Polanyi, R. Hehir, Rev. Sci. Instrum. 33, 1050 (1962).
[CrossRef]

Reynolds, L.

L. Reynolds, J. Molcho, C. Johnson, A. Ishimaru, “Optical Cross Sections of Human Erythrocytes,” Proc. of the 27th Annual Conf. on Eng. in Med. and Biol. (1974), Vol. 16.

L. Reynolds, “Optical Diffuse Reflectance and Transmittance from an Anisotropically Finite Blood Medium,” Ph.D. Thesis, Electrical Engineering Department, University of Washington (1975).

L. Reynolds, C. Johnson, “Three-Dimensional Optical Diffusion Theory for Spectrophotometric Instrumentation Design,” Proc. of the 24th Annual Conf. on Eng. in Med. and Biol. (1971), Vol. 13.

G. Petersen, N. McCormick, L. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. (in press).

L. Reynolds, “Three Dimensional Reflection and Transmission Equations for Optical Diffusion in Blood,” M.S. Thesis, Electrical Engineering Department, University of Washington (1970).

Sekelj, P.

N. Anderson, P. Sekelj, Phys. Med. Biol. 12, 185 (1967).
[CrossRef] [PubMed]

Shuster, A.

A. Shuster, Astrophysics 21, 1 (January1905).
[CrossRef]

Stewart, D.

C. Johnson, R. Palm, D. Stewart, Assoc. Adv. Med. Instrum. 5, No. 2 (March–April 1971).

Twersky, V.

V. Twersky, Opt. Soc. Am. 60, 1084 (1970).
[CrossRef]

V. Twersky, J. Math. Phys. 3, No. 4, 724 (1962).
[CrossRef]

Woodroof, E.

E. Woodroof, S. Koorajian, J. Assoc. Adv. Med. Instrum. 7, No. 5, 287 (December1973).

Zdrojkowski, R.

R. Zdrojkowski, N. Pisharoty, IEEE Trans. Biomed. Eng. BME-17, 122 (April1970).
[CrossRef]

R. Longini, R. Zdrojkowski, IEEE Trans. Biomed. Eng. BME-15, 4 (January1968).
[CrossRef]

Zweifel, P.

K. Case, P. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8.

Am. J. Cardiol. (1)

J. Cole, W. Martin, P. Cheung, C. Johnson, Am. J. Cardiol. 29, 383 (1972).
[CrossRef] [PubMed]

Assoc. Adv. Med. Instrum. (1)

C. Johnson, R. Palm, D. Stewart, Assoc. Adv. Med. Instrum. 5, No. 2 (March–April 1971).

Astrophysics (1)

A. Shuster, Astrophysics 21, 1 (January1905).
[CrossRef]

IEEE Trans. Biomed. Eng. (3)

R. Longini, R. Zdrojkowski, IEEE Trans. Biomed. Eng. BME-15, 4 (January1968).
[CrossRef]

R. Zdrojkowski, N. Pisharoty, IEEE Trans. Biomed. Eng. BME-17, 122 (April1970).
[CrossRef]

C. Johnson, IEEE Trans. Biomed. Eng. BME-17, 129 (April1970).
[CrossRef]

J. Assoc. Adv. Med. Instrum. (1)

E. Woodroof, S. Koorajian, J. Assoc. Adv. Med. Instrum. 7, No. 5, 287 (December1973).

J. Math. Phys. (1)

V. Twersky, J. Math. Phys. 3, No. 4, 724 (1962).
[CrossRef]

J. Opt. Soc. of Am. (1)

P. Kubelka, J. Opt. Soc. of Am. 38, 448 (May1948).
[CrossRef]

Med. Biol. Eng. (4)

F. Janssen, Med. Biol. Eng. 10, 231 (1972).
[CrossRef] [PubMed]

A. Cohen, R. Longini, Med. Biol. Eng. 9, 61 (January1971).
[CrossRef] [PubMed]

W. Blom, V. H. deVilleneuve, Med. Biol. Eng. 537 (July1974).

F. Janssen, Med. Biol. Eng. 10, 9 (1972).
[CrossRef] [PubMed]

Opt. Soc. Am. (1)

V. Twersky, Opt. Soc. Am. 60, 1084 (1970).
[CrossRef]

Phys. Med. Biol. (1)

N. Anderson, P. Sekelj, Phys. Med. Biol. 12, 185 (1967).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

M. Polanyi, R. Hehir, Rev. Sci. Instrum. 33, 1050 (1962).
[CrossRef]

Other (9)

K. Case, P. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8.

R. Pierce, “An Experimental Determination of the Average Scattering and Absorption Cross Sections of Human Red Blood Cells for Near Infrared Light,” M.S. Thesis, Electrical Engineering Department, University of Washington (1972).

