Absolute optical measurements with multi-distance FD-NIRS data

In an infinite medium with an intensity-modulated point source at the origin, the \(ln\left[{r\phi}_{AC}\right]\) (logarithmic amplitude) and \(\theta\) (phase) measured with FD-NIRS are linear functions of \(r\) (see Fig. 1). The intercepts of these linear functions depend on a number of instrumental factors, whereas the slopes of the \(ln\left[{r\phi}_{AC}\right]\) and \(\theta\) with respect to \(r (S_{AC}\) and \(S_\theta\), respectively) only depend on the optical properties of the medium:

 \(S_{AC}=\frac{d}{dr}ln\left[r\phi_{AC}\left(r,\omega\right)\right]=-\sqrt{\frac{3\mu_a\left(\mu_s^\prime+\mu_a\right)}{2}}\sqrt{\sqrt{1+\left(\frac{\omega}{c_n\mu_a}\right)^2}+1}\),                   (1)

 \(S_\theta=\frac{d}{dr}\theta\left(r,\omega\right)=\sqrt{\frac{3\mu_a\left(\mu_s^\prime+\mu_a\right)}{2}}\sqrt{\sqrt{1+\left(\frac{\omega}{c_n\mu_a}\right)^2}-1}\).                   (2)
 

Fig. 1. Linear dependence of \(ln\left[{r\phi}_{AC}\right]\) and phase on the source-detector distance \(r\) in an infinite medium.

The system of two Equations (1) and (2) can be inverted to yield expressions for the absorption and reduced scattering coefficients in terms of \(S_{AC}\) and \(S_\theta\): 

\(\mu_a=\frac{\omega}{2c_n}\left(\frac{S_\theta}{S_{AC}}-\frac{S_{AC}}{S_\theta}\right)\),                   (3) 

\(\mu_s^\prime=-\frac{2c_n}{3\omega}S_{AC}S_\theta-\frac{\omega}{2c_n}\left(\frac{S_\theta}{S_{AC}}-\frac{S_{AC}}{S_\theta}\right)\),                   (4) 

which show how frequency-domain methods can determine the absolute values of \(\mu_a\) and \(\mu_s^\prime\) from multi-distance measurements. As a practical note, the units of \(S_{AC}\) are cm^-1, and they are not affected by the units of \(\phi_{AC}\) because \(\frac{d}{dr}ln\left[r\phi_{AC}\right]=\frac{1}{r}+\frac{1}{\phi_{AC}}\frac{d\phi_{AC}}{dr}\), so that the units of \(\phi_{AC}\) cancel out. By contrast, to apply Eqs. (3) and (4) the units of \(\theta\) must be radians, so that the units of \(S_\theta\) are rad/cm. 

It is also possible to perform multi-frequency measurements and obtain the optical coefficients with a nonlinear fit of the frequency dependence of the phase and AC amplitude. The advantage of this approach is that it can be applied to data collected at a single source-detector distance thereby being less sensitive to tissue inhomogeneities. Nonetheless, calibration measurements on phantoms of known optical properties are required to correct for frequency-dependent instrumental factors. 

To a first approximation, Eqs. (3) and (4) can also be used in a semi-infinite geometry by modifying the AC slope \((S_{AC})\) to represent the slope of \(ln\left[r^2\phi_{AC}\right]\) instead of \(ln\left[r\phi_{AC}\right]\).