Diffusion theory: continuous-wave (CW), frequency domain (FD), and time domain (TD)

Light propagation in scattering media can be described in terms of photons that experience absorption and scattering events as they travel within the medium. Photons are deflected by scattering events and annihilated by absorption events.  The absorption and scattering processes are defined by their probability per unit distance travelled by the photon, as expressed by the absorption \(\mu_a\) and scattering \(\mu_s\). The scattering process is further defined by the probability density of scattering as a function of the angle \(\theta\) between the directions of scattered and incident photons, the so-called phase function p\((\theta)\). In the case of highly scattering media, where \(\mu_s\gg\mu_a\), what matters is the mean cosine of the scattering angle, g=cosθ, and a key optical parameter is the reduced scattering coefficient, \(\mu_s^\prime=\mu_s(1-g)\), which may be interpreted as the inverse distance over which photons lose memory of their initial direction of propagation. It is this highly scattering regime that is relevant for light propagation in many biological tissues and forms the basis for diffuse optical techniques. In diffuse optics, the light distribution in tissue is described in terms of the fluence rate \((\phi)\), which represents the optical energy that flows per unit time per unit area along all possible directions at a given position \(\mathbf{r}\) and time \(t\). Diffusion theory provides an analytical description of the fluence rate through the diffusion equation:

\(\frac{\partial\phi\left(\mathbf{r},t\right)}{\partial t}=\mathbf{\nabla}\bullet\left[D\mathbf{\nabla}\phi\left(\mathbf{r},t\right)\right]-c\mu_a\phi\left(\mathbf{r},t\right)+cS_0\left(\mathbf{r},t\right)\)                   (1)

where \(c\) is the speed of light in the medium, \(D=c/\left[3\left(\mu_s^\prime+\mu_a\right)\right]\) is the optical diffusion coefficient, and \(S_0\) is a spherically symmetric source term. The diffusion equation, in combination with appropriate boundary conditions, provides an analytical basis for diffuse optics. Three broad classes of techniques in diffuse optics may be distinguished on the basis of the temporal dependence of the optical illumination: continuous wave (CW) (constant source emission), frequency domain (FD) (modulated source emission), and time domain (TD) (pulsed source emission). The relevant source emission properties (power in CW and FD, energy in TD) and the optical fluence rate in tissue are illustrated in Figure 1 as a function of time.

Figure 1. Illustration of the time dependence of source emission and fluence rate \((\phi)\) in tissue for (a) continuous-wave (CW), (b) frequency-domain (FD), and (c) time-domain (TD) techniques. The relevant source emission property is power in CW (P_CW) and FD (P_FD(ω)e^-iwt at angular frequency ω), whereas it is energy of the illumination pulse in TD (Q_TD). DC, AC, and q represent the mean value, amplitude, and phase, respectively, of the modulated signal in FD.

Continuous wave – Constant illumination

In continuous-wave (CW) techniques, the light source emits a constant power \((P_{CW})\), and the fluence rate resulting from a point source embedded in a homogeneous, infinite medium depends only on the distance \((r)\) from the source, as given by the CW solution to Eq. (1):

\(\phi_{CW}\left(r\right)=\frac{c\ P_{CW}}{4\pi D}\frac{e^{-r\mu_{eff}}}{r}\),                   (2)

where \(\mu_{eff}=\sqrt{3\mu_a\left(\mu_s^\prime+\mu_a\right)}\) is the effective attenuation coefficient. Differentiation of Eq. (2) with respect to \(\mu_a\) yields a relationship between a change in absorption and a change in the fluence rate that is referred to as the modified Beer-Lambert law:

 \(\Delta\mu_a=-\frac{1}{rDPF}\frac{\Delta\phi cw}{\phi cw}\),                   (3)

where the DPF, or differential pathlength factor, is given by \({\rm DPF}_{inf}=\sqrt{3\mu_s^\prime}/\left(2\sqrt{\mu_a}\right)\) in the infinite geometry represented by Eq. (2), and by \({\rm DPF}_{seminf}=r\mu_{eff}^2/\left[2\mu_a\left(1+r\mu_{eff}\right)\right]\) in a semi-infinite geometry with source and detector (separated by distance \(r)\) located on the plane boundary of the medium. Equation (3) is widely used in CW spectroscopy of tissue.

Frequency domain – Sinusoidally modulated illumination

In frequency-domain (FD) techniques, the light source emits a power that is sinusoidally, temporally modulated at a frequency \(\omega [P_{FD}\left(\omega\right)]\), and the fluence rate resulting from a point source embedded in a homogeneous, infinite medium is given by the FD solution to Eq. (1):

\(\phi_{FD}\left(r,\omega\right)=\frac{c\ P_{FD}\left(\omega\right)\ }{4\pi D}\frac{e^{-r\mu_{eff}\sqrt{1-i\left(\frac{\omega}{c\mu_a}\right)}}}{r}\).                   (4)

The fluence rate of Eq. (4) should be interpreted as the complex amplitude of a sinusoidal oscillation at frequency \(\omega\) associated with an implied sinusoidal factor \(e^{-i\omega t}\). Such complex amplitude represents the magnitude and the phase of the oscillatory fluence rate that is measured in the FD-NIRS. The ability of FD techniques to measure two independent quantities, namely the magnitude and the phase of the oscillatory fluence rate, allows for absolute measurements of both absorption and scattering properties of the medium. Similarly to the modified Beer-Lambert law in CW techniques, it is also possible to obtain absorption changes from changes in either magnitude or phase measured in the frequency domain.

Time domain – Pulsed illumination

In time-domain (TD) techniques, the light source emits a pulse (typically in the picoseconds range), that is short compared to the typical propagation times of light from the source to the detector, which is on a scale of nanoseconds for source-detector distances of centimeters. The relevant source emission property in TD techniques is the pulse energy \((Q_{TD})\). The fluence rate resulting from a point source embedded in a homogeneous, infinite medium is given by the TD solution to Eq. (1):

\(\phi_{TD}\left(r,t\right)=\frac{Q_{TD}\ c}{\left(4\pi Dt\right)^{3/2}}\ e^{-\frac{3r^2}{4Dt}-\mu_act}\).                   (5)

TD data provides a richer information content than CW and FD data. In fact, FD and TD are related by a temporal Fourier transformation, which means that the information content of TD data corresponds to the information content of FD data at all modulation frequencies. The TD fluence rate of Eq. (5) is representative of the photon time-of-flight distribution, whose zeroth moment (area under the curve) is representative of the CW case, and the first moment (mean value) is closely related to the phase measured in FD techniques. In addition to allowing absolute measurements of absorption and scattering properties, TD techniques allow for time-gating or measurements of individual moments of the photon time-of-flight distribution, with relevance to both spectroscopy and imaging applications.