Numerical reproduction of hemodynamics change by acupuncture on Taichong (LR-3) based on the lumped-parameter approximation model of the systemic arteries

Abstract

Background

The aim of this study was to develop a mathematical model of blood flow in the systemic circulation to emulate the change in hemodynamics by acupuncture therapy to elucidate the mechanism of the therapy. For this purpose, as a first step, a simple model of arterial blood flow was presented to reproduce previously reported change in the blood flow volume by the acupuncture needle stimulation of Taichong (LR-3).

Methods

This model was based on the lumped-parameter approximation of arterial blood flow together with linear resistance of peripheral circulation. It has been reported that blood flow in the left arm was enhanced after the stimulation, yielding the peripheral vascular resistance-regulated blood flow dominated by the sympathetic nervous system. In addition to the peripheral resistance, another parameter that possibly regulates the blood flow is the cross-sectional area of the vessel. These two factors were changed to numerically examine their contributions to the blood flow based on the hypothesis that they could be changed by the stimulation. The numerical result was compared with the experimental result to confirm the validity of the hypothesis that the blood flow in the arm is regulated by the peripheral resistance.

Results

This model is extremely simple and the physical parameters introduced for the simulation were gleaned from different reports in the literature. It was demonstrated, however, that regulation of the peripheral resistance rather than of the cross-sectional area could reproduce the experimentally observed change in the blood flow. Moreover, the relationship between the changes in the flow volume and the systemic vascular resistance quantitatively matched the experimental data.

Conclusion

The present model has a potential to emulate hemodynamic change by acupuncture therapy by incorporating physiological correlation of stimulation of an acupoint and regulation of parameters that affect the hemodynamics.

1. Introduction

Acupuncture therapy is an important component of traditional Chinese medicine along with moxibustion and herbal medicine. In traditional Chinese medicine, it is believed that a meridian network is spread over the entire human body and that “Qi” or “blood” runs over the meridians to maintain life.¹ Acupuncture involves restoration of the normal flow of Qi or blood by stimulating appropriate “acupoints,” about 400 of which exist in the human body, using needles (acupuncture needles).

Recently, the effectiveness of acupuncture therapy has been reported from a modern medical point of view. For example, in a randomized controlled trial of patients suffering from chronic neck pain, the symptoms of patients receiving additional acupuncture improved faster than those receiving only primary care.² Witt et al3, 4, 5, 6 reported randomized acupuncture trials in patients with osteoarthritis of the knee, migraine, tension-type headache, and chronic low back pain. In those studies, acupuncture was found to be more effective in patients with some types of pain, compared with those who did not receive any treatment. Yamazaki et al⁷ showed that acupuncture could enhance cognitive function and possibly prevent dementia. In 1999, Guidelines on basic training and safety in acupuncture was published by the World Health Organization,⁸ and the therapy has been approved for treatment of various diseases including diseases of the nervous system. Therefore, currently in the United States, Europe, and Asia, acupuncture is being used together with modern medicine in actual medical practice.1, 9, 10

Acupuncture therapy, however, presently lacks a scientific basis because its body of the theory is based on the existence of hypothetical organs and the concept of meridians, acupoints, and Qi that cannot yet be validated by modern medicine and science. Thus, the distrust of acupuncture therapy cannot be easily dispelled. Therefore, scientific elucidation of underlying acupuncture mechanism is essential for the therapy to be generally accepted in the future. There are only few studies in which quantitative measurement of blood flow in response to stimulation of a specific acupoint was conducted at clinical sites11, 12, 13; whereas, numerous studies were conducted on hemodynamic change following acupuncture therapy. Takayama et al¹¹ measured blood pressure (BP), cardiac index (CI) and systemic vascular resistance index (SVRI), and the diameter and blood flow volume of left brachial and radial arteries in response to acupuncture therapy applied to Taichong (LR-3), focusing on the effect of the stimulation of the acupoint on the hemodynamics. The authors of that study demonstrated that blood flow volume in the arm decreased during twisting of the acupuncture needle and then increased to a level higher than the preacupuncture level at 180 seconds after the stimulation, while BP and CI were not changed. They also demonstrated that SVRI at 180 seconds after the stimulation was decreased. Vasoconstriction of digital blood vessels, unlike the case of other vessels, is regulated only by the sympathetic nervous system,¹⁴ and change in peripheral blood circulation at the fingernail bed by acupuncture has been reported.¹⁵ Therefore, Takayama et al¹¹ argued that the steep decrease and subsequent gradual increase in the blood flow volume were due to the change in peripheral vascular resistance induced by tension and sedation of the sympathetic nervous system.

