何川 何培芳

（1.四川大学华西医院，四川 成都610041；2.三台华西医院，四川 三台621100）

The law of the heart working in systole and diastole is one of the most fundamental topics in cardiac physiology.On this topic，there is no generally accepted precise expression in language of physics and mathematics.

The heart is a special chemical-mechanical transducer.After a series of complicated physical and chemical changes，ultimately，foods change into chemical energy substrates that are transported to myocardial cells，the biochemical reactions taking place（During depolarization of the cardiac muscle，Ca2＋ions or calcium sparks arouse the chemical energy substrates burning／detonation among the thick filaments cross-bridges，thick and thin filaments），releasing the energy （the systolic forces）and bring the changes in the cross-bridges’configuration and the relative position of thick and thin filaments［1－3，5］.Thus the mechanical activity of the heart occurs，and the ventricular volumes alternate the changes in the narrowing／contraction and expansion／relaxation.As a result，the intraventricular pressure changes ceaselessly，increasing the ventricular pressure（ejection of blood into the artery，forming apressure gradient of the arterial blood flow），and decreasing the ventricular pressure（suction the venous blood backflow into ventricular cavity.This is the root cause of the venous blood flow pressure gradient），providing dynamic pressure，which keeps the blood flowing in the closed circulation system［1－4］.So the heart plays a role of“pump”，maintaining the body’s blood circulation. The chemical-mechanical transduction processes of the heart working are so complex that using aphenomenon in the experiments on behalf of its power conversion rule is extremely difficult，only expressed in integral form can be accurate.

It is well known that the life phenomenon has its unique biological law，but it is also subject to relevant rules of the general motion of matter as expressed scientifically in physics，chemistry and mathematics［5，6］.