Study On The Analytic Solutions For The Photon Diffusion Equations And Transport Equations In Biological Tissue

Biomedical Photonics is usually viewed as the science and technology which combines biology,medicine,physics,mathematics,computer and other disciplines.As one of the most important cornerstones of Biomedical Photonics,tissue optics focuses on finding the light energy which reaches a target chromophore per unit area per unit time at certain position and developing methods or technologies by which the optical properties of tissue can be measured.So tissue optics can be applied to many biomedical applications,such as optical imaging,optical biopsy,optical health care,etc.Thus,the time-dependent or independent photons distribution in the tissue is the basic task and should be made great efforts.Neutron transport equation is the most effective mathematical model and always used to describe the neutron distribution in the medium.Many analytical and numerical methods,such as Case method,PN approximation,and FN method etc.had been reported to solve the integrodifferential equation and made great success.The transport theory became the basis of Biomedical Photonics consequently.Firstly,this paper dives into the photon diffusion equation,and the homogeneous biological tissue models are built in Cartesian,cylindrical,spherical domains with one,two or three dimensions and with infinite,semi-infinite,finite space to simulate the various shapes of tissue.Several methods,such as the method of standard basis,the method of eigenfunctions expansion,and the method of images,are applied to derive the Green’s functions of the photon diffusion equation with the Dirichlet,Neumann,and Robin boundary conditions respectively or in certain group.The comprehensive and systematic study will improve the approximation theory of photon transport.Secondly,the Case’s method is used to find the Green’s functions of photon transport integrodifferential equation in the infinite homogeneous biological tissue with isotropic scattering,general anisotropic scattering with and without azimuthal symmetry.The orthogonalities and norms of the Case’s singular eigenfunctions(CSE)are derived for discrete and continuous spectrum.The Caseology study will improve the exact theory of photon transport.Next,the Fourier transform and inversion are applied to the same cases of scattering solved by the Case’s method.In complex domain,two kinds of generalized singular eigenfunctions(GSE)and the general solution form for three-term recurrences are extended from the general anisotropic scattering with azimuthal symmetry to without azimuthal symmetry,thus the Green’s functions can be obtained by the Fourier transform inversion integral.The consistencies between the CSE solution and the Fourier transform solution are proved perfectly in a more mathematically unified manner.The consistency between the GSEs and CSEs are shown from both expression forms and intrinsic behaviors.When the GSE1 approaches the discrete eigenvalues outside of the branch cut on the real axis,it will be equivalent to the discrete CSE,and the GSE1 approaches the continuous spectrum on the branch cut,it will be equivalent to the continuous CSE.For the Green’s function,the contributions from the poles in Fourier transform inversion are equivalent to the contributions of discrete eigenvalues in Case’s method,and correspondingly,the branch cut contribution is equivalent to that of the continuous spectrum.Lastly,considering m = 0 and m≥0,it is the first time to name the(associated)Chandrasekhar polynomials of the first and second kind normatively.The Christoffel-Darboux(C-D)identities or Liouville-Ostrogradski(L-O)formulas about the(associated)Chandrasekhar the polynomials and(associated)Legendre polynomials(or functions)are derived in both complex domain and real domain(|x|≤1 and |x|>1 respectively).Considering that the two kinds of m-dependent Chandrasekhar orthogonal polynomials that play vital roles in these analytical solutions are very sensitive to the typical optical parameters of biological tissue(such as single particle albedo and anisotropy factor)as well as the degrees or orders,four numerical evaluation methods,determinant eigenvalues,forward recurrence,backward recurrence and linear system of equations,were benchmarked to find the stable,reliable and feasible numerical evaluation methods in high degrees and high orders.The guiding rules are given in the end.