The subscripts “” and “” refer to dynamic changes in and resulting from dynamic changes in volume and flow, hence due to and . The hemodynamic model describes explicitly how given and yield , , , and . Specifically, the model derives temporal dynamics (time-domain description) of hemoglobin changes, as well as sinusoidal hemodynamic oscillations as a function of the angular frequency (frequency-domain description). Here, we adopt the frequency-domain version of the model for describing oscillating hemodynamic signals such as those related to cardiac pulsation and respiration. We adopted the phasor representation of oscillatory quantities, where phasors are indicated in boldface. By definition, phasors (boldface) are dynamic quantities. The model expressions for , , (i.e., the phasors that describe the oscillations of the oxy-, deoxy-, and total hemoglobin concentrations, respectively) as a function of and (i.e., the phasors that describe the oscillations of CBV and CBF) are as follows:13,16Display Formula
(9)where and are the complex transfer functions associated with blood circulation in the capillary bed [, approximated by a resistor-capacitor (RC) low-pass filter, which contains as a parameter the capillary transit time ] and in the venous compartment [, approximated by a time-shifted Gaussian low-pass filter, which contains as parameters the capillary transit time and the venous transit time ], ctHb is the hemoglobin concentration in blood, is the Fåhraeus factor (ratio of capillary-to-large vessel hematocrit), and the superscripts , , and for CBV, cbv, and oxygen saturation of hemoglobin indicate their values associated with the arterial, capillary, and venous compartments, respectively. The total, steady state blood volume is given by , where , , and correspond to the baseline contributions of blood volume in the arterial, capillary, and venous compartment, respectively. Boldface lowercase, quantities correspond to dynamic changes relative to baseline values. We have set since dynamic dilation and recruitment of capillaries in brain tissue are negligible.30–35 Note that the only measurable parameters in Eqs. (1)–(9) are , , , , and . Finally, we observe that one can define an overall blood volume phasor, , as follows: Display Formula
(10)so that, as seen in Eq. (9), the phasor of total hemoglobin concentration, , has the same phase as .