It is difficult to noninvasively observe the correlation between stimulation of an acupoint and the change in peripheral vascular resistance other than presuming it from resultant change in the flow volume or SVRI in response to the stimulation. Numerical simulation is expected to provide additional information not only for evaluating the contribution of the change in the vascular resistance to the hemodynamics, but also for systemic elucidation of the mechanism of the therapy in future by combining it with a correlation model of stimulation and the physiological parameters. In this study, as an initial step, a simple mathematical model of blood flow in a systemic arterial system was presented to reproduce the changes in the blood flow by stimulation of Taichong.¹¹ The arterial system was simplified to the aorta, arteries of the four limbs, and the superior mesenteric artery. Blood flow in each artery was expressed by the lumped-parameter approximation and each peripheral circulation was modeled as a linear resistance. Cross-sectional area of the artery of the left arm and its peripheral vascular resistance were chosen as parameters that affect the blood flow in the arm, and they were changed to investigate their contributions to the blood flow. The result of this study was compared with that of Takayama et al¹¹ to confirm the validity of their hypothesis that the blood flow is dominated by the peripheral vascular resistance of that arm.

2. Methods

2.1. Lumped-parameter approximation model

Fig. 1 shows a schematic of the lumped-parameter approximation model of the systemic arterial system introduced in this study. This model consists of arteries modeled by straight rigid pipes, with each peripheral circulation being modeled by a linear resistance. The upstream end of the model is connected to the left ventricle. Each artery is defined as a part divided by points of pressure definition points, P_S and P₁–P₈. The arteries are numbered as follows: Number 1 is the ascending aorta and the aortic arch, Number 2 extends from the subclavian artery to the radial artery of the left arm, Number 3 extends from the brachiocephalic artery to the radial artery of the right arm, Number 4 is the thoracic aorta and the abdominal aorta above the branching of the superior mesenteric artery, Number 5 is the superior mesenteric artery, Number 6 is the remaining portion of the abdominal aorta below the branching of the super mesenteric artery, Number 7 extends from the common iliac artery to the anterior tibial artery of the left leg, and Number 8 extends from the common iliac artery to the anterior tibial artery of the right leg. These arteries were chosen by referring to major branches of canine arterial tree.¹⁶ Pressures are defined at both ends of each artery, and flow volume is defined at the middle assuming that the blood flows uniformly. Table 1 presents the length L, proximal radius R_u, and distal radius R_d of the arteries.17, 18

Fig. 1.

Schematic of arterial system model. Pressure is defined at circles and flow volume is defined at squares.

Table 1.

Size of arteries introduced in the present model17, 18

Artery ID	Artery	Length (L), m	Proximal radius (Ru), m	Distal radius (Rd), m
1	Ascending aorta–Aortic arch	0.099	0.0147	0.0107
2	Left subclavian artery–Left radial artery	0.691	0.0042	0.0014
3	Brachiocephalic artery–Right radial artery	0.725	0.0062	0.0014
4	Thoracic aorta–Abdominal aorta	0.229	0.0100	0.0059
5	Superior mesenteric artery	0.059	0.0044	0.0044
6	Abdominal aorta	0.116	0.0058	0.0052
7	Left common iliac artery–Left anterior tibial artery	0.988	0.0037	0.0013
8	Right common iliac artery–Right anterior tibial artery	0.992	0.0037	0.0013

Data were gleaned from the reports by Stergiopulos et al¹⁷ and Wang and Paker¹⁸.

2.2. Governing equations

Blood is assumed to be an incompressible Newtonian fluid with viscosity (μ) of 3.5 × 10⁻³ Pa·s (3.5 cP), which was chosen as an intermediate value between 3.0 × 10⁻³ Pa·s (3.0 cP) and 4.0 × 10⁻³ Pa·s (4.0 cP)19, 20; the flow is laminar steady flow. Neglecting the influence of gravity, the relationship between flow volume Q and pressure difference P_u–P_d of each artery is expressed as follows:

$$Q_i=\frac{A_i^2}{8\pi \mu L_i}\left(P_{u_i}-P_{d_i}\right)$$ (1)

where the subscript i represents the ID of an artery listed in Table 1; P_u and P_d are pressures at the proximal and the distal ends of the artery, respectively, and A_i is the mean cross-sectional area of the artery. A_i is obtained using the following equation as the cross-sectional area of a straight cylinder with the same length and volume as the artery modeled as a circular truncated cone:

$$A_i=\frac{\pi }{3}\left(R_{u_i}^2+R_{u_i}{R}_{d_i}+R_{d_i}^2\right)$$ (2)

