]> Time-domain Imaging

Time-domain Imaging

TD2pt() and TD3pt() calculate the unperturbed fluence and the forward matrix respectively for a time-domain imager in the first-Born approximation. These routines are safe to call if you in advance that you are only calculating a time-domain forward problem.

See genBornMat() for a description of the muVec flags.

Function Summary

Syntax:	Phi0 = TD2pt(SD, Medium, MeasList);
Inputs:	SD	SD structure
	Medium	Medium structure
	MeasList	Measurement List that corresponds to experimental data
Outputs:

Syntax:	[Phi0,A] = TD3pt(SD, Medium, MeasList, muVec);
Inputs:	SD	SD structure
	Medium	Medium structure
	MeasList	Measurement List that corresponds to experimental data
	muVec	Flags to indicate optical perturbations (see genBornMat() for a description of the muVec flags)
Outputs:	Phi0	Fluence for a homogeneous medium
Outputs:	A	The forward matrix

Detailed Descriptions

TD2pt() calculates detected signal for a given measurement pair of a time-domain imager assuming homogeneous optical properties. The detected signal is calculated from the Green's function

Φ^{0} (r_{src}, r_{det}, t) = \int_{t}^{t + T_{gate}} ⅆ t^{'} G (r_{det} - r_{src}, t^{'})

where, for time-domain imaging, the Green's function is

G (r, t) = {(4 D t)}^{- 3 / 2} \exp (\frac{- r^{2}}{4 D t} - v μ_{a} t),

the diffusion coefficient $D$ is given by

D = \frac{v}{3 (μ_{s}^{'} + μ_{a})},

and $v = c / n$ is the local speed of light. By definition, $t_{src},$ the time when the light enters the medium, defines $t = 0 .$

TD3pt() calculates the forward matrix $A$ , which maps perturbations in the optical properties to perturbations in the measured fluences in the first Born approximation. Only the first Born approximation is supported by TD3pt(). For absorbing perturbations, the forward matrix is given in terms of the homogeneous Green's functions as

A_{a} (r_{src}, r_{vox}, r_{det}, t_{det}) = (\frac{- {Δμ}_{a} (r_{vox})}{V (r_{vox})}) \int_{t_{det}}^{t_{det} + t_{gate}} ⅆ t^{'} \int_{0}^{t^{'}} ⅆ t^{″} G (r_{vox} - r_{src}, t^{″}) G (r_{det} - r_{vox}, t^{'} - t^{″})

where $V$ is the volume of the voxel and ${Δμ}_{a}$ is the perturbation, relative to the background (average) absorption.

For scattering perturbations, the forward matrix can be written in terms of dot products of the gradients of the Green's function

A_{a} (r_{src}, r_{vox}, r_{det}, t_{det}) = (\frac{v ΔD (r_{vox})}{V (r_{vox}) D (r_{vox})}) \int_{t_{det}}^{t_{det} + t_{gate}} ⅆ t^{'} \int_{0}^{t^{'}} ⅆ t^{″} ∇G (r_{vox} - r_{src}, t^{″}) \cdot ∇G (r_{det} - r_{vox}, t^{'} - t^{″})

and $ΔD$ is the perturbation, relative to the background (average) diffusion coefficient.

TD2pt() and TD3pt() are really just wrapper functions; the actual calculation of the fluence is done by a series of low-level routines. To insure future compatibility, though, user code should always call TD2pt() or TD3pt() instead of calling the low-level routines.