解点源在无限大各向同性介质中辐射传输方程的P3近似

Analytical solution of P3 approximation to radiative transfer equation for an infinite homogenous media

辐射传输方程在球坐标下的P3近似是一个非线性偏微分方程组,其齐次解为球Bessel函数.需要将球Bessel函数分解为指数函数,才能用参数变异法求出它的特解.除r=0外,可以利用Marshak近似边界条件确定常数,但考虑球Bessel函数在r=0的奇异性,无法利用Marshak和其它近似边界条件,因此我们直接利用能量守恒以及当介质的吸收系数比约化散射系数小得多时P3近似等于P1近似这个特点来确定全解中的常数.研究辐射传输方程的P3近似理论的解析解可以为测量大吸收和小间距的生物组织光学属性提供理论依据.

P3 approximation to raditive transfer equation in the spherical coordinate is nonlinear partial differential equation system, which its solution is spherical Bessel function. After breaking down the spherical Bessel function, we provide the particular solution of it by method of variation of parameters. Especially, since full solution discontinues at r = 0, the approximation boundary condition of Marshak and the other does not satisfy, therefore, we determine the constants of full solution by direct application of energy conservation, and by the fact that P3 approximation reduces to P1 approximation when absorption coefficient is much lower than reduced scattering coefficient. Analytic solution of P3 approximation to radiative transfer equation can provide theory basis for measuring the optical property of high absorption tissue at small source-detector separation.