3.2 Forward Problem-Solving
The linear relationship between the measured data on the surface of the imaging area
and the internal fluorescence distribution in the imaging area based on the photon
propagation model is the core of the FMT forward problem. In recent years,
researchers have proposed various mathematical solution methods including the
analytic method, statistical method, and numerical analysis method to solve the
forward problem of FMT [52]. Numerical analysis method is the main solving
method currently used in optical molecular imaging reconstruction. Its computational efficiency is high and its applicability is wide. Numerical analysis methods
include the finite difference method (FDM) [70], boundary element method (BEM),
finite element method (FEM) [74], and meshless method (MM) [75]. FDM uses
equidistant grid points and regular grids to solve the forward problem, which is more
efficient than irregular grids. However, FDM has difficulty in dealing with geometrically complex imaging spaces and boundary conditions. In contrast, FEM is the
mainstream solution to FMT forward problems in recent years. The main advantage
of FEM is its effectiveness in dealing with complex geometric problems. In addition,
the system matrices obtained by FEM are usually sparse and positive definite, which
leads to a more stable solution and high computational efficiency, which is also
beneficial to FMT reconstruction [76, 77]. However, the main drawback of FEM is
that it is difficult to generate the FEM grid. In contrast, BEM only needs to discretize
the imaging surface and the boundaries of the heterogeneous tissue within the space
without the need to mesh the entire imaging space. Therefore, compared with FEM,
BEM can effectively reduce the computational dimension and complexity to
improve computational efficiency. However, fast and stable 3D mesh generation
for complex geometry problems remains a challenging issue. In order to overcome
the problem of 3D mesh generation, An et al. proposed a meshless method and
applied it to solve the forward problem of FMT [78]. The method only needs to
obtain nodes that are relatively independent from each other to discretize the imaging
space and does not require a cumbersome gridding process.
3.3 Inverse Problem-Solving
In FMT preclinical and clinical trials, the fluorescence signal is usually only measured from the imaging surface. However, the dimension of the measurement data on
the imaging space surface is usually much less than the number of internal nodes in
the imaging space. Therefore, the inverse problem of FMT is ill-conditioned
[55]. Moreover, because of the high scattering properties of photons in the imaging
space, the inverse problem is also ill-posed, and it is difficult to find the exact
solution [79–81]. At the same time, the noise generated during the experiment also
affects the accuracy of the FMT reconstruction [82].
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J. Tian et al.
The linear relationship between the measured data on the surface of the imaging area
and the internal fluorescence distribution in the imaging area based on the photon
propagation model is the core of the FMT forward problem. In recent years,
researchers have proposed various mathematical solution methods including the
analytic method, statistical method, and numerical analysis method to solve the
forward problem of FMT [52]. Numerical analysis method is the main solving
method currently used in optical molecular imaging reconstruction. Its computational efficiency is high and its applicability is wide. Numerical analysis methods
include the finite difference method (FDM) [70], boundary element method (BEM),
finite element method (FEM) [74], and meshless method (MM) [75]. FDM uses
equidistant grid points and regular grids to solve the forward problem, which is more
efficient than irregular grids. However, FDM has difficulty in dealing with geometrically complex imaging spaces and boundary conditions. In contrast, FEM is the
mainstream solution to FMT forward problems in recent years. The main advantage
of FEM is its effectiveness in dealing with complex geometric problems. In addition,
the system matrices obtained by FEM are usually sparse and positive definite, which
leads to a more stable solution and high computational efficiency, which is also
beneficial to FMT reconstruction [76, 77]. However, the main drawback of FEM is
that it is difficult to generate the FEM grid. In contrast, BEM only needs to discretize
the imaging surface and the boundaries of the heterogeneous tissue within the space
without the need to mesh the entire imaging space. Therefore, compared with FEM,
BEM can effectively reduce the computational dimension and complexity to
improve computational efficiency. However, fast and stable 3D mesh generation
for complex geometry problems remains a challenging issue. In order to overcome
the problem of 3D mesh generation, An et al. proposed a meshless method and
applied it to solve the forward problem of FMT [78]. The method only needs to
obtain nodes that are relatively independent from each other to discretize the imaging
space and does not require a cumbersome gridding process.
3.3 Inverse Problem-Solving
In FMT preclinical and clinical trials, the fluorescence signal is usually only measured from the imaging surface. However, the dimension of the measurement data on
the imaging space surface is usually much less than the number of internal nodes in
the imaging space. Therefore, the inverse problem of FMT is ill-conditioned
[55]. Moreover, because of the high scattering properties of photons in the imaging
space, the inverse problem is also ill-posed, and it is difficult to find the exact
solution [79–81]. At the same time, the noise generated during the experiment also
affects the accuracy of the FMT reconstruction [82].
14
J. Tian et al.