Mathematical Analysis of the Meridian System in Traditional Chinese Medicine

WAN Xiao-Hai

EMD Serono Research & Development Institute,Billerica,Massachusetts 01821-3936,USA

Keywords

Meridian system

Pumping power

Fluid mechanics

Human physiology

Work of breathing

ABSTRACT

Energy expenditure of the meridian system in traditional Chinese medicine(TCM)for living human bodies is one of the fundamental questions regarding the physical properties of the meridian system.As a first attempt to address this,24 hand/foot meridians were modeled as a continuous channel with the sustainable movement of Qi and blood inside.Then,the fluid mechanics energy equation for steady and incompressible flow was applied to estimate the pumping power of meridian circulation,based on the meridian length,blood viscosity and circulation velocity.Additional findings based on the pumping power estimation results include:(1)new information to corroborate the determination of the production era of Yellow Emperor’s Inner Classics(Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》);(2)a new definition for work of breathing(WOB)from a TCM perspective.This paper could lead to future research aimed at modernizing the understanding of the meridian system,Qi and TCM using quantitative methods.

1 Introduction

The meridian system is an important concept in traditional Chinese medicine(TCM),and it is about a path through which the life-energy,known as Qi flows.I hypothesize that physically,this life-energy Qi represents effective pumping energy,which is defined as,for a given region of interest in the meridian system,the pumping power needed to maintain the flow of Qi.Meridian system is a complex network to connect human body viscera(e.g.,lung,kidney and heart)with the rest of the body.An illustration of the system can be reviewed in Figure 1.As this system is complex,and there is a lack of proper ways to study it quantitatively using modern technology,the meridian system has long been regarded as an abstract concept.Although there is increasing worldwide interest in acupuncture and alternative medicines,there is still much that needs to be learned about the different aspects of the meridian system.

In this paper,I have applied principles of fluid mechanics to study the pumping power of the meridian system,focusing on the resting human body without inclination,such as while sleeping.My findings include:(1)pumping power estimation of circulation in twelve hand/foot meridians;(2)new information using a mathematical analysis to corroborate the determination of the production era ofYellow Emperor’s Inner Classics(Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》);(3)a new definition for work of breathing(WOB)from a TCM perspective.

This study is organized as follows.In Section 2,existing data on the meridian system is summarized fromYellow Emperor’s Inner Classics(Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》).Section 3 provides details of the pumping power analysis,and Section 4 discusses and summarizes the findings based on the analysis.Section 5 contains further discussion and the conclusions of the paper.

2 Data

According to TCM theory,and more specifically,the principle of Yin and Yang,Qi’s property is regarded as Yang as it represents a moving action.Moreover,blood,which is a general term for living and mobile entities that have material representation in the meridian,not to be confused with blood that flows in arteries and veins,is regarded as Yin.Indeed,what is flowing in the meridian is this blood driven by Qi,or Yin driven by Yang.

There are 12 hand/foot meridians on each side(left and right)of the human body,and one middle meridian in the front(Yin)and back(Yang).Together,12×2 + 2=26 meridians are of common interest in TCM.The total number of meridians will become 28 if the Yang Heel Vessel(Yang HV)and Yin Heel Vessel(Yin HV)meridians are also included.

2.1 Unit conversion

2.1.1 Time unitAs mentioned in reference[1],13 500 standard breathing periods occur in a 24 hour period,with a unit referred to as “Xi” in Chinese.Therefore,one can deduce that one standard breathing period takes 24×3 600/13 500=6.4 seconds(s),which includes inspiration,expiration and a pause in between.Note that a breathing period in modern medicine usually includes around 2 s for inhalation,and 3 s for exhalation[2],which would add up to 5 s.Therefore,we can deduce that the pause between inspiration and expiration lasts about 1.4 s.

2.1.2 Length unitInYellow Emperor’s Inner Classics(Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》),length unit “Chi”has a long history of use in China and could refer to quite different physical lengths during different historical periods[3,4].According to reference[4],conversion of 1 “Chi” to centimeters(cm)and feet(ft)are summarized in Table 1,which includes the Chinese dynasties before or around the time when the bookYellow Emperor’s Inner Classics(Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》)was conceived.

