Approximated peripheral arterial stiffness βart estimation method in an fMRI environment

We previously proposed a model, called the log-linearised peripheral arterial viscoelastic model, that can evaluate peripheral sympathetic nerve activity by estimating arterial stiffness (βart′) using cardiac cycles, continuous arterial pressure (Pbt), and arterial wall diameter (Plt)¹³. This model has been adopted to support sympathetic nerve activity evaluation in real-time in endoscopic thoracic sympathectomy¹⁴ when the patient is under general anaesthesia. The cardiac cycles are obtained from electrocardiography, and Pbt and Plt are estimated using continuous sphygmomanometer and photo-plethysmography, respectively. The model is defined by the following equation:

where μ is the inertia, η is the viscosity, βart′ is the peripheral arterial stiffness, Pbβart0′ is the constant pressure component, and Pbβartnl′Plt is the nonlinear stiffness pressure component originating in the vein. In addition, t represents time and the dot operator on Plt represents the time derivative. This model is applicable to cases where Pbt and Plt are available, but no fMRI-compatible device allows the measurement of Pbt. We instead used fMRI-compatible sphygmomanometer to measure systolic blood pressure (PSYS) and diastolic blood pressure (Pdia) on a beat-to beat basis. Therefore, we approximated the peripheral arterial stiffness¹⁵ using the measurable parameters. Please refer to Supplemental Material S2 for the approximation process. The proposed model is given by the following equation:

where Plmax and Plmin are the maximum and minimum values of the photo-plethysmogram within a heartbeat. The model focuses on the linearly approximated relationship between Pbt and Plt, where βart is its slope. Using this method, βart can be calculated for each heartbeat. Here, the measured electrocardiogram (ECG) was used to determine the R-R interval for extracting PSYS, Pdia, Plmax, and Plmin per heartbeat.

Figure 1 shows an example of the Lissajous curve between the radial artery pressure and photo-plethysmogram during a single heartbeat. First, we can assume that the photo-plethysmogram, P_l(t), is proportional to the arterial volume. As depicted by point a in the figure, the arterial volume becomes minimum when the artery pressure is the lowest (P_b(t) = P_dia). The arterial volume increases with a phase delay during the increase of the arterial pressure from diastolic pressure to systolic pressure (from point a to point b). Here, the increase rate depends on the arterial stiffness, and the phase delay of increase depends on the arterial inertia and viscosity. The arterial volume continues to increase to its maximum due to the phase delay, even after the arterial pressure reaches its maximum and, in turn, starts to decrease (from point b to point c). Finally, the arterial volume decreases and returns to its minimum (from point c to point a). The small circle that appears during this decrease is caused by the increase of arterial pressure due to a reflecting pulse wave. Equation 1 approximates the Lissajous curve and estimates βart′, but the approximated method estimates βart using the linear regression of two points, (Plmax, ln PSYS) and (Plmin, ln Pdia), which allows peripheral arterial stiffness to be estimated using the present fMRI-compatible devices. Hereafter, we denote βart as the arterial stiffness estimated by the proposed model (Eq. (²) and βart′ as that estimated by the previous model (Eq. (¹)).

Example of a measured Lissajous curve between radial artery pressure and the associated photo-plethysmography data. The previously used log-linearised peripheral arterial viscoelastic model estimates peripheral arterial stiffness (βart′) by fitting the model to a Lissajous curve of the photo-plethysmogram (Plt) and arterial pressure (Pbt), as represented by the dashed curve. However, the model proposed in this paper linearly approximates the relationship between Plt and Pbt, as represented by the solid black line, so that βart corresponds to the slope. a denotes the point at which the photo-plethysmogram and arterial pressure are at their minimum (P_l(t) = P_{l min}, P_b(t) = P_dia). b denotes the point at which the arterial pressure is at its maximum (P_b(t) = P_sys). c denotes the point at which the arterial volume is at its maximum (P_l(t) = P_{l max}). The solid line represents the slope between points (ln P_dia, P_{l min}) and (ln P_sys, P_{l max}) corresponding to the peripheral arterial stiffness.