The equation of continuity is expressed as follows to denote that inflow and outflow volumes are equal at each junction of arteries:

$$\sum _{i\in J}{Q}_i=0$$ (3)

where J is the set of IDs of arteries sharing a pressure definition point P_j.

The pressure drop across a peripheral circulation ΔP is expressed as follows, taking into account its resistance R_p:

$$\Delta P=P_{d_i}-P_e=R_{p_i}{Q}_i$$ (4)

The peripheral vascular resistance was estimated to be 4.3 × 10⁹ Pa·s/m³ from the pressure difference between an artery and a venule.21, 22 Because it could be an overestimated value due to pressure drops within the artery or in the venule, R_p in this model was assumed to be 1.0 × 10⁹ Pa·s/m³ for each peripheral circulation to prevent extraordinary laterality of blood distribution. Venous pressure downstream of the peripheral circulation, P_e, was set to 0 Pa considering the facts that pressure at the right atrium is 6 mmHg or less and pulsation of the pressure in venules is much smaller than that in the aorta.23, 24 Nomenclatures used in the governing equations are listed in Table 2.

Table 2.

Nomenclature

Notation	Description
A	Mean cross-sectional area [Eq. (2)]
L	Length of artery
Pd	Distal pressure of artery
Pe	Venous pressure
PS	Supply pressure
Pu	Proximal pressure of artery
Q	Flow volume
Rd	Distal radius of artery
Rp	Peripheral vascular resistance
Ru	Proximal radius of artery
μ	Viscosity of blood

In this study, the contributions of the cross-sectional area of the artery of the left arm, A₂, and its peripheral vascular resistance, R_p2, to blood distribution in the body were examined numerically. For this purpose, blood distribution to the arteries was first obtained using the values listed in Table 1 as the control condition, and A₂ and R_p2 were changed in the range of ±25% from their original values to examine the hemodynamic change from the control condition. Takayama et al¹¹ showed that neither BP nor cardiac output (CO) was changed through the therapy. It is mathematically impossible, however, to use both supply pressure and supply flow volume simultaneously as the upstream boundary condition. Therefore, the authors conducted the aforementioned simulations by applying a constant pressure of P_S = 13.3 kPa (100 mmHg), which corresponds to the mean pressure of the left ventricle,23, 24 and a constant flow volume of Q₁ = 8.33 × 10⁻⁵ m³/s (5 L/min), which corresponds to CO,25, 26, 27 as the upstream boundary condition to examine the influence of the boundary condition on the blood distribution.

2.3. Systemic vascular resistance

Takayama et al¹¹ introduced SVRI as a parameter to discuss hemodynamics. SVRI (dyn·s·m²/cm⁵) is obtained using the following equation with CI (L/min/m²), mean arterial pressure (MAP, mmHg), and central venous pressure (CVP, mmHg)28, 29:

$$\text{SVRI}=\frac{\text{MAP}-\text{CVP}}{\text{CI}}\times 80$$ (5)

where CI is the CO (L/min) normalized by body surface area (m²). In this study, systemic vascular resistance (SVR, MPa·s/m³), expressed in Eq. (6), is introduced as the representative of Eq. (5) so that the body surface area need not be taken into account²⁹:

$$\text{SVR}=\frac{\text{MAP}-\text{CVP}}{\text{CO}}\times 8$$ (6)

Actually, in this model, P_S (Pa) is substituted for MAP, P_e (0 Pa) for CVP, and Q₁ (m³/s) for CO, and thus, Eq. (6) is rewritten as follows:

$$\text{SVR}=\frac{P_S}{Q_1}$$ (7)