L. Reynolds, J. Molcho, C. Johnson, A. Ishimaru, “Optical Cross Sections of Human Erythrocytes,” Proc. of the 27th Annual Conf. on Eng. in Med. and Biol. (1974), Vol. 16.

J. Dave, “Stokes Parameters of the Radiation Scattered by a Homogeneous Sphere,” IBM Prog. 360D-17.4.002, IBM Scientific Center, Palo Alto, Calif. (1968).

G. Petersen, N. McCormick, L. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. (in press).

M. Moaveni, “A Multiple Scattering Field Theory Applied to Whole Blood,” Ph.D. Thesis, Electrical Engineering Department, University of Washington (1970).

L. Reynolds, “Three Dimensional Reflection and Transmission Equations for Optical Diffusion in Blood,” M.S. Thesis, Electrical Engineering Department, University of Washington (1970).

L. Reynolds, “Optical Diffuse Reflectance and Transmittance from an Anisotropically Finite Blood Medium,” Ph.D. Thesis, Electrical Engineering Department, University of Washington (1975).

L. Reynolds, C. Johnson, “Three-Dimensional Optical Diffusion Theory for Spectrophotometric Instrumentation Design,” Proc. of the 24th Annual Conf. on Eng. in Med. and Biol. (1971), Vol. 13.

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Figures (9)

Fig. 1
Fig. 1

Diagrammatic representation of the receiving aperture weighting function for reflectance calculations.

Fig. 2
Fig. 2

Comparison of the diffusion theory and experimental results for reflectance from whole human blood vs radial separation distance between two optical fibers at a wavelength of 0.685 μm, 100% oxygen saturation, and a hematocrit of H = 0.41.

Fig. 3
Fig. 3

Comparison of the diffusion theory and experimental results for reflectance from whole human blood vs hematocrit for two different fiber separation distances at a wavelength of 0.685 μm and 100% oxygen saturation.

Fig. 4
Fig. 4

Schematic representation of a twenty-eight fiber ring catheter configuration (Z-13).

Fig. 5
Fig. 5

Comparison of the catheter model and experimental results for oxygen saturation vs reflectance ratio from whole human blood (catheter Z-13).

Fig. 6
Fig. 6

Comparison of the catheter model and experimental results for oxygen saturation vs reflectance ratio from whole human blood (catheter Z-2).

Fig. 7
Fig. 7

Comparison of the catheter model and experimental results for oxygen saturation vs reflectance ratio from whole human blood (catheter V-10).

Fig. 8
Fig. 8

Schematic representation of a thirty-five fiber ring catheter configuration (Z-2).

Fig. 9
Fig. 9

Schematic representation of an eighteen-fiber 2-lm catheter configuration (V-10).

Tables (2)

Tables Icon

Table I Comparison of Mie Scattering Calculation for a Sphere of 2.79-μm Radius and Experimentally Derived Cross Sections for the Red Blood Cell at 100% Oxygen Saturationa

Tables Icon

Table II Simulation of the Scattering and Absorption Characteristics of a Red Blood Cell by Mie Scattering (Relative Index of Refraction η = ηr + i)

Equations (44)

