3444:
3468:
2498:
3480:
22:
3456:
2171:
2944:
652:
1739:
3152:
353:
3264:. To calculate the optical broad-beam response, the impulse response of a pencil beam is convolved with the beam function. As shown by Equation 4, this is a 2-D convolution. To calculate the response of a light beam on a plane perpendicular to the z axis, the beam function (represented by a
2160:
2493:{\displaystyle I_{\phi }(r,r'')={\begin{cases}1,&{\mbox{if }}R\geq r+r'',\\{\tfrac {1}{\pi }}\cos ^{-1}\left({\tfrac {r^{2}+r''^{2}-R^{2}}{2rr''}}\right),&{\mbox{if }}\left|r-r''\right|\leq R<r+r'',\\0,&{\mbox{if }}R<\left|r+r''\right|.\end{cases}}\qquad (11)}
1900:
3368:
1157:
127:. Time invariance implies that a photon beam delayed by a given time produces a response shifted by the same delay. Linearity indicates that a given response will increase by the same amount if the input is scaled and obeys the property of
2651:
931:
2662:
131:. Translational invariance means that if a beam is shifted to a new location on the tissue surface, its response is also shifted in the same direction by the same distance. Here, only spatial convolution is considered.
3424:
L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, "Convolution for responses to a finite diameter photon beam incident on multi-layered tissues," Computer
Methods and Programs in Biomedicine 54, 141–150 (1997).
1506:
1999:
1225:
462:
1517:
2517:. Errors will result if absorption due to the first interactions is not recorded separately from absorption due to subsequent interactions. The total impulse response can be expressed in two parts:
1361:
164:
1777:
2955:
453:
408:
3421:
L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, "Monte Carlo modeling of photon transport in multi-layered tissues," Computer
Methods and Programs in Biomedicine 47, 131–146 (1995).
1405:
40:
697:
2010:
2523:
2939:{\displaystyle C(r,z)=G_{1}(0,z)S(r)+2\pi S_{0}\int _{0}^{\infty }G_{2}(r'',z)\,exp\leftI_{0e}\left({\frac {4rr''}{R^{2}}}\right)r''\,dr''.\qquad (13)}
103:. Photon transport methods can be directly used to compute broad-beam responses by distributing photons over the cross section of the beam. However,
3378:
88:
2513:
First photon-tissue interactions always occur on the z axis and hence contribute to the specific absorption or related physical quantities as a
135:
3443:
939:
705:
3347: − 1) matrices, so the time complexity is O. Comparing O and O, it is apparent that direct convolution will be faster if
1231:, it only needs to be calculated once for all depths. Thus this form of the broad-beam response is more computationally advantageous.
2656:
where the first term results from the first interactions and the second, from subsequent interactions. For a
Gaussian beam, we have
58:
1434:
647:{\displaystyle C(x,y,z)=\int _{-\infty }^{\infty }\!\int _{-\infty }^{\infty }\ G(x'',y'',z)S(x-x'',y-y'')\,dx''\,dy''.\qquad (2)}
1932:
1734:{\displaystyle C(r,z)=2\pi S(r)\int _{0}^{\infty }G(r'',z)\exp \leftI_{0}\left({\frac {4rr''}{R^{2}}}\right)r''\,dr'',\qquad (7)}
348:{\displaystyle C(x,y,z)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\ G(x-x',y-y',z)S(x',y')\,dx'\,dy'.\qquad (1)}
3434:
3393:
1895:{\displaystyle S(r')={\begin{cases}S_{0},&{\text{if }}r'\leq R\\\,0,&{\text{if }}r'>R\end{cases}}\qquad (8)}
3214:
holds. For a
Gaussian beam, no simple upper integration limits exist because it theoretically extends to infinity. At
1254:
3147:{\displaystyle C(r,z)=G_{1}(0,z)S(r)+2\pi S_{0}\int _{0}^{\infty }G_{2}(r'',z)I_{\phi }(r,r'')r''\,dr''.\qquad (14)}
3500:
1165:
1752:
128:
1373:
3324:
3257:
124:
3307: − 1). The calculation of each of these elements (except those near boundaries) includes
116:
3467:
3256:
There are two common methods used to implement discrete convolution: the definition of convolution and
2514:
2213:
1806:
3261:
96:
84:
3472:
3373:
2155:{\displaystyle C(r,z)=2\pi S_{0}\int _{0}^{\infty }G(r'',z)I_{\phi }(r,r'')r''\,dr'',\qquad (10)}
660:
415:
370:
3426:
100:
3188:
direction does not affect the convolution. To convolve reliably for physical quantities at
3460:
3397:
3369:
Radiative transfer equation and diffusion theory for photon transport in biological tissue
115:
In order for convolution to be used to calculate a broad-beam response, a system must be
154:, from a pencil beam in Cartesian space and a collimated light source with beam profile
3484:
3448:
3328:
158:, a broad-beam response can be calculated using the following 2-D convolution formula:
3494:
1245:
147:
2646:{\displaystyle C(r,z)=G_{1}(0,z){\frac {\delta (r)}{2\pi r}}+G_{2}(r,z),\qquad (12)}
3479:
3268:
matrix) is convolved with the impulse response on that plane (represented by an
1764:
143:
104:
92:
3390:
80:
134:
Responses from photon transport methods can be physical quantities such as
3284:. The calculation efficiency of these two methods depends largely on
699:
has cylindrical symmetry, its convolution integrals can be rewritten as:
76:
1152:{\displaystyle C(r,z)=\int _{0}^{\infty }G(r'',z)r''\leftdr''\qquad (4)}
3391:
Optical
Imaging Laboratory at Washington University in St. Louis (MCML)
139:
72:
926:{\displaystyle C(r,z)=\int _{0}^{\infty }\ S(r')r'\leftdr'\qquad (3)}
120:
3402:
3162:
For a top-hat beam, the upper integration limits may be bounded by
358:
Similar to 1-D convolution, 2-D convolution is commutative between
107:
can be used in certain cases to improve computational efficiency.
3331:. Using the FFT method, the major steps are the FFT and IFFT of (
1227:. Because the inner integration of Equation 4 is independent of
15:
1501:{\displaystyle S_{0}={\frac {2P_{0}}{\pi R^{2}}}.\qquad (6)}
3291:
In direct convolution, the solution matrix is of the size (
2476:
1994:{\displaystyle S_{0}={\frac {P_{0}}{\pi R^{2}}}.\qquad (9)}
1878:
36:
3248:
can be used approximately for
Gaussian beams as well.
2434:
2365:
2291:
2259:
2225:
1378:
3432:
2958:
2665:
2526:
2174:
2013:
1935:
1780:
1520:
1437:
1376:
1257:
1168:
942:
708:
663:
465:
418:
373:
167:
3192:in response to a top-hat beam, we must ensure that
95:incident on a scattering medium are referred to as
31:
may be too technical for most readers to understand
3146:
2938:
2645:
2492:
2154:
1993:
1894:
1733:
1500:
1399:
1355:
1219:
1151:
925:
691:
646:
447:
402:
347:
3222:, a Gaussian beam and a top-hat beam of the same
3199:in photon transport methods is large enough that
1414:denotes the intensity at the center of the beam.
511:
3233:have comparable convolution results. Therefore,
1356:{\displaystyle S(r')=S_{0}\exp \left.\qquad (5)}
8:
3416:Biomedical Optics: Principles and Imaging.
1220:{\displaystyle r'={\sqrt {x'^{2}+y'^{2}}}}
3184:. Thus, the limited grid coverage in the
3119:
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3055:
3045:
3040:
3030:
2984:
2957:
2911:
2891:
2869:
2856:
2841:
2814:
2788:
2762:
2752:
2747:
2737:
2691:
2664:
2609:
2573:
2552:
2525:
2433:
2364:
2329:
2315:
2298:
2290:
2274:
2258:
2224:
2208:
2179:
2173:
2127:
2093:
2060:
2055:
2045:
2012:
2004:Substituting Eq. 8 into Eq. 4, we obtain
1969:
1955:
1949:
1940:
1934:
1856:
1847:
1824:
1813:
1801:
1779:
1706:
1686:
1664:
1654:
1639:
1620:
1569:
1564:
1519:
1511:Substituting Eq. 5 into Eq. 4, we obtain
1476:
1461:
1451:
1442:
1436:
1396:
1387:
1377:
1375:
1329:
1310:
1282:
1256:
1208:
1190:
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1065:
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1024:
1019:
973:
968:
941:
885:
832:
831:
813:
807:
787:
782:
739:
734:
707:
688:
662:
619:
607:
524:
516:
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497:
464:
444:
417:
399:
372:
320:
308:
225:
217:
207:
199:
166:
59:Learn how and when to remove this message
43:, without removing the technical details.