Table 1 Conversion of 1 “Chi” to centimeters(cm)and feet(ft)in the Shang,Zhou,Qin and Han dynasties

2.2 Meridians' length

According to reference[1],the length data for each of the 12 meridians are listed in Table 2.

3 Analysis

3.1 Meridians' circulation time and speed

The Qi takes 270 standard breathing periods(or 270×6.4/3 600=0.48 hours(h))to complete one full cycle through the 28(and more)meridians[1,5].Therefore,in a single day and night,the Qi completes 13 500/270=50 cycles.On average,the circulation speedVavgis 162/0.48=337.5 “Chi”/h,or 0.052(Shang dynasty),0.061(Zhou dynasty),0.071(Qin dynasty),0.066 - 0.073(Han dynasty)ft/s,if the length unit“Chi” is converted based on the unit conversion rule for each dynasty in Table 1.

The two middle meridians and the Yin/Yang HV meridians belong to eight extraordinary meridians,and they do not have a corresponding visceral manifestation in TCM theory.For analysis purposes,I choose to focus on Qi’s flow within the 12(or 24 for both the left and right side)meridians,which all have visceral manifestations.

Although it may seem not intuitive,Qi’s flow is assumed to be sequential,from the left side(Yang)to the right side(Yin)and continues without stopping such that life-supporting subsystems can run.On each side of the body,according to midnight-noon and ebb-flow doctrine,the flow happens sequentially in the following order:(1)hand-Yin-lung;(2)hand-Yang-large intestine;(3)foot-Yang-stomach;(4)foot-Yin-spleen;(5)hand-Yin-heart;(6)hand-Yang-small intestine;(7)foot-Yang-bladder;(8)foot-Yin-kidney;(9)hand-Yin-heart sac;(10)hand-Yangtriple energizer;(11)foot-Yang-gall; and(12)foot-Yin-liver.The incoming connection between the left and right sides is assumed to be the following:(1)left-foot-Yin-liver; and(2)right-hand-Yin-lung.The outgoing connection between the right and left sides is assumed to be the following:(1)right-foot-Yin-liver; and(2)left-hand-Yin-lung.This completes the description of the sequential flow of Qi within the 12 meridians.In the following analysis,I assumed that the average circulation speedVavgis the same as those estimated based on 28 meridians data,even though additional meridians such as the eight extraordinary meridians and others are not considered.

Table 2 Length data for 12 meridians(unit:“Chi”)

Note that the assumption of the sequential flow of Qi from the left side to the right side does affect the pumping power estimation.As an example,if the flow of Qi is indeed parallel,and it happens concurrently at both the left and right sides of the body,then this would lead to a different interpretation of one full circulation cycle through the 28 meridians in TCM theory.As a result,it would impact the calculation of the meridian circulation time and speed,as demonstrated in this section.Nevertheless,it seems to be reasonable to assume that the one full circulation cycle in TCM theory is meant for Qi’s flow sequentially through all branches of the meridian system,namely a Lagrangian specification of its flow field.

3.2 Pumping power estimation

As the flow of Qi is assumed to be sequential,it is modeled as fluid flow inside a circular pipe.The energy equation for steady flow inside the pipe was calculated,in terms of the heads as(Eq.5-74 in reference[6]):

wherePi(i=1,2)is the static pressure,ρiis the density of the blood,gis the gravitational acceleration constant on earth,Viis the flow velocity,ziis the hydrostatic head,hpump,uis useful head delivered to the flow,hturbine,eis the extracted head removed from the flow,andhLis the irreversible head loss between meridian location 1 and 2 due to all components of the meridian system,other than the pump or turbine.I further assumed that there was no pump(for the power analysis using the energy equation)and turbine in the meridian system.As this paper focuses on the resting human body without inclination,such as sleeping on a flatbed at nighttime,there is no hydrostatic head,zi.If the flow is steady with a constant velocity,Vavg,the power equation is:

If the density of the blood is assumed to be the same asρbetween the meridian location 1 and 2,the power equation becomes:

where ΔP=P1-P2is the pressure loss between the meridian locations 1 and 2 according to the fluid mechanics terminology[6].