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[ 2 - ( Σ a / D ) ] ρ s ( r , θ , z ) = - ( 1 / D ) S ( r , θ , z )
S ( r , θ , z ) = Q 0 ( r , θ , z ) - 3 D · Q 1 ( r , θ , z ) ;
Q ( r , θ , z , μ ) = q 0 Σ d ( μ ) exp ( - Σ z ) R ( r ) ,
Σ = H v [ ( 1 - H ) σ s + σ a ] = Σ s + Σ a .
Q m = 2 π - 1 1 Q ( r , θ , z , μ ^ ) P m ( μ ^ ) d μ ^
Q 0 = 2 π - 1 1 Q ( r , θ , z , μ ^ ) = q 0 Σ s exp ( - Σ z ) R ( r ) ,
Q 1 = 2 π - 1 1 Q ( r , θ , z , μ ^ ) μ ^ d μ ^ = q 0 Σ s μ ¯ exp ( - Σ z ) R ( r ) z ^ ,
ρ s + g n · ρ s = 0 ,
ρ s ( r , θ , z ) = 1 D v S ( r , θ , z ) G ( r , θ , z r , θ , z ) d v .
( 2 - Σ a D ) G ( r , θ , z r , θ , z ) = - δ ( r - r ) 2 π r δ ( z - z ) δ ( θ - θ ) ,
G - g n ^ · G = 0
G ( r , z r z ) = 2 π n = 1 Γ n ρ n ( z ) ρ n ( z ) G n ( r r ) .
ρ n ( z ) = sin ( k n z + γ n ) ,
Γ n = k n 2 k n d + sin 2 γ n - sin [ 2 ( k n d + γ n ) ] .
G n ( r r ) = { I 0 ( λ n r ) K 0 ( λ n r ) , r > r , K 0 ( λ n r ) I 0 ( λ n r ) , r < r ,
tan k n d = 2 g k n g 2 k n 2 - 1 ,
γ n = tan - 1 ( g k n )
λ n 2 = k n 2 + Σ a D .
1 - Σ s Σ a + Σ s 1 ,
g = ( 0.7104 ) / Σ t ,
Σ t = Σ s ( 1 - u ¯ ) + Σ a .
ρ s ( r , z ) = 2 Σ s π ( 1 D + 3 μ ¯ Σ ) q 0 n = 1 Γ n ρ n ( z ) 0 2 π d θ 0 d × exp ( - Σ z ) sin ( k n z + γ n ) d z 0 b R ( r ) [ I 0 ( λ n r ) K 0 ( λ n r ) K 0 ( λ n r ) I 0 ( λ n r ) ] r d r ,
R ( r ) = U ( r ) - U ( r - b ) ,
0 d exp ( - Σ z ) sin ( k n z + γ n ) d z = z n ( k n 2 + Σ 2 ) ,
z n = { Σ sin γ n [ 1 + exp ( - Σ d ) ( k n Σ sin k n d - cos k n d ) ] + k n cos γ n [ 1 - exp ( - Σ d ) ( Σ k n sin k n d + cos k n d ) ] } .
0 r K 0 ( λ n r ) I 0 ( λ n r ) r d r = r λ n K 0 ( λ n r ) I 1 ( λ n r ) ,
r b I 0 ( λ n r ) K 0 ( λ n r ) r d r = r λ n I 0 ( λ n r ) K 1 ( λ n r ) - b λ n I 0 ( λ n r ) K 1 ( λ n b ) .
0 b K 0 ( λ n r ) I 0 ( λ n r ) r d r = b λ n K 0 ( λ n r ) I 1 ( λ n b ) .
ρ s ( r , z ) = 4 Σ s ( 1 D + 3 μ ¯ Σ ) q 0 n = 1 Γ n ρ n ( z ) z n R n ( λ n r ) ( k n 2 + Σ 2 ) ,
R n ( λ n λ ) = { 1 λ n 2 [ 1 - λ n b I 0 ( λ n r ) K 1 ( λ n b ) ] , r < b , b λ n [ K 0 ( λ n r ) I 1 ( λ n b ) ] , r > b .
Γ s z = Γ z + - Γ z - ,
Γ z - = 1 2 ( 1 2 ρ s + 1 3 Σ t ρ s z - 1 Σ t z ^ · Q 1 )
Γ z + = 1 2 ( 1 2 ρ s - 1 3 Σ t ρ s z + 1 Σ t z ^ · Q 1 ) ,
Γ s z = - 1 3 Σ t ρ s z + 1 Σ t z ^ · Q 1 ,
R diff = 0 2 π 0 a Γ z - r d r d θ 0 2 π 0 b q 0 r d r d θ | z = 0
R diff = 8 Σ s b 2 ( 1 D + 3 μ ¯ Σ ) n = 1 Γ n z n ( k n 2 + Σ 2 ) ( 1 2 sin γ n + D k n cos γ n ) R a ,
R a = { 1 λ n 2 [ a 2 2 - a b I 1 ( λ n a ) K 1 ( λ n b ) ] , a b , 1 λ n 2 [ b 2 2 - a b K 1 ( λ n a ) I 1 ( λ n b ) ] , a b .
R = [ R diff ( r a ) - R diff ( r b ) ] A A 1 .
σ d ( μ ) = v H ( 1 - H ) Σ d ( μ )
σ s = σ s + + σ s - ,
σ s + = 2 π 0 1 σ d ( μ ) d μ
σ s - = 2 π - 1 0 σ d ( μ ) d μ
R T = i = 1 m j = 1 n [ R diff ( r a i , j ) - R diff ( r b i , j ) ] A A j ,
O S = A - B ( R i / R r ) ,

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