3. Results

3.1. Pressure boundary condition

Table 3 presents the data on pressure at each definition point, the pressure drop across each artery segment, the flow volume in each segment, and the relative flow volume to Q₁ when a constant pressure of P_S = 13.3 kPa (100 mmHg),23, 24 which corresponds to mean BP in the left ventricle, is applied to the upstream end of artery Number 1. Here, Q₁ = 6.07 × 10⁻⁵ m³/s (3.64 L/min), which is in the range of the systemic circulation between 5.60 × 10⁻⁵ m³/s (3.36 L/min) and 7.78 × 10⁻⁵ m³/s (4.67 L/min), which were calculated taking into account the CO at rest ranging from 6.83 × 10⁻⁵ m³/s (4.1 L/min) to 9.50 × 10⁻⁵ m³/s (5.7 L/min)25, 26, 27 and distributions to cerebral and cardiovascular circulations of about 14% and 4%, respectively, of the CO.³⁰ This result, accordingly, indicates that 1.0 × 10⁹ Pa·s/m³ of the peripheral vascular resistance is an acceptable value. The resultant SVR in this condition is 219 MPa s/m³. The flow volume in the left arm, Q₂, is 1.23 × 10⁻⁵ m³/s and the pressure drop across the artery, P₁–P₂, is 1046 Pa, which are focused on hereafter. The hemodynamics is discussed as the change from the values shown in this table.

Table 3.

Blood distribution at the control condition shown in Table 1 under the pressure boundary condition

Pressure (Pa)	Artery ID	Pressure drop (Pa)	Flow volume (m3/s)	Ratio from Q1 (%)
P1 13,298	1	PS–P1 2.0	Q1 6.07 × 10−5	100
P2 12,252	2	P1–P2 1,046.0	Q2 1.23 × 10−5	20.2
P3 12,984	3	P1–P3 313.5	Q3 1.30 × 10−5	21.4
P4 13,281	4	P1–P4 17.3	Q4 3.55 × 10−5	58.4
P5 13,262	5	P4–P5 18.6	Q5 1.33 × 10−5	21.9
P6 13,256	6	P4–P6 25.0	Q6 2.22 × 10−5	36.6
P7 11,098	7	P6–P7 2,157.6	Q7 1.12 × 10−5	18.3
P8 11,091	8	P6–P8 2,164.9	Q8 1.12 × 10−5	18.3

Fig. 2 shows the changes in each value of pressure and flow volume when the cross-sectional area, A₂, of the left arm was changed in the range of ±25% with its peripheral vascular resistance, R_p2, being fixed at the control condition. In Fig. 2A, P_S remains constant because it is the given value. P₂ decreases by 5.8% for a 25% decrease of A₂ and increases by 2.9% for a 25% increase of A₂, whereas P₁ and P₃–P₈ are independent of the change in A₂. In Fig. 2B, Q₁ and Q₂ decrease by 1.2% and 5.8%, respectively, for the 25% decrease of A₂ and increase by 0.59% and 2.9%, respectively, for the 25% increase of A₂, whereas Q₃–Q₈ are independent of the change in A₂. Fig. 3 shows the change in each pressure and flow volume when R_p2 was changed in the range of ±25% with A₂ being fixed at the control condition. In Fig. 3A, P₂ decreases by 2.6% for a 25% decrease of R_p2 and increases by 1.6% for a 25% increase of R_p2, whereas P₁ and P₃–P₈ are independent of the change in R_p2. In Fig. 3B, Q₁ and Q₂ increase by 6.0% and 29.9%, respectively, for the 25% decrease of R_p2, and decrease by 3.8% and 18.7%, respectively, for the 25% increase of R_p2, whereas Q₃–Q₈ are independent of the change in R_p2.

Fig. 2.

Contribution of cross-sectional area of the left arm, A₂, to blood flow under the pressure boundary condition. Tangential axis is the ratio of A₂ from its control condition. (A) Pressure; (B) flow volume.

Fig. 3.

Contribution of peripheral vascular resistance of the left arm, R_p2, to blood flow under the pressure boundary condition. Tangential axis is the ratio of R_p2 from its control condition. (A) Pressure; (B) flow volume.

From these results, it can be said that the influence of R_p2 on the change in Q₂ is more significant than the changes in others. It is interesting that these changes only had an influence on the blood flow in artery Numbers 1 and 2, but not in artery Numbers 3–8.

3.2. Flow-volume boundary condition

It was described in the previous section that Q₁ changed with the changes in A₂ or R_p2 although no statistically significant change in the CO was found in the experiment by Takayama et al.¹¹ Therefore, in this section, a constant flow volume Q₁ is applied instead of P_S to the upstream boundary.