If the cross-sectional shape of the meridian is assumed to be circular with a constant diameterD,the volume flow rate of the meridian therefore is:

whereVavgis the average circulation speed,andAcis the cross-sectional area of the meridian.So,the required pumping power to overcome this pressure loss is determined by:

If the meridian’s circulation is mostly laminar,according to Poiseuille’s law[6],the pressure loss(due to frictional force)for a constant circulation speedVavgis(Eq.8-20 in reference[6]):

it turns out that the corresponding pumping power is:

whereμis the dynamic blood viscosity,andLis the meridian length.It is interesting to see that diameterDcancels out in this formulation,and pumping power is a function ofVavg,μandL.

In our case,L=138 "Chi" for the 24 hand/foot meridians.Therefore the corresponding pumping power delivered to the meridian is:

To continue with the analysis,the blood dynamic viscosityμassumes a value of(3 ～ 4)×10-3Pa·s[7].Therefore,the pumping power was calculated to be(4.4 ～ 5.9)×10-4(Shang dynasty),(7.1 ～ 9.5)×10-4(Zhou dynasty),(1.1 ～ 1.5)×10-3(Qin dynasty),and(0.9 ～ 1.2)×10-3-(1.2 ～ 1.6)×10-3(Han dynasty)watts.Furthermore,if 28 meridians are considered,and one assumes that the sequential channel model still holds true,L=162 “Chi”.Then,the pumping power for this total meridian length is(5.2 ～ 6.9)×10-4(Shang dynasty),(8.4 ～ 11.2)×10-4(Zhou dynasty),(1.3 ～ 1.8)×10-3(Qin dynasty),and(1.1 ～ 1.4)×10-3-(1.4 ～ 1.9)×10-3(Han dynasty)watts.The pumping power required to keep Qi flowing in the meridian system has been estimated to have a possible range of 4.4×10-4- 1.9×10-3watts.

4 Further Findings

Based on the estimates in Section 3,further findings are presented below.

4.1 New information about the Yellow Emperor’s Inner Classics( Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》)production era

Yellow Emperor’s Inner Classics(Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》)is considered to date the earliest book of medicine in China.It provides a foundation of TCM theory.However,due to its long history,it is not known exactly when the book was written.Based on TCM literature,this book was first recorded in another book calledHan Shu · Treatise on Literature(Han Shu · Yi Wen Zhi,《漢書·藝文志》),which itself is considered to be the earliest history directory book in China.Therefore,it is clear thatYellow Emperor’s Inner Classics(Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》)must have been written during the Han dynasty or even earlier.Interestingly,the estimate of pumping power in Section 3 could shed some light on this.Here,I will start with information about the power of breathing,a quantity which is measured in modern medical studies to check lung function.

In a normal person at rest,the power of breathing per liter of ventilation has been measured to be about 2.4 J/min[8]or 0.040 08 watts.For a standard breathing period that usually lasts 6.4 s(Section 2),with an associated tidal volume of 500 mL[9],the power of breathing turns out to be 3.13×10-3watts.

According to the knowledge of human physiology[10,11],the WOB is divided into elastic work and resistive work.Elastic work is the energy required to overcome elastic forces,including lung elastic recoil and surface tension of alveoli,and is stored as elastic potential energy.Resistive work is due to the energy required to overcome frictional forces between tissue and between gas molecules.

About 65% of the total work is elastic work,while 35% of the total work is resistive[10,11].