150:. Given a specific physical quantity,
3439:
3379:Monte Carlo method for photon transport
3355:, but the FFT method will be faster if
3323: − 1 additions, so the
2312:
1912:denotes the intensity inside the beam.
1205:
1187:
1400:{\displaystyle {\tfrac {1}{e^{2}}}\,}
41:make it understandable to non-experts
7:
3385:Links to other Monte Carlo resources
1248:, the intensity profile is given by
3339: − 1) × (
3299: − 1) × (
1919:is related to the total beam power
3046:
2753:
2061:
1570:
974:
740:
525:
520:
506:
501:
226:
221:
208:
203:
14:
3478:
3466:
3454:
3442:
3260:(FFT and IFFT) according to the
657:Because the broad-beam response
20:
3134:
2926:
2633:
2480:
2142:
1981:
1882:
1721:
1488:
1343:
1139:
913:
634:
335:
3288:, the size of the light beam.
3141:
3135:
3108:
3091:
3078:
3061:
3014:
3008:
3002:
2990:
2974:
2962:
2933:
2927:
2785:
2768:
2721:
2715:
2709:
2697:
2681:
2669:
2640:
2634:
2627:
2615:
2585:
2579:
2570:
2558:
2542:
2530:
2504:Errors in numerical evaluation
2487:
2481:
2202:
2185:
2149:
2143:
2116:
2099:
2086:
2069:
2029:
2017:
1988:
1982:
1889:
1883:
1795:
1784:
1771:, the source function becomes
1728:
1722:
1595:
1578:
1557:
1551:
1536:
1524:
1495:
1489:
1421:is related to the total power
1350:
1344:
1272:
1261:
1146:
1140:
999:
982:
958:
946:
920:
914:
762:
751:
724:
712:
685:
667:
641:
635:
604:
570:
564:
536:
487:
469:
342:
336:
305:
283:
277:
237:
189:
171:
87:), are commonly used to model
1:
3252:Implementation of convolution
71:Photon transport theories in
2949:For a top-hat beam, we have
111:General convolution formulas
3403:Oregon Medical Laser Center
3258:fast Fourier transformation
366:with a change of variables
89:light propagation in tissue
3517:
692:{\displaystyle C(x,y,z)\,}
448:{\displaystyle y''=y-y'\,}
403:{\displaystyle x''=x-x'\,}
3414:L.-H. Wang and H.-I. Wu.
1753:modified Bessel function
1407:radius of the beam, and
3148:
2940:
2647:
2494:
2156:
1995:
1896:
1735:
1502:
1401:
1357:
1221:
1153:
927:
693:
648:
449:
404:
349:
91:. The responses to a
3359:is relatively large.