Table 4 presents the data on the pressure at each definition point, the pressure drop across each artery segment, the flow volume in each segment, and the relative flow volume to Q₁ when a constant flow volume of Q₁ = 8.33 × 10⁻⁵ m³/s (5 L/min),25, 26, 27 which corresponds to CO, is applied at artery Number 1. Here, P_S is 18,256 Pa (137 mmHg), which is higher than the mean BP of 13.3 kPa due to higher Q₁ than that calculated in the previous section. The flow volume in the left arm, Q₂, is 1.68 × 10⁻⁵ m³/s and the pressure drop across the artery, P₁–P₂, is 1,435.8 Pa, which are focused on hereafter. Although these are higher than the values presented in Table 3, it was confirmed that the relative flow volume to Q₁ and SVR was the same. The hemodynamics is discussed as the change from the values shown in Table 4.

Table 4.

Blood distribution at the control condition shown in Table 1 under the flow-volume boundary condition

Pressure (Pa)	Artery ID	Pressure drop (Pa)	Flow volume (m3/s)	Ratio from Q1 (%)
P1 18,253	1	PS–P1 2.78	Q1 8.33 × 10−5	100.0
P2 16,817	2	P1–P2 1,435.8	Q2 1.68 × 10−5	20.2
P3 17,823	3	P1–P3 430.3	Q3 1.78 × 10−5	21.4
P4 18,229	4	P1–P4 23.8	Q4 4.87 × 10−5	58.4
P5 18,204	5	P4–P5 25.5	Q5 1.82 × 10−5	21.9
P6 18,195	6	P4–P6 34.3	Q6 3.05 × 10−5	36.6
P7 15,233	7	P6–P7 2,961.6	Q7 1.52 × 10−5	18.3
P8 15,233	8	P6–P8 2,971.6	Q8 1.52 × 10−5	18.3

Fig. 4 shows the changes in each pressure and flow volume when A₂ was changed in the range of ±25% with R_p2 being fixed in the control condition. In Fig. 4A, P₂ decreases by 4.1% for a 25% decrease of A₂ and increases by 2.0% for a 25% increase of A₂, whereas P_S, P₁, and P₃–P₈ increase by 1.0% for the 25% decrease of A₂ and decrease by 0.58% for the 25% increase of A₂. In Fig. 4B, Q₁ remains constant because it is the given value. Q₂ decreases by 4.1% for the 25% decrease of A₂ and increases by 2.0% for the 25% increase of A₂, whereas Q₃–Q₈ increase by 1.0% for the 25% decrease of A₂ and decrease by 0.58% for the 25% increase of A₂. Fig. 5 shows the change in each pressure and flow volume when R_p2 was changed in the range of ±25% with A₂ being fixed at the control condition. In Fig. 5A, P₂ decreases by 7.9% for a 25% decrease of R_p2 and increases by 5.4% for a 25% increase of R_p2, whereas P_S, P₁, and P₃–P₈ decrease by 5.8% for the 25% decrease of R_p2 and increase by 4.0% for the 25% increase of R_p2. In Fig. 5B, Q₂ increases by 22.8% for the 25% decrease of R_p2 and decreases by 15.7% for the 25% increase of R_p2, whereas Q₃–Q₈ decrease by 5.8% for the 25% decrease of R_p2 and increase by 4.0% for the 25% increase of R_p2.

Fig. 4.

Contribution of cross-sectional area of the left arm, A₂, to blood flow under the flow-volume boundary condition. Tangential axis is the ratio of A₂ from its control condition. (A) Pressure; (B) flow volume.

Fig. 5.

Contribution of peripheral vascular resistance of the left arm, R_p2, to blood flow under the flow-volume boundary condition. Tangential axis is the ratio of R_p2 from its control condition. (A) Pressure; (B) flow volume.

Comparing these results with those presented in the previous section, a similar tendency is observed, that is, the influence of R_p2 on the change in Q₂ is more significant, but the area of influence is different. That is, with the constant supply flow volume, changes in A₂ and R_p2 influence all the parameters other than Q₁ although the area of influence was limited to Arteries 1 and 2 for the constant supply pressure.