InYellow Emperor’s Inner Classics · Miraculous Pivot(Huang Di Nei Jing·Ling Shu,《黃帝內(nèi)經(jīng)·靈樞》),relationship between breathing and meridian circulation is well documented.For example,1 “Xi” is comprised of one inspiration and one expiration.During one inspiration,meridian circulates by 3“Cun”.Similarly,during one expiration,meridian also circulates by 3 “Cun”.Therefore,it is stated that 1 “Xi” leads to 6 “Cun” circulations of the meridian,implying that there is no movement of meridian circulation during the pause between breathing.As Qi takes 270 “Xi” to complete one full cycle through the meridian system,it can be verified that the distance of such circulation equals 270×6 “Cun”=162 “Chi”,which matches with the 28 meridians’length in Section 2.2.Therefore,based on TCM theory,it is reasonable to assume that the energy spent during breathing includes the energy to sustain the circulation of the meridian system.

Furthermore,if the resistive work during breathing is indeed close to the work to overcome viscous force in the meridian,one would expect that:

This would dictate that the length unit “Chi” used extensively inYellow Emperor’s Inner Classics(Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》)should allow for the matching of the estimated pumping power and breathing power measured in modern times.Indeed,1.1×10-3watts coincides exactly with the estimated pumping power in Section 3 with the Han dynasty and a blood dynamic viscosity value of 3×10-3Pa·s.It can be seen in Figure 2,that the pumping power estimates from the Shang dynasty are not as close as those estimates from the Zhou,Qin and Han dynasties,regarding the power of breathing.Therefore,one can deduce that “Chi,” as written in the book,was more likely the length unit commonly used in the Zhou,Qin and Han dynasties as opposed to the Shang dynasty.It is a surprising,yet powerful finding that this mathematical analysis could shed some light on questions usually addressed by archaeology,language science and the astronomical calendar.

Although one of the estimates of the Han dynasty can be considered closest to the estimate from the power of breathing,the Zhou and Qin dynasties cannot be excluded since the range of their estimates is also close to the line representing 35% of breathing power.

The horizontal line represents 35% of the breathing power.

4.2 New definition for WOB

Another finding that could be deduced from the pumping power estimate is the WOB.WOB is defined as the energy spent to inhale and exhale a breathing gas,and is an important indicator for scientific and medical research,especially in respiratory-related medicine.The pumping power of the meridian system takes up a substantial amount(～ 35%)of the power of breathing.Therefore,from a TCM perspective,WOB should be defined as the energy spent to allow Qi to flow in the meridian system,plus the non-meridian excitation energy during a breathing period.WOB,as discussed earlier,is comprised of two components,one is the resistive work or pumping power of the meridian system,and the other is elastic work,or non-meridian excitation power,such as stretching of the lung.

5 Discussion and Conclusions

This paper has presented a novel approach to analyze the meridian system using the principles of fluid mechanics.Although it is not a new idea to study meridian systems using computational fluid dynamic techniques,it is,to the best of our knowledge,the first time that the meridian system has been holistically analyzed using fluid mechanic principles.This paper has demonstrated the scientific validity of such an analysis and could provide insights into the physical properties of the meridian system in humans.This work could also be the basis for future research on TCM using quantitative methods.

A limitation of this paper is using averaged estimations due to the lack of individual subject level data,a potential extension of the current work is to measure the viscosity of blood(or interstitium related to the meridian)and breathing power on the same subjects to assess potential subject level variability,and fine-tune the relationship between the meridian pumping power and breathing power.This could lead to a more precise determination of theYellow Emperor’s Inner Classics(Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》)production era.This paper also does not consider geometry of the meridian system and further work is required to understand the impact of gravitational force on pumping power estimation.

In conclusion,this paper has estimated the pumping energy of 12 hand/foot meridians in the range of 4.4×10-4- 1.9×10-3watts.Based on the pumping energy estimate,it is deduced thatYellow Emperor’s Inner Classics(Huang Di Nei Jing,《黃帝內(nèi)經(jīng)》)was more likely conceived in the Zhou,Qin and Han dynasties as opposed to the Shang dynasty.A new definition for WOB was also introduced from a TCM perspective.

Competing Interests

The author is currently employed by EMD Serono Research & Development Institute(Merck KGaA subsidiary in the United States)as a global group leader in biostatistics.This paper reflects the personal work and views of the author and does not relate to or represent the company’s views or policies,nor does it relate to the resources at the company.The author declares no conflict of interest.