Research On The Theory And Numerical Calculation Of Photoacoustic Wave Produced By A Spheroidal Biological Particle

Recently, photoacoustic imaging technology in biomedicine is developing very rapidly. What especially impressive is the dramtic improvement in the imaging spatial resolution, which makes it possible for people to obtain the photoacoustic images for individual biological particles such as red blood cells,cell nuclei, and melanosomes. Based on the fact that the geometric shapes of most of biological particles are closer to spheroids (or ellipsoids), rather than spheres (for example red blood cells are closer to oblate spheroids while the cell nuclei and the melanosomes are closer to prolate spheroids), this paper not only develops an analytic theory for describing the photoacoustic wave generation from a single spheroidal droplet but also implements the numerical calculation.In this paper, at first, the photoacoustic equation for a spheroidal droplet has been developed in the thermal confine condition starting from the inviscous flow mechanics equation.We use two totally different methods to further develop the theory:one is the geometric calculation method, developped under the condition that the spheroid droplet and its surrounding medium own identical sound speeds and mass densities; the other is the spheroid wave function method, developped under the condition that the spheroid droplet and the surrounding fluid own different sound speeds or mass densities or both. In the latter method, we consider the corresponding boundary condition.As for the geometric calculation method, this paper presents the derivation of this method based on photoacoustic waves Green’s function method. Using this method, the analytical formulas for describing the wave profile along the rotation axis of a spheroidal droplet (including a prolate spheroid and an oblate spheroid) are derived. Furthermore, characteristics of pulsed photoacoustic waves’space domain waveform and time domain waveform of a prolate spheroidal droplet, an oblate spheroidal droplet and a spherical droplet are discussed.This study is considered to be important due to the following two factors:1) Since the acoustic properties of a biological particle and its surrounding medium are usually similar, the geometric calculation method can be applied to approximately model the photoacoustic pulse waves generated by a biological particle;2) For the general cases of the discontinuity in acoustic properties beween the spheroidal droplet and its surrounding medium, this solution can still be useful for testing the standard solution.As for the spheroidal wave function method, this paper combines the separation-variable-method of solving photoacoustic Helmholtz equation in the spheroidal coordinates with the natural boundary conditions well decribed in the spheroidal coordinates. First, this paper deduces the general analytical solution of the photoacoustic waves produced by a spheroidal droplet in the frequency domain, which can be written in the weighted summation of spheroidal wave functions for every modes with the weight coefficient to be determined by the boundary conditions. And then, by taking the Fourier transform, the time-domain solution is obtained. To prove the correctness of the solutions, this paper adopts two theoretical methods:one is the asymptotical analysis method which demonstrated that the analytical solutions in three kinds of extreme cases are reduced to the analytical solutions of a spherical optically-thin droplet, an infinitely long optically-thin cylinder, and an infinite optically-thin layer respectively. This analysis also shows that the well-known theories of photoacoustic waves layer can be actually unified into the theory of photoacoustic wave generation by a spheroidal droplet. The other method is to decompose the photoacoustic waves in time domain from the point of view of the partial transmission and reflection at the droplet’s interface, which proves that the photoacoustic waves outside the spheroidal droplet can be decomposed into the superposition of continuous transmission wave, while the photoacoustic wave inside the spheroidal droplet can be expressed as the sum of the successive reflection wave plus the source term.For the numerical calculation, we apply the spheroidal function method to model the photoacoustic waves produced by the biological particle based on the existing MATLAB package for calculating the spheroid wave functions. A red blood cell is modeled as an oblate spheroid droplet and a MCF7cell nucleus is modeled as a prolate spheroidal droplet. This paper first applies the geometrical calculation method as well to calculate the photoacoustic spectrum of a red blood celland a MCF7cell nucleus, and the comparison with the spheroidal wave functions calculation proves the correctness of the matlab program. Then, the continuous photoacoustic field distribution of the red blood cells and MCF7cell nuclei at different modulation frequency are simulated by the spheroidal wave function method and the angular distribution nonuniform of them are analyzed as well based on the mode decomposition of the spheroidal wave function. Finally, the spectra of near-filed and far-filed, and the photoacoustic pulse corresponding to the far-filed spectra of the red blood cells and MCF7cell nuclei are calculated by the spheroidal wave function method. Meanwhile, the far-filed spectra at polarization angle of00,450,900are compared with equal volume spherical photoacoustic spectra, which show that the photoacoustic field distribution of the red blood cells and MCF7cell nucleus can not be modeled by that of spherical droplets unless the modulation fresquency is less than100MHz.