With the development of molecular imaging, Cherenkov optical imaging technology has been widely concerned. Most studies regard the partial boundary flux as a stochastic variable and reconstruct images based on the steady-state diffusion equation. In this paper, time-variable will be considered and the Cherenkov radiation emission process will be regarded as a stochastic process. Based on the original steady-state diffusion equation, we first propose a stochastic partial differential equation model. The numerical solution to the stochastic partial differential model is carried out by using the finite element method. When the time resolution is high enough, the numerical solution of the stochastic diffusion equation is better than the numerical solution of the steady-state diffusion equation, which may provide a new way to alleviate the problem of Cherenkov luminescent imaging quality. In addition, the process of generating Cerenkov and penetrating ^{18}

Molecular imaging has developed rapidly since the 21st century. Currently available molecular imaging techniques include optical imaging, magnetic resonance imaging (MRI), positron emission tomography (PET), single-photon emission computed tomography (SPECT), and other nuclear medical imaging and ultrasonic molecular imaging [

Cherenkov optical bio-tomography is based on the Cherenkov transmission characteristics in the tissue inversion reconstruction. Different tissue and organs have different scattering and absorption of Cherenkov light. Heterogeneous models based on different optical properties of different tissues can simulate the transmission of Cherenkov

In order to adapt to this Brownian motion, we firstly propose a stochastic partial differential equation model by introducing stochastic term in time. Then we simulate the numerical solution of the stochastic partial differential model. Furthermore, we compare the numerical solution of the stochastic partial differential equation with the original steady-state diffusion equation. Finally, we compare the numerical solution of the stochastic partial differential equation with numerical simulation results of the Cherenkov effect, which is obtained by the GEANT4 software.

This paper is organized as follows. In

The original form of the time-dependent diffusion equation [

Most image reconstruction methods are based on the below steady-state diffusion equation:

_{a} is the absorption coefficient and µ_{s} is scattering coefficient,

As the boundary condition of the stochastic partial differential equation, the simplest is to use homogeneous boundary conditions, which assumes the photon cannot be (

As for the numerical solution of the stochastic differential equation, Yan [

In

Notice that _{n} =

_{h} : _{h} → _{h} is the discrete analogure of

_{h} : ^{2}(Ω) → _{h} is defined by

Let

Denote by _{n}). The backward Euler method is

We can rewrite

^{n} and Φ (t_{n}) be respectively the solutions of

Φ_{0} ∈ ^{2} (Ω), 0 ≤ γ < β ≤ 1, then there exists a constant _{n} ∈ [0,

By the definition of the mild solution of ^{−t A},

Defining _{n} = Φ^{n} − Φ (_{n}) and _{n} = _{h} − _{n}), we have

For _{1}, using lemma 2.8 in [

For _{2}, we have

For I,_{21}, noticing that

For _{22}, in a similar way, we have

Thus

The proof of Theorem 2.1 is completed.

By using the finite element analysis method [_{h} are obtained by piecewise linear interpolation, element analysis and total synthesis, and then the numerical solution of the stochastic partial differential equation is solved by the

Finite element method (FEM) is an effective method for solving the numerical solution of partial differential equations. For the stochastic partial differential equations of the mixed boundary conditions, such as Robin boundary condition in this paper, firstly, using the finite element approximation theory, the three-dimensional spatial variables of stochastic parabolic equation are discretized, and the space is divided into several positive tetrahedron units; and then the backward Euler method is used to complete the discretization of the time variable. The stochastic process W is approximated by the Wiener process, and the finite element approximate solution of the original stochastic parabolic equation can be solved.

Furthermore, it is the expectation of the solution of stochastic parabolic equation that is influenced by stochastic factors. Therefore, the finite element numerical solution of stochastic parabolic equation must undergo repeated experiments to approximate the expectancy by using the average value of many numerical simulation results.

In this paper, we do a numerical simulation of a homogeneous model, which is desirable for the corresponding optical parameters of muscle. It is found in that [_{a} take 0.01 ^{−1}, and the anisotropy coefficient

In

For the steady state equation and the stochastic partial differential equation, we take Robin boundary condition. The steady state equation has no initial value, while the stochastic partial differential equation sets the initial value of the ^{1} module as 10^{6}, the initial value of

Take a straight line, for example,

For stochastic partial differential equations with stochastic terms, let the time range

With

Equation | The number of son tetrahedras | Relative error |
---|---|---|

5381 | 0.0054 | |

Stochastic partial differential equation | 6732 | 0.0045 |

10,291 | 0.0027 | |

5381 | 2.2285e-04 | |

Steady-state diffusion equations | 6732 | 1.1403e-04 |

10,291 | 1.4629e-04 |

The next step is to study the relation between the Cherenkov imaging process of the stochastic partial differential equation and Monte Carlo simulation.

In order to compare the effect of the stochastic partial differential equation and the steady-state diffusion equation on the Cherenkov imaging process, simulation software is an economical solution. There is a large amount of Monte Carlo simulation packages available. Among these codes, GEANT4 is the most commonly used option for Cherenkov, partly because of its flexibility in the description of complex detectors and its accurate physics models. For the research of Cherenkov, GEANT4 can simulate the physical process of photon and charged particles in matter, and GEANT4 has reliable electromagnetic physical model and flexible detector design, which is the most preferred simulation tool [

Glaster et al. [^{18}

On this basis, we designed the following geometric model (^{18} ^{7}. The particle energy is set to 500

The distance between the detector and the radiation source is 1

Cherenkov spectrum is a special kind of continuous spectrum of visible light, the wavelength range between 300–750 ^{7}, so the initial energy was calculated to be about

It is also known from ^{2} area, which can be used to calculate the curve of energy changing with time at any detector. A curve in which the energy of the detector received by the GEANT4 Simulation (2,10,1) is changed over time. The results obtained from the GEANT4 simulation are compared with those of the numerical solution of the diffusion theory in the form of stochastic partial differential equations (

In this study, we propose and study the numerical solution of the stochastic partial differential, based on the GEANT4 simulation of the Cherenkov of ^{18}

It is known from

When the time resolution is high enough, the numerical solution of the stochastic diffusion equation is better than the numerical solution of the steady-state diffusion equation, which may provide a new way to alleviate the problem of Cherenkov luminescent imaging quality. This study shows that the form of stochastic partial differential equation is more helpful to simulate the radiation-induced optical transmission in biological media.

We would like to thank the reviewer for the many useful comments.

_{2}gas with an efficient method