3.3. Systemic vascular resistance

Fig. 6 shows the dependence of SVR calculated from Fig. 2, Fig. 3, Fig. 4, Fig. 5 on A₂ and R_p2. The vertical axis represents the ratio of SVR from 219 MPa s/m³ at the control condition, and the horizontal axis represents the ratios of A₂ and R_p2 from their control values (it is the ratio of A₂ in Fig. 2, Fig. 4 and the ratio of R_p2 in Fig. 3, Fig. 5). From this figure, it can be seen that the change in SVR is independent of the upstream boundary condition. SVR increases by 1.2% for a 25% decrease of A₂ and decreases by 0.58% for a 25% increase of A₂. It decreases by 5.7% for a 25% decrease of R_p2 and increases by 3.9% for a 25% increase of R_p2. This figure indicates that the contribution of R_p2 to blood flow is superior to that of A₂.

Fig. 6.

Dependence of systemic vascular resistance, SVR, on cross-sectional area, A₂, and peripheral vascular resistance, R_p2, of the left arm. A₂ was changed in Fig. 2, Fig. 4, and R_p2 was changed in Fig. 3, Fig. 5.

4. Discussion

4.1. Influence of A₂ and R_p2 on hemodynamics

When the cross-sectional area, A₂, of the left arm was changed in the range of ±25% with its peripheral vascular resistance, R_p2, being fixed, the magnitude of the corresponding change in the pressures and the flow volumes was up to 5.8% under the pressure boundary condition, as shown in Fig. 2, and up to 4.7% under the flow-volume boundary condition, as shown in Fig. 4. Because they are much smaller than the change in A₂, it can be said that the cross-sectional area contributes less to the blood flow regardless of the upstream boundary condition. Actually, the changes in the pressures and the flow volumes calculated with the present model were less than 1%, corresponding to the maximum change in the cross-sectional area of 3.5% observed by Takayama et al.¹¹

By contrast, when R_p2 was changed in the range of ±25% with A₂ being fixed, the magnitude of the change in Q₂ was 29.9%, whereas those of pressures and other flow volumes were up to 6.0% under the pressure boundary condition, as shown in Fig. 3. The magnitude of the change in Q₂ was 22.5%, whereas those of pressures and other flow volumes were up to 8.1% under the flow-volume boundary condition, as shown in Fig. 5. This result indicates that the influence of R_p2 on Q₂ is more significant than the others, and thus, it supports the hypothesis by Takayama et al¹¹ (i.e., the peripheral vascular resistance dominantly regulates the blood flow in the brachial and radial arteries). Together with this fact, because the vasoconstriction of fingers is regulated only by the sympathetic nervous system,¹⁴ the hemodynamics can be said to be regulated by the peripheral vascular resistance controlled by the tension and sedation of the sympathetic nervous system.

4.2. Comparison of SVR with experimental data

The maximum magnitude of the change in SVR was 5.7% for a ±25% change in R_p2 and 1.2% for the same amount of change in A₂, regardless of the upstream boundary condition. This result indicates that the increase in SVR is also due to the increase in the peripheral vascular resistance, which supports the hypothesis by Takayama et al.¹¹ The reason why SVR is independent of the upstream boundary conditions is explained as follows: Assuming the arterial system shown in Fig. 1 to be one vessel, the relationship between the pressure drop across the vessel, ΔP, and flow volume in it, Q, is expressed as follows using the resistance, R

$$\Delta P=RQ$$ (8)

R is expressed as follows because all the venous pressures, P_e, were fixed to 0 Pa in this study.

$$R=\frac{P_S}{Q_1}$$ (9)

Because SVR = P_S/Q₁ as shown in Eq. (7), SVR equals R independent of the upstream boundary conditions. This indicates that SVR is uniquely specified by the configuration of the arterial system, including the peripheral vascular resistances.

Fig. 7 shows the dependence of Q₂ on SVR when R_p2 was changed under the pressure and the flow-volume boundary conditions. SVR in this figure represents the ratio from the value at the control condition. SVRI decreased by 4.3% and the flow volume in the radial artery increased by 19.6% at 180 seconds after the stimulation, as shown in Table 5.¹¹ In Fig. 7, a 4% decrease in SVR correlates with an increase of about 21% in Q₂ for the pressure boundary condition and an increase of about 16% for the flow-volume boundary condition. The numerical result well agrees with the experimental result despite the size of each artery and the value of the peripheral vascular resistance being based on different reports in the literature. Therefore, this result indicates that the present model can be adopted to explain the change in the flow volume by regulation of the peripheral vascular resistance.

Fig. 7.

