Approaches for modelling autocorrelation function and data processing in time-domain diffuse correlation spectroscopy

Time-domain diffuse correlation spectroscopy (TD-DCS) is a non-invasive optical technique for measuring tissue blood flow. Recovering the blood flow index (<i>αD</i> _b) requires accurate modelling of the normalised electric field autocorrelation function (<i>g</i> ₁), and an optimised data processing approach to minimise noise. We quantitatively compared four modelling approaches for <i>g</i> ₁: (i) using momentum transfer (<i>Y</i>) and pathlengths (<i>L</i>) from Monte Carlo (MC) simulations, (ii) using <i>L</i> only, (iii) applying an analytical solution of the photon diffusion equation (DE) in time domain, and (iv) applying an analytical solution of the correlation diffusion equation (CDE) in steady state. The second and third approaches use solutions in near-infrared spectroscopy (NIRS) for modelling <i>g</i> ₁ in DCS by assuming <i>Y</i> = <i>μ'</i> _s <i>L</i>. We computed <i>g</i> ₁ curves using the first approach, considered the gold standard, and recovered <i>αD</i> _b using the other three approaches for various source-detector distances (<i>ρ</i>) and scattering coefficients (<i>μ'</i> _s). Also, we investigated how the correlator time bin width (<i>T</i> _bin), which is an adjustable parameter in data processing, affects the standard deviation of <i>g</i> ₁ (or the normalised intensity autocorrelation function <i>g</i> ₂). We used a more convenient version of the noise equation expressed as a function of <i>g</i> ₁ (or <i>g</i> ₂), which removes the need to know the decay rate. When using photons detected after ∼0.5 ns, all four approaches produced nearly identical <i>g</i> ₁ curves. Using all detected photons, the DE solution produced negligible errors (up to ∼2%) in the recovered <i>αD</i> _b across various <i>ρ</i> and <i>μ'</i> _s, while using <i>L</i> from MC simulations resulted in larger errors (up to ∼9% at <i>ρ</i> = 5 mm and ∼1.5% at <i>ρ</i> = 30 mm). The analysis of the probability distributions <i>P</i>(<i>Y</i>) and <i>P</i>(<i>μ'</i> _s <i>L</i>) explained these differences. As expected, the standard deviation of <i>g</i> ₁ (or <i>g</i> ₂) can be reduced during data processing by increasing <i>T</i> _bin. To achieve the lowest standard deviation, <i>T</i> _bin should be longer than the inverse of the photon count rate, indicating that the optimal <i>T</i> _bin may vary across different time gates. The results provide quantitative insights into modelling <i>g</i> ₁ (or <i>g</i> ₂), and provide a direct guideline for minimising the standard deviation of <i>g</i> ₁ (or <i>g</i> ₂) in data processing.