3351:is much smaller than
3149:
2941:
2648:
2495:
2157:
1996:
1897:
1736:
1503:
1402:
1358:
1222:
1154:
928:
694:
649:
450:
405:
350:
125:translation invariant
3315:multiplications and
3218: >>
2956:
2663:
2524:
2515:Dirac delta function
2172:
2011:
1933:
1778:
1751:is the zeroth-order
1518:
1435:
1374:
1255:
1235:Common beam profiles
1166:
940:
706:
661:
463:
416:
371:
165:
3262:convolution theorem
3244: −
3210: −
3180: −
3050:
2757:
2065:
1574:
1032:
978:
795:
744:
529:
510:
230:
212:
3396:2009-12-02 at the
3374:Monte Carlo method
3276:matrix). Normally
3144:
3036:
2936:
2743:
2643:
2509:First interactions
2490:
2475:
2438:
2369:
2353:
2268:
2229:
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1991:
1892:
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1731:
1560:
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1394:
1353:
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85:Monte Carlo method
3501:Scattering theory
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1323:
1215:
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101:impulse responses
97:Green's functions
69:
68:
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3446:
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3427:Download article
3280:is greater than
3158:Truncation error
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2894:
2890:
2884:
2881:
2877:
2874:
2868:
2862:
2859:
2855:
2850:
2844:
2839:
2834:
2830:
2827:
2823:
2820:
2813:
2808:
2805:
2801:
2797:
2794:
2791:
2787:
2784:
2781:
2777:
2774:
2770:
2765:
2761:
2755:
2750:
2746:
2740:
2736:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2694:
2690:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2654:
2653:
2642:
2639:
2636:
2632:
2629:
2626:
2623:
2620:
2617:
2612:
2608:
2604:
2598:
2595:
2592:
2587:
2584:
2581:
2578:
2572:
2569:
2566:
2563:
2560:
2555:
2551:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2510:
2507:
2505:
2502:
2501:
2500:
2489:
2486:
2483:
2477:
2472:
2468:
2463:
2460:
2456:
2453:
2449:
2445:
2442:
2432:
2430:
2427:
2424:
2423:
2420:
2416:
2413:
2409:
2406:
2403:
2400:
2397:
2393:
2388:
2385:
2381:
2378:
2374:
2363:
2361:
2357:
2349:
2346:
2342:
2339:
2332:
2328:
2324:
2318:
2314:
2310:
2306:
2301:
2297:
2289:
2285:
2280:
2277:
2273:
2266:
2263:
2257:
2256:
2253:
2249:
2246:
2242:
2239:
2236:
2233:
2223:
2221:
2218:
2215:
2214:
2212:
2207:
2204:
2200:
2197:
2193:
2190:
2187:
2182:
2178:
2163:
2162:
2151:
2148:
2145:
2141:
2137:
2134:
2130:
2125:
2122:
2118:
2114:
2111:
2107:
2104:
2101:
2096:
2092:
2088:
2085:
2082:
2078:
2075:
2071:
2068:
2063:
2058:
2054:
2048:
2044:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2002:
2001:
1990:
1987:
1984:
1980:
1972:
1968:
1964:
1958:
1954:
1948:
1943:
1939:
1923:
1916:
1909:
1903:
1902:
1891:
1888:
1885:
1879:
1874:
1871:
1867:
1864:
1855:
1853:
1850:
1846:
1845:
1842:
1839:
1835:
1832:
1823:
1821:
1816:
1812:
1808:
1807:
1805:
1800:
1797:
1793:
1790:
1786:
1783:
1760:
1757:
1748:
1742:
1741:
1730:
1727:
1724:
1720:
1716:
1713:
1709:
1704:
1701:
1696:
1689:
1685:
1679:
1676:
1672:
1669:
1663:
1657:
1653:
1648:
1642:
1637:
1632:
1628:
1625:
1619:
1614:
1611:
1607:
1603:
1600:
1597:
1594:
1591:
1587:
1584:
1580:
1577:
1572:
1567:
1563:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1509:
1508:
1497:
1494:
1491:
1487:
1479:
1475:
1471:
1464:
1460:
1456:
1450:
1445:
1441:
1425:
1418:
1411:
1390:
1386:
1382:
1364:
1363:
1352:
1349:
1346:
1342:
1338:
1332:
1327:
1322:
1318:
1315:
1309:
1304:
1301:
1297:
1293:
1290:
1285:
1281:
1277:
1274:
1270:
1267:
1263:
1260:
1241:
1238:
1236:
1233:
1211:
1207:
1203:
1199:
1193:
1189:
1185:
1179:
1175:
1172:
1160:
1159:
1148:
1145:
1142:
1137:
1134:
1130:
1126:
1121:
1118:
1114:
1109:
1103:
1100:
1096:
1093:
1090:
1086:
1083:
1079:
1076:
1073:
1068:
1061:
1058:
1054:
1049:
1045:
1039:
1035:
1030:
1027:
1022:
1018:
1013:
1008:
1005:
1001:
998:
995:
991:
988:
984:
981:
976:
971:
967:
963:
960:
957:
954:
951:
948:
945:
934:
933:
922:
919:
916:
911:
908:
904:
900:
895:
892:
888:
883:
879:
876:
870:
867:
863:
860:
857:
853:
850:
846:
843:
840:
835:
828:
825:
821:
816:
812:
805:
801:
793:
790:
785:
781:
776:
771:
768:
764:
760:
757:
753:
750:
742:
737:
733:
729:
726:
723:
720:
717:
714:
711:
687:
684:
681:
678:
675:
672:
669:
666:
655:
654:
643:
640:
637:
633:
629:
626:
622:
617:
614:
610:
606:
602:
599:
595:
592:
589:
585:
582:
578:
575:
572:
569:
566:
563:
560:
556:
553:
549:
545:
542:
538:
535:
527:
522:
519:
515:
508:
503:
500:
496:
492:
489:
486:
483:
480:
477:
474:
471:
468:
442:
439:
435:
432:
429:
425:
422:
397:
394:
390:
387:
384:
380:
377:
356:
355:
344:
341:
338:
334:
330:
327:
323:
318:
315:
311:
307:
303:
300:
296:
292:
289:
285:
282:
279:
276:
273:
269:
266:
262:
259:
256:
252:
249:
245:
242:
239:
236:
228:
223:
220:
216:
210:
205:
202:
198:
194:
191:
188:
185:
182:
179:
176:
173:
170:
117:time invariant
112:
109:
67:
66:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
3513:
3502:
3499:
3498:
3496:
3486:
3481:
3476:
3474:
3469:
3464:
3462:
3452:
3450:
3445:
3440:
3436:
3428:
3423:
3420:
3417:
3413:
3412:
3408:
3404:
3401:
3399:
3395:
3392:
3389:
3388:
3384:
3380:
3377:
3375:
3372:
3370:
3367:
3366:
3362:
3360:
3358:
3354:
3350:
3346:
3343: +
3342:
3338:
3335: +
3334:
3330:
3326:
3322:
3319: ×
3318:
3314:
3311: ×
3310:
3306:
3303: +
3302:
3298:
3295: +
3294:
3289:
3287:
3283:
3279:
3275:
3271:
3267:
3263:
3259:
3251:
3249:
3247:
3240:
3237: ≤
3236:
3229:
3225:
3221:
3217:
3213:
3206:
3203: ≤
3202:
3195:
3191:
3187:
3183:
3176:
3173: ≤
3172:
3165:
3157:
3138:
3131:
3127:
3124:
3120:
3115:
3112:
3104:
3101:
3097:
3094:
3086:
3082:
3075:
3072:
3068:
3065:
3056:
3052:
3041:
3037:
3031:
3027:
3023:
3020:
3017:
3011:
3005:
2999:
2996:
2993:
2985:
2981:
2977:
2971:
2968:
2965:
2959:
2952:
2951:
2950:
2930:
2923:
2919:
2916:
2912:
2907:
2904:
2899:
2892:
2888:
2882:
2879:
2875:
2872:
2866:
2860:
2857:
2853:
2848:
2842:
2837:
2832:
2828:
2825:
2821:
2818:
2811:
2806:
2803:
2799:
2795:
2792:
2789:
2782:
2779:
2775:
2772:
2763:
2759:
2748:
2744:
2738:
2734:
2730:
2727:
2724:
2718:
2712:
2706:
2703:
2700:
2692:
2688:
2684:
2678:
2675:
2672:
2666:
2659:
2658:
2657:
2637:
2630:
2624:
2621:
2618:
2610:
2606:
2602:
2596:
2593:
2590:
2582:
2576:
2567:
2564:
2561:
2553:
2549:
2545:
2539:
2536:
2533:
2527:
2520:
2519:
2518:
2516:
2508:
2503:
2484:
2470:
2466:
2461:
2458:
2454:
2451:
2447:
2443:
2440:
2428:
2425:
2418:
2414:
2411:
2407:
2404:
2401:
2398:
2395:
2391:
2386:
2383:
2379:
2376:
2372:
2359:
2355:
2347:
2344:
2340:
2337:
2330:
2326:
2322:
2316:
2308:
2304:
2299:
2295:
2287:
2283:
2278:
2275:
2271:
2264:
2261:
2251:
2247:
2244:
2240:
2237:
2234:
2231:
2219:
2216:
2210:
2205:
2198:
2195:
2191:
2188:
2180:
2176:
2168:
2167:
2166:
2146:
2139:
2135:
2132:
2128:
2123:
2120:
2112:
2109:
2105:
2102:
2094:
2090:
2083:
2080:
2076:
2073:
2066:
2056:
2052:
2046:
2042:
2038:
2035:
2032:
2026:
2023:
2020:
2014:
2007:
2006:
2005:
1985:
1978:
1970:
1966:
1962:
1956:
1952:
1946:
1941:
1937:
1929:
1928:
1927:
1922:
1915:
1908:
1886:
1872:
1869:
1865:
1862:
1851:
1848:
1840:
1837:
1833:
1830:
1819:
1814:
1810:
1803:
1798:
1791:
1788:
1781:
1774:
1773:
1772:
1770:
1766:
1758:
1756:
1754:
1747:
1725:
1718:
1714:
1711:
1707:
1702:
1699:
1694:
1687:
1683:
1677:
1674:
1670:
1667:
1661:
1655:
1651:
1646:
1640:
1635:
1630:
1626:
1623:
1617:
1612:
1609:
1605:
1601:
1598:
1592:
1589:
1585:
1582:
1575:
1565:
1561:
1554:
1548:
1545:
1542:
1539:
1533:
1530:
1527:
1521:
1514:
1513:
1512:
1492:
1485:
1477:
1473:
1469:
1462:
1458:
1454:
1448:
1443:
1439:
1431:
1430:
1429:
1424:
1417:
1410:
1388:
1384:
1380:
1369:
1347:
1340:
1336:
1330:
1325:
1320:
1316:
1313:
1307:
1302:
1299:
1295:
1291:
1288:
1283:
1279:
1275:
1268:
1265:
1258:
1251:
1250:
1249:
1247:
1246:Gaussian beam
1240:Gaussian beam
1239:
1234:
1232:
1230:
1209:
1201:
1197:
1191:
1183:
1177:
1173:
1170:
1143:
1135:
1132:
1128:
1124:
1119:
1116:
1112:
1107:
1101:
1098:
1094:
1091:
1088:
1084:
1081:
1077:
1074:
1071:
1066:
1059:
1056:
1052:
1047:
1043:
1037:
1033:
1028:
1025:
1020:
1016:
1011:
1006:
1003:
996:
993:
989:
986:
979:
969:
965:
961:
955:
952:
949:
943:
936:
935:
917:
909:
906:
902:
898:
893:
890:
886:
881:
877:
874:
868:
865:
861:
858:
855:
851:
848:
844:
841:
838:
833:
826:
823:
819:
814:
810:
803:
799:
791:
788:
783:
779:
774:
769:
766:
758:
755:
748:
735:
731:
727:
721:
718:
715:
709:
702:
701:
700:
682:
679:
676:
673:
670:
664:
638:
631:
627:
624:
620:
615:
612:
608:
600:
597:
593:
590:
587:
583:
580:
576:
573:
567:
561:
558:
554:
551:
547:
543:
540:
533:
517:
513:
498:
494:
490:
484:
481:
478:
475:
472:
466:
459:
458:
457:
440:
437:
433:
430:
427:
423:
420:
395:
392:
388:
385:
382:
378:
375:
365:
361:
339:
332:
328:
325:
321:
316:
313:
309:
301:
298:
294:
290:
287:
280:
274:
271:
267:
264:
260:
257:
254:
250:
247:
243:
240:
234:
218:
214:
200:
196:
192:
186:
183:
180:
177:
174:
168:
161:
160:
159:
157:
153:
149:
148:transmittance
145:
141:
137:
132:
130:
129:superposition
126:
122:
118:
110:
108:
106:
102:
98:
94:
90:
86:
83:(such as the
82:
78:
74:
63:
60:
52:
42:
38:
32:
29:This article
27:
18:
17:
3415:
3356:
3352:
3348:
3344:
3340:
3336:
3332:
3320:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3290:
3285:
3281:
3277:
3273:
3269:
3265:
3255:
3245:
3238:
3234:
3227:
3223:
3219:
3215:
3211:
3204:
3200:
3193:
3189:
3185:
3181:
3174:
3170:
3169:, such that
3163:
3161:
2948:
2655:
2512:
2164:
2003:
1920:
1913:
1906:
1904:
1768:
1765:top-hat beam
1762:
1759:Top-hat beam
1745:
1743:
1510:
1422:
1415:
1408:
1370:denotes the
1367:
1365:
1243:
1228:
1161:
656:
363:
359:
357:
155:
151:
133:
114:
70:
55:
49:October 2021
46:
30:
3473:Mathematics
3418:Wiley 2007.
144:reflectance
105:convolution
93:pencil beam
3409:References
1767:of radius
136:absorption
81:Statistics
3087:ϕ
3047:∞
3038:∫
3024:π
2826:−
2804:−
2754:∞
2745:∫
2731:π
2594:π
2577:δ
2396:≤
2380:−
2323:−
2284:
2276:−
2265:π
2235:≥
2181:ϕ
2095:ϕ
2062:∞
2053:∫
2039:π
1963:π
1838:≤
1610:−
1602:
1571:∞
1562:∫
1546:π
1470:π
1300:−
1292:
1117:ϕ
1099:ϕ
1072:−
1029:π
1017:∫
975:∞
966:∫
891:ϕ
866:ϕ
839:−
792:π
780:∫
741:∞
732:∫
594:−
577:−
526:∞
521:∞
518:−
514:∫
507:∞
502:∞
499:−
495:∫
434:−
389:−
261:−
244:−
227:∞
222:∞
219:−
215:∫
209:∞
204:∞
201:−
197:∫
3495:Category
3461:Medicine
3394:Archived
3363:See also
3128:″
3116:″
3105:″
3069:″
2920:″
2908:″
2883:″
2822:″
2776:″
2462:″
2436:if
2415:″
2387:″
2367:if
2348:″
2313:″
2248:″
2227:if
2199:″
2136:″
2124:″
2113:″
2077:″
1866:′
1858:if
1834:′
1826:if
1792:′
1715:″
1703:″
1678:″
1627:″
1586:″
1317:′
1269:′
1206:′
1188:′
1174:′
1136:″
1120:″
1102:″
1085:″
1060:″
1007:″
990:″
910:′
894:′
869:′
852:′
827:′
770:′
759:′
628:″
616:″
601:″
584:″
555:″
544:″
441:′
424:″
396:′
379:″
329:′
317:′
302:′
291:′
268:′
251:′
152:G(x,y,z)
77:Medicine
3485:Science
3449:Physics
3435:Portals
140:fluence
73:Physics
35:Please
2165:where
1905:where
1763:For a
1744:where
1366:Here,
1244:For a
1162:where
797:
746:
531:
232:
156:S(x,y)
123:, and
121:linear
79:, and
3266:b × b
146:, or
3226:and
2444:<
2402:<
1870:>
411:and
362:and
3327:is
3242:max
3208:max
3197:max
3178:max
3167:max
2272:cos
1926:by
1599:exp
1428:by
1289:exp
99:or
39:to
3497::
3272:×
3139:14
2931:13
2638:12
2485:11
2147:10
1755:.
456::
142:,
138:,
119:,
75:,
3437::
3429:.
3357:b
3353:a
3349:b
3345:b
3341:a
3337:b
3333:a
3329:O
3321:b
3317:b
3313:b
3309:b
3305:b
3301:a
3297:b
3293:a
3286:b
3282:b
3278:a
3274:a
3270:a
3246:R
3239:r
3235:r
3231:0
3228:S
3224:R
3220:R
3216:r
3212:R
3205:r
3201:r
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