Contribution of systemic vascular resistance, SVR, to flow volume in the left arm, Q₂.

Table 5.

Change in hemodynamics measured by Takayama et al.¹¹

Parameter	Time course
	Before	During	After 30 s	After 60 s	After 180 s
Flow volume in radial artery	100	47.1*	105.9	121.6†	119.6 (+19.6%)†
Flow volume in brachial artery	100	76.7†	105.5	108.2	119.2 (+19.2%)†
Mean arterial pressure	100				97.8 (–2.2%)
Cardiac index	100				101.7 (+1.7%)
Systemic vascular resistance index	100				95.7 (–4.3%)†

The data were reproduced from the report by Takayama et al.¹¹

^*p < 0.01 versus before acupuncture.

^†p < 0.05.

4.3. Influence of upstream boundary condition on hemodynamics

The amount of change in Q₂ was same as that in Q₁ for the pressure boundary condition, that is, both of them increased by 3.67 × 10⁻⁶ m³/s when R_p2 decreased by 25% in Fig. 3B. By contrast, the absolute value of the amount of the change in Q₂ was the same as that in the sum of Q₃ and Q₄, that is, Q₂ increased by 3.83 × 10⁻⁶ m³/s and Q₃ + Q₄ decreased by the same amount when R_p2 decreased by 25% in Fig. 5B. As for the pressure distribution, although the change in R_p2 influenced only P₂ under the pressure boundary condition, as shown in Fig. 3A, it influenced all the pressures under the flow-volume boundary condition and their magnitudes were much larger than the pressure boundary condition.

As shown in Table 5, CI increased by 1.7% and the mean BP decreased by 2.2% when SVRI decreased by 4.3% at 180 seconds after the stimulation, although they had no statistical significance. In this study, a 4% decrease in SVR corresponds to an 18% decrease in peripheral vascular resistance, as shown in Fig. 6. This results in an increase of approximately 4% in Q₁ for the pressure boundary condition and a decrease of about 4% in P_S for the flow-volume boundary condition, as shown in Fig. 8. Either of these trends matches the experimental results by Takayama et al¹¹ with respect to the change in SVR. By contrast, however, P_S and Q₁ for each simulation cannot be changed because they were given as the boundary condition. Therefore, at present, we cannot conclude which boundary condition is appropriate, and thus, measurement of blood flow in another limb (e.g., a leg) is necessary to discuss it from the viewpoint of the area of the influence of R_p2 within the arterial system.

Fig. 8.

Contribution of peripheral vascular resistance of the left arm, R_p2, to cardiac output, Q₁, and supply pressure, P_S. P_S is fixed in Fig. 3B and Q₁ is fixed in Fig. 5A.

4.4. Limitations

The findings of this study were in agreement with the experimental data¹¹ with respect to the relationships of Q₂ versus SVR, and Q₁ or P_S versus SVR. Because of limited available quantitative clinical data, however, the validity of the present model was limited to the value of Q₁ at the control condition under the pressure boundary condition, which was in the physiological range of the systemic circulation, and to satisfaction of the mass conservation at the present state. Thus, the authors are applying this model to hemodynamic change by stimulation of Zusanli (ST-36)¹³ to further ensure its validity.

Takayama et al¹¹ have discussed the physiological relationship between the stimulation of Taichong and change in the blood flow volume. They argued that the sympathetic nervous system is involved in this change via regulation of the peripheral vascular resistance, but this is still a hypothesis. The present analysis is based on the hypothesis that the change in the resistance is induced by stimulation. It is known that physical stimulation of an acupoint in different ways or even stimulation of different acupoints has the same efficacy although a difference in efficiency remains.13, 31, 32, 33 Therefore, it is presumed that not only the peripheral nervous system, but also the central nervous system and endocrine system are involved in signal transduction of the stimulation. Elucidation of the correlation will be introduced in a future work.

Hemodynamic change in the whole body by therapy is sometimes discussed based on partial hemodynamic change observed at limited sites, assuming that the observed change occurs equally in the whole body. This is considered to be due to the difficulty in simultaneous measurement of the changes in blood flow in the whole body resulting from the therapy. The present model has a potential to emulate the changes in blood distribution by acupuncture therapy for elucidation of the mechanism of the therapy by incorporating correlation of stimulation of an acupoint and response of physiological parameters affecting the hemodynamics.

Conflicts of interest

All contributing authors declare no conflicts of interest.