# Can OCT Angiography Be Made a Quantitative Blood Measurement Tool?
## 7 OCTA Measurements of Hemodynamics
### Flow Quantification  
In 2010, [PERSON] et al. [19] made an early effort at providing an autocorrelation model to measure transverse particle flow speed. Though they focused on intensity transients, here we generalize their initial work. The basic principle of their model is that when particles pass through the imaging beam, they create OCT signal transients that may provide information about the speed of the underlying particles. However, with a large coherence volume, the individual transients may overlap in time. The complex signal at position \((x,z)\) is expressed as a superposition of particle contributions:  
\[S(x,z,t)=\sum_{k=1}^{G(x,z)}M_{k}(x,z)REC(x,z,t-t_{k}), \tag{18}\]  
\[REC(x,z,t)=\left\{ \begin{array}{cc}1,&0\leq t\leq\tau_{0}(x,z)\\ 0,&\text{otherwise}\end{array} \right., \tag{19}\]  
where k is index of the kth particle, \(G(x,z)\) is the total number of particles passing through the imaging beam within the signal acquisition period, \(M_{k}(x,z)\) is the complex amplitude of the kth particle transient, \(t_{k}\) denotes the time when a particle begins to pass through the beam, and \(\tau_{0}(x,z)\) is the position-dependent transit time of the particle. After expressing the complex OCT signal in terms of particle contributions, the normalized autocorrelation function of \(S(x,z,t)\) is given by:  
\[\frac{R(x,z,\tau)}{R(x,z,0)}=\left\{ \begin{array}{cc}1-\frac{\tau}{\tau_{0} (x,z)},&\tau\leq\tau_{0}(x,z)\\ 0,&\tau\geq\tau_{0}(x,z)\end{array} \right., \tag{20}\]  
where \(R(x,z,\tau)\) is the autocorrelation function of \(S(x,z,t)\) with time lag \(\tau\). Note that this is equivalent to the normalized autocorrelation of \(REC(t)\). The slope of the normalized autocorrelation function in Equation (20) is proportional to the transverse speed (\(\sim\)1/\(\tau_{0}\)). Note that Equation (20) can be further generalized to accommodate other transient shapes. In 2012, [PERSON] et al. [PERSON] et al. (2012) proposed an alternative model to relate the autocorrelation to speed. For small particles undergoing isotropic motion through a coherence volume, they proposed that the autocorrelation decay is determined by axial and transverse point spread functions, while for large particles, the spatial characteristics of the particles themselves dominate the autocorrelation as described above. In [76], for small particles, the autocorrelation function at time lag \(\tau\) in cylindrical coordinates \((\varrho,\varphi,z)\) is:  
\[\mathrm{R_{d}}(\tau)=\frac{2|\mathrm{K}|^{2}}{\pi^{2}\mathrm{w_{e}^{4}}} \sqrt{\frac{\pi}{(\frac{\mathrm{v_{e}^{2}}}{\mathrm{w_{e}^{2}}}+\frac{\mathrm{ v_{e}^{2}}}{\mathrm{w_{z}^{2}}})}}\mathrm{P_{A}}\exp\left[-\frac{(\mathrm{v_{e} \tau)^{2}}}{\mathrm{w_{e}^{2}}}-\frac{(\mathrm{v_{z}\tau)^{2}}}{\mathrm{w_{z}^ {2}}}\right]\exp[\mathrm{i}(\frac{4\pi\mathrm{m}}{\lambda_{0}})\mathrm{v_{z} \tau}], \tag{21}\]  
where \(\mathrm{w_{e}}\) is the transverse beam profile, \(\mathrm{w_{z}}\) is the axial resolution, \(\mathrm{K}\) is an arbitrary complex constant [77], \(\mathrm{P_{A}}\) is the power in the random process which describes the field, \(\mathrm{v}\) is the particle's speed, \(\mathrm{n}\) denotes the refractive index, and \(\lambda_{0}\) is the central wavelength.