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Light Transport in Tissue


Conclusions

The adding-doubling method has been implemented with boundary conditions and scattering functions similar to those for tissue. The adding-doubling method is one-dimensional and with a modest number of quadrature angles (N=16) very accurate results may be obtained (0.01%). The adding-doubling method is not as flexible as the Monte Carlo method, but it is a valuable standard against which other one-dimensional methods may be compared. The tabulated values (Tables 3.1-3.6) of reflection and transmission presented in this chapter will be used to evaluate the accuracy of the one-dimensional delta-Eddington approximation in the Chapter 4.


Table 3.1: Total reflection from a slab for normal incidence as a function of optical depth ($\tau =2^n$) and albedo (a). Scattering is isotropic (g=0), the boundary conditions are matched, and all digits shown are significant. The diffusion method was used for the italicized entries, all others were obtained with the doubling method.
n a=1 a=.99 a=.95 a=.9 a=.8 a=.6 a= .4 a=.2
-15 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000
-14 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000
-13 .0001 .0001 .0001 .0000 .0000 .0000 .0000 .0000
-12 .0001 .0001 .0001 .0001 .0001 .0001 .0000 .0000
-11 .0002 .0002 .0002 .0002 .0002 .0002 .0001 .0000
-10 .0005 .0005 .0005 .0004 .0004 .0003 .0002 .0001
-9 .0010 .0010 .0009 .0009 .0008 .0006 .0004 .0002
-8 .0019 .0019 .0019 .0018 .0016 .0012 .0008 .0004
-7 .0039 .0038 .0037 .0035 .0031 .0023 .0015 .0008
-6 .0077 .0077 .0074 .0069 .0061 .0046 .0030 .0015
-5 .0154 .0152 .0146 .0137 .0121 .0090 .0059 .0029
-4 .0303 .0300 .0286 .0269 .0236 .0173 .0112 .0055
-3 .0589 .0582 .0553 .0518 .0450 .0323 .0207 .0099
-2 .1117 .1101 .1039 .0965 .0824 .0573 .0356 .0167
-1 .2025 .1989 .1851 .1690 .1401 .0927 .0553 .0250
0 .3413 .3329 .3017 .2674 .2108 .1295 .0734 .0320
1 .5175 .4975 .4287 .3616 .2659 .1510 .0820 .0349
2 .6909 .6450 .5124 .4081 .2840 .1553 .0833 .0352
3 .8218 .7287 .5344 .4148 .2853 .1554 .0834 .0352
4 .9036 .7513 .5355 .4149 .2853 .1554 .0834 .0352
5 .9498 .7527 .5355 .4149 .2853 .1554 .0834 .0352
6 .9743 .7527 .5355 .4149 .2853 .1554 .0834 .0352
7 .9870 .7527 .5355 .4149 .2853 .1554 .0834 .0352
8 .9935 .7527 .5355 .4149 .2853 .1554 .0834 .0352
9 .9968 .7527 .5355 .4149 .2853 .1554 .0834 .0352
10 .9984 .7527 .5355 .4149 .2853 .1554 .0834 .0352
11 .9992 .7527 .5355 .4149 .2853 .1554 .0834 .0352
12 .9996 .7527 .5355 .4149 .2853 .1554 .0834 .0352
13 .9998 .7527 .5355 .4149 .2853 .1554 .0834 .0352
14 .9999 .7527 .5355 .4149 .2853 .1554 .0834 .0352


