Extended Finite Element Method with Simplified Spherical Harmonics Approximation for the Forward Model of Optical Molecular Imaging

<table>Surface detector readings <svg height="16.575001" id="M182" style="vertical-align:-1.97925pt" version="1.1" viewbox="0 0 15.5 16.575001" width="15.5" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,14.05)"><path d="M434 650l-5 -28q-60 -8 -75 -20.5t-25 -69.5l-71 -395q-20 -107 -57.5 -168t-106.5 -94q-40 -18 -61 -20l-10 32q77 23 113 108q21 50 41 160l68 377q10 57 -1.5 69.5t-76.5 20.5l6 28h261z" id="x1D43D"></path></g>
<g transform="matrix(.012,-0,0,-.012,7.825,5.887)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g>
</svg> using MC method, XFEM, and FEM at the sampling points. Green line denotes the results of MC method, red and blue lines are that of FEM and XFEM with <svg height="15.65" id="M183" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 23.887501 15.65" width="23.887501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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t30 -51.5t49 -42.5t67 -16q56 0 88 32.5t32 87.5q0 51 -31.5 82t-98.5 67q-80 44 -110 73q-55 53 -55 124q0 75 56 126.5t150 51.5q53 0 126 -23z" id="x53"></path></g><g transform="matrix(.017,-0,0,-.017,7.916,11.362)"><path d="M46 650h247q112 0 169 -41q64 -49 64 -141q0 -91 -56.5 -143t-138.5 -62q-23 -3 -45 -1l-74 19v-157q0 -63 15.5 -77t83.5 -19v-28h-271v28q60 5 74 20t14 77v400q0 65 -12.5 79t-69.5 18v28zM212 559v-245q31 -12 73 -12q59 0 102 38t43 124q0 77 -42.5 115.5
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<g transform="matrix(.012,-0,0,-.012,17.488,15.45)"><path d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
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</svg> approximation.</table>

Computational and Mathematical Methods in Medicine

fig7

Figure 7

Figure 7: Extended Finite Element Method with Simplified Spherical Harmonics Approximation for the Forward Model of Optical Molecular Imaging 