the forward problem is used to calculate the photon propagation model of the
fluorescence transmitted in the imaging space to obtain the linear relationship
between the fluorescence measurement data on the surface of the tissue and the
fluorescence distribution inside the bio-tissue. After the linear relationship is
obtained by solving the photon transfer model, various methods are used to solve
the linear model, and the distribution of fluorescence inside the imaging space is
obtained, which is called the inverse problem [52].
3.1 Photon Propagation Model
The process of transmitting fluorescence from a light source to a biological body
through a specific biological tissue is extremely complicated and includes various
physical processes such as scattering of light, inter-tissue reflection, refraction,
diffusion, and absorption. For FMT imaging, imaging is usually performed in the
visible and near-infrared optical bands, and the scattering and absorption effects of
this band of light inside the biological tissues are the main forms of our study.
Therefore, the FMT photon propagation model can be simplified to a photon
stochastic propagation model that contains only the scattering and absorption effects
without considering the reflection and refraction of different tissues. Current mainstream mathematical theory to solve these problems is mainly based on Boltzmann’s
radiative transfer equation (RTE) [58], which is equivalent to photon propagation as
transport of photon flux in a medium from particle fluctuation to energy transport
and to study transport of light energy in biological tissue problems.
In three-dimensional biological tissue, the RTE solution is transformed into a
six-dimensional space-time problem. There are few methods in solving mathematical and computer problems, and they are usually not able to directly close the
analytical solution. Moreover, because of its unknowns, it can be solved precisely
only in rare cases. Usually it cannot get a closed analytical solution. At the same
time, it is extremely difficult to solve RTE directly, while the exact solution will only
exist in rare cases. Therefore, it is common practice to replace itself with a simplified
approximation of the radiation transfer equation [59], such as diffusion equation
(DE), which is a widely used RTE-based simplified model [60–65]. It utilizes the
first-order spherical harmonic function to expand important function items in the
RTE equation and performs the approximate processing, which significantly reduces
the computational complexity and is suitable for the visible and near-infrared bands
of FMT imaging. In recent years, researchers have proposed such high-order
approximations as RTE [66–71]. Compared with diffusion equations, higher-order
approximation models can significantly improve FMT accuracy. The SN model [72],
PN model [73], and SPN model [69] are three commonly used RTE high-order
approximation models and usually give more accurate RTE solutions to the more
diffusive equations. By these approximation methods, the traditional RTE equation
can be transformed into several coupled higher-order partial differential equations
for easy calculation and solution.
Fluorescence Molecular Imaging of Medicinal Chemistry in Cancer
13
fluorescence transmitted in the imaging space to obtain the linear relationship
between the fluorescence measurement data on the surface of the tissue and the
fluorescence distribution inside the bio-tissue. After the linear relationship is
obtained by solving the photon transfer model, various methods are used to solve
the linear model, and the distribution of fluorescence inside the imaging space is
obtained, which is called the inverse problem [52].
3.1 Photon Propagation Model
The process of transmitting fluorescence from a light source to a biological body
through a specific biological tissue is extremely complicated and includes various
physical processes such as scattering of light, inter-tissue reflection, refraction,
diffusion, and absorption. For FMT imaging, imaging is usually performed in the
visible and near-infrared optical bands, and the scattering and absorption effects of
this band of light inside the biological tissues are the main forms of our study.
Therefore, the FMT photon propagation model can be simplified to a photon
stochastic propagation model that contains only the scattering and absorption effects
without considering the reflection and refraction of different tissues. Current mainstream mathematical theory to solve these problems is mainly based on Boltzmann’s
radiative transfer equation (RTE) [58], which is equivalent to photon propagation as
transport of photon flux in a medium from particle fluctuation to energy transport
and to study transport of light energy in biological tissue problems.
In three-dimensional biological tissue, the RTE solution is transformed into a
six-dimensional space-time problem. There are few methods in solving mathematical and computer problems, and they are usually not able to directly close the
analytical solution. Moreover, because of its unknowns, it can be solved precisely
only in rare cases. Usually it cannot get a closed analytical solution. At the same
time, it is extremely difficult to solve RTE directly, while the exact solution will only
exist in rare cases. Therefore, it is common practice to replace itself with a simplified
approximation of the radiation transfer equation [59], such as diffusion equation
(DE), which is a widely used RTE-based simplified model [60–65]. It utilizes the
first-order spherical harmonic function to expand important function items in the
RTE equation and performs the approximate processing, which significantly reduces
the computational complexity and is suitable for the visible and near-infrared bands
of FMT imaging. In recent years, researchers have proposed such high-order
approximations as RTE [66–71]. Compared with diffusion equations, higher-order
approximation models can significantly improve FMT accuracy. The SN model [72],
PN model [73], and SPN model [69] are three commonly used RTE high-order
approximation models and usually give more accurate RTE solutions to the more
diffusive equations. By these approximation methods, the traditional RTE equation
can be transformed into several coupled higher-order partial differential equations
for easy calculation and solution.
Fluorescence Molecular Imaging of Medicinal Chemistry in Cancer
13