Table 3.2: Total transmission by a slab for normal incidence as a function of optical depth ($\tau =2^n$) and albedo (a). Scattering is isotropic (g=0), the boundary conditions are matched, and all digits shown are significant. The diffusion method was used for the italicized entries, all others were obtained with the doubling method.
n a=1 a=.99 a=.95 a=.9 a=.8 a=.6 a= .4 a=.2
-15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
-14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9999 .9999
-13 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999
-12 .9999 .9999 .9999 .9999 .9998 .9998 .9998 .9998
-11 .9998 .9998 .9997 .9997 .9997 .9997 .9996 .9996
-10 .9995 .9995 .9995 .9995 .9994 .9993 .9992 .9991
-9 .9990 .9990 .9990 .9989 .9988 .9986 .9984 .9982
-8 .9980 .9980 .9980 .9979 .9977 .9973 .9969 .9965
-7 .9961 .9961 .9959 .9957 .9953 .9945 .9937 .9930
-6 .9923 .9922 .9919 .9914 .9906 .9891 .9875 .9860
-5 .9846 .9844 .9838 .9830 .9814 .9782 .9751 .9721
-4 .9697 .9693 .9680 .9663 .9630 .9567 .9506 .9449
-3 .9411 .9403 .9375 .9340 .9272 .9146 .9030 .8924
-2 .8883 .8867 .8806 .8733 .8595 .8349 .8136 .7951
-1 .7975 .7941 .7808 .7654 .7378 .6928 .6577 .6296
0 .6587 .6510 .6226 .5916 .5414 .4714 .4251 .3923
1 .4825 .4657 .4093 .3565 .2860 .2111 .1730 .1502
2 .3091 .2755 .1869 .1285 .0751 .0394 .0272 .0215
3 .1782 .1221 .0408 .0160 .0047 .0012 .0006 .0004
4 .0964 .0296 .0020 .0002 .0000 .0000 .0000 .0000
5 .0502 .0019 .0000 .0000 .0000 .0000 .0000 .0000
6 .0257 .0000 .0000 .0000 .0000 .0000 .0000 .0000
7 .0130 .0000 .0000 .0000 .0000 .0000 .0000 .0000
8 .0065 .0000 .0000 .0000 .0000 .0000 .0000 .0000
9 .0032 .0000 .0000 .0000 .0000 .0000 .0000 .0000
10 .0016 .0000 .0000 .0000 .0000 .0000 .0000 .0000
11 .0008 .0000 .0000 .0000 .0000 .0000 .0000 .0000
12 .0004 .0000 .0000 .0000 .0000 .0000 .0000 .0000
13 .0002 .0000 .0000 .0000 .0000 .0000 .0000 .0000
14 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000


Table 3.3: Total reflection for normal incidence as a function of optical depth ($\tau =2^n$) and albedo (a) for three different anisotropies (g=0, g=0.5 and g=0.875) using a Henyey-Greenstein phase function and matched boundary conditions. All digits shown are significant.
  g=0 g=.5 g=.875
n a=.6 a=.9 a=.99 a=.6 a=.9 a=.99 a=.6 a=.9 a=.99
-15 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000
-14 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000
-13 .0000 .0000 .0001 .0000 .0000 .0000 .0000 .0000 .0000
-12 .0001 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000
-11 .0002 .0002 .0002 .0000 .0001 .0001 .0000 .0000 .0000
-10 .0003 .0004 .0005 .0001 .0002 .0002 .0000 .0000 .0000
-9 .0006 .0009 .0010 .0002 .0003 .0003 .0000 .0000 .0001
-8 .0012 .0018 .0019 .0004 .0006 .0007 .0001 .0001 .0001
-7 .0023 .0035 .0038 .0008 .0012 .0013 .0001 .0002 .0002
-6 .0046 .0069 .0077 .0016 .0024 .0026 .0003 .0004 .0005
-5 .0090 .0137 .0152 .0031 .0048 .0053 .0005 .0008 .0009
-4 .0173 .0269 .0300 .0060 .0096 .0107 .0010 .0016 .0018
-3 .0323 .0518 .0582 .0114 .0190 .0216 .0019 .0033 .0037
-2 .0573 .0965 .1101 .0208 .0375 .0438 .0035 .0064 .0076
-1 .0927 .1690 .1989 .0353 .0720 .0878 .0059 .0125 .0157
0 .1295 .2674 .3329 .0527 .1298 .1707 .0089 .0238 .0327
1 .1510 .3616 .4975 .0658 .2045 .3053 .0116 .0422 .0691
2 .1553 .4081 .6450 .0698 .2612 .4698 .0128 .0658 .1417
3 .1554 .4148 .7287 .0700 .2770 .6001 .0129 .0826 .2584
4 .1554 .4150 .7513 .0700 .2778 .6561 .0129 .0864 .3753
5 .1554 .4150 .7527 .0700 .2778 .6644 .0129 .0866 .4312
6 .1554 .4150 .7527 .0700 .2778 .6646 .0129 .0866 .4396
7 .1554 .4150 .7527 .0700 .2778 .6646 .0129 .0866 .4397
8 .1554 .4150 .7527 .0700 .2778 .6646 .0129 .0866 .4397
9 .1554 .4150 .7527 .0700 .2778 .6646 .0129 .0866 .4397


Table 3.4: Total transmission for normal incidence as a function of optical depth ($\tau =2^n$) and albedo (a) for three different anisotropies (g=0, g=.5 and g=0.875) using a Henyey-Greenstein phase function and matched boundary conditions. All digits shown are significant.
  g=0 g=.5 g=.875
n a=.6 a=.9 a=.99 a=.6 a=.9 a=.99 a=.6 a=.9 a=.99
-15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
-14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
-13 .9999 .9999 .9999 .9999 1.0000 1.0000 .9999 1.0000 1.0000
-12 .9998 .9999 .9999 .9999 .9999 1.0000 .9999 1.0000 1.0000
-11 .9997 .9997 .9998 .9998 .9999 .9999 .9998 .9999 1.0000
-10 .9993 .9995 .9995 .9995 .9998 .9998 .9996 .9999 1.0000
-9 .9986 .9989 .9990 .9990 .9995 .9997 .9992 .9998 .9999
-8 .9973 .9979 .9980 .9980 .9990 .9993 .9984 .9995 .9998
-7 .9945 .9957 .9961 .9961 .9980 .9986 .9967 .9990 .9997
-6 .9891 .9914 .9922 .9921 .9960 .9972 .9935 .9980 .9994
-5 .9782 .9830 .9844 .9843 .9920 .9944 .9870 .9960 .9988
-4 .9567 .9663 .9693 .9685 .9839 .9886 .9741 .9921 .9975
-3 .9146 .9340 .9403 .9372 .9675 .9770 .9487 .9841 .9950
-2 .8349 .8733 .8867 .8758 .9341 .9533 .8993 .9679 .9898
-1 .6928 .7654 .7940 .7602 .8672 .9057 .8068 .9354 .9790
0 .4714 .5916 .6510 .5629 .7391 .8145 .6458 .8702 .9558
1 .2111 .3565 .4657 .2955 .5233 .6603 .4067 .7432 .9057
2 .0394 .1285 .2755 .0743 .2505 .4527 .1539 .5212 .8002
3 .0012 .0160 .1221 .0041 .0544 .2438 .0197 .2335 .6081
4 .0000 .0002 .0296 .0000 .0025 .0860 .0003 .0410 .3522
5 .0000 .0000 .0019 .0000 .0000 .0120 .0000 .0012 .1263
6 .0000 .0000 .0000 .0000 .0000 .0002 .0000 .0000 .0171
7 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0003
8 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000
9 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000


Table: Total reflection as a function of optical depth ($\tau =2^n$) and albedo (a) for three different anisotropies (g=0, g=.5, and g=0.875) using a Henyey-Greenstein phase function. The slab ( $n_{\protect\mathrm{slab}}=1.4$) is bounded by glass slides ( $n_{\rm glass}=1.5$) and air ( $n_{\protect\mathrm{air}}=1.0$). All digits shown are significant.
  g=0 g=.5 g=.875
n a=.6 a=.9 a=.99 a=.6 a=.9 a=.99 a=.6 a=.9 a=.99
-20 .0789 .0789 .0789 .0789 .0789 .0789 .0789 .0789 .0789
-19 .0789 .0789 .0789 .0789 .0789 .0789 .0789 .0789 .0789
-18 .0789 .0789 .0790 .0789 .0789 .0789 .0789 .0789 .0789
-17 .0789 .0790 .0790 .0789 .0789 .0790 .0789 .0789 .0789
-16 .0789 .0790 .0790 .0789 .0790 .0790 .0789 .0789 .0789
-15 .0790 .0790 .0790 .0789 .0790 .0790 .0789 .0789 .0790
-14 .0790 .0790 .0790 .0789 .0790 .0790 .0789 .0789 .0790
-13 .0790 .0790 .0790 .0789 .0790 .0790 .0789 .0790 .0790
-12 .0790 .0790 .0790 .0790 .0790 .0790 .0789 .0790 .0790
-11 .0790 .0791 .0791 .0790 .0790 .0791 .0789 .0790 .0790
-10 .0790 .0792 .0793 .0790 .0791 .0792 .0789 .0790 .0790
-9 .0791 .0795 .0797 .0790 .0792 .0794 .0789 .0790 .0790
-8 .0792 .0800 .0805 .0790 .0795 .0799 .0788 .0790 .0791
-7 .0795 .0811 .0821 .0790 .0800 .0808 .0787 .0791 .0793
-6 .0801 .0833 .0852 .0791 .0811 .0826 .0786 .0792 .0797
-5 .0812 .0876 .0915 .0792 .0833 .0861 .0781 .0794 .0804
-4 .0832 .0959 .1036 .0794 .0875 .0932 .0774 .0799 .0820
-3 .0870 .1115 .1268 .0796 .0956 .1071 .0759 .0808 .0850
-2 .0931 .1391 .1693 .0799 .1102 .1333 .0730 .0825 .0910
-1 .1013 .1817 .2405 .0795 .1342 .1805 .0680 .0855 .1031
0 .1084 .2326 .3422 .0769 .1649 .2554 .0604 .0899 .1267
1 .1107 .2689 .4552 .0717 .1867 .3485 .0516 .0940 .1701
2 .1106 .2795 .5506 .0686 .1880 .4323 .0463 .0915 .2336
3 .1106 .2803 .6088 .0684 .1857 .4933 .0453 .0816 .2871
4 .1106 .2803 .6259 .0684 .1855 .5222 .0453 .0776 .3068
5 .1106 .2803 .6270 .0684 .1855 .5268 .0453 .0775 .3124
6 .1106 .2803 .6270 .0684 .1855 .5269 .0453 .0775 .3132
7 .1106 .2803 .6270 .0684 .1855 .5269 .0453 .0775 .3132
8 .1106 .2803 .6270 .0684 .1855 .5269 .0453 .0775 .3132


Table: Total transmission as a function of optical depth ($\tau =2^n$) and albedo (a) for three different anisotropies (g=0.0, g=0.5 and g=0.875) using a Henyey-Greenstein phase function. The slab ( $n_{\protect\mathrm{slab}}=1.4$) is bounded by glass slides ( $n_{\protect\mathrm{glass}}=1.5$) and air ( $n_{\protect\mathrm{air}}=1$). All digits shown are significant.
  g=0 g=.5 g=.875
n a=.6 a=.9 a=.99 a=.6 a=.9 a=.99 a=.6 a=.9 a=.99
-20 .9211 .9211 .9211 .9211 .9211 .9211 .9211 .9211 .9211
-19 .9211 .9211 .9211 .9211 .9211 .9211 .9211 .9211 .9211
-18 .9211 .9211 .9211 .9211 .9211 .9211 .9211 .9211 .9211
-17 .9211 .9211 .9211 .9211 .9211 .9211 .9211 .9211 .9211
-16 .9210 .9211 .9211 .9211 .9211 .9211 .9211 .9211 .9211
-15 .9210 .9210 .9210 .9210 .9210 .9211 .9210 .9211 .9211
-14 .9210 .9210 .9210 .9210 .9210 .9210 .9210 .9211 .9211
-13 .9210 .9210 .9210 .9210 .9210 .9210 .9210 .9210 .9211
-12 .9209 .9209 .9209 .9209 .9210 .9210 .9209 .9210 .9210
-11 .9207 .9208 .9208 .9208 .9209 .9209 .9208 .9210 .9210
-10 .9203 .9205 .9206 .9204 .9207 .9208 .9206 .9209 .9210
-9 .9195 .9200 .9202 .9198 .9203 .9205 .9202 .9207 .9209
-8 .9180 .9189 .9193 .9186 .9196 .9200 .9194 .9204 .9208
-7 .9150 .9166 .9176 .9162 .9180 .9190 .9177 .9197 .9205
-6 .9090 .9123 .9142 .9113 .9150 .9170 .9143 .9184 .9200
-5 .8971 .9036 .9075 .9017 .9091 .9129 .9077 .9158 .9189
-4 .8738 .8865 .8942 .8826 .8972 .9048 .8945 .9105 .9167
-3 .8289 .8535 .8688 .8456 .8738 .8890 .8686 .9001 .9124
-2 .7458 .7916 .8218 .7758 .8287 .8587 .8188 .8792 .9035
-1 .6032 .6829 .7414 .6521 .7448 .8030 .7269 .8378 .8856
0 .3940 .5131 .6208 .4587 .6017 .7100 .5710 .7574 .8491
1 .1673 .2970 .4697 .2251 .3966 .5793 .3486 .6115 .7766
2 .0297 .1029 .3049 .0530 .1784 .4241 .1261 .3888 .6488
3 .0009 .0126 .1453 .0028 .0377 .2503 .0153 .1548 .4734
4 .0000 .0002 .0360 .0000 .0017 .0925 .0002 .0255 .2800
5 .0000 .0000 .0023 .0000 .0000 .0130 .0000 .0007 .1031
6 .0000 .0000 .0000 .0000 .0000 .0003 .0000 .0000 .0141
7 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0003
8 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000

S. A. Prahl."Light Transport in Tissue," PhD thesis, University of Texas at Austin, 1988.