（3）

The log-normal distribution uses the following parameters,

μ: mean of the natural logarithm of the data.

σ: standard deviation of the natural logarithm of the data.

The BSDs of the MBs generated by each manual operation ten times are shown in Figure 6 (a, b, c, d). The results indicate that the size distribution of microbubbles varied with the number of operations, especially between 20-60 µm. The diameter of the bubbles is mainly distributed in the range of 10-100 µm, with a wide distribution range. These bubbles may affect the detection results of c-TCD after entering the vein because the lifetime of the bubbles decreases as the bubble size decreases, and large bubbles cannot pass through the capillaries and fail to reach the foramen ovale. Eventually, the inhomogeneous mixture of gas-liquid will cause hemodynamic changes.

Figure 7a compares the bubble size distribution averaged of 10 experiments performed by each person, with the curves being messy between 15-80 μm, which also shows that the difference in results between each individual is particularly obvious. In addition, the mean bubble size, Sauter mean diameter, and standard deviation value of 10 manual operations for each volunteer are shown graphically in Figure 7(b, c, and d). As can be seen, the reproducibility is relatively poor. The mean bubble size is between 21-34 μm, the D₃₂ is mainly between 50-300 μm, and the maximum standard deviation was close to 17 μm in 60 experiments. This initial inference has been strongly validated that the velocity and strength of each manual operation vary when pushing with the syringe leading to this phenomenon, especially between individuals. Through the analysis of Figure 7 and Table 2, the results show that the stronger the gas-liquid mixing, the more uniform the state, and the mean bubble size and standard deviation of the bubbles may decrease. These values are unstable with large fluctuations, which provides a more accurate basis for the instability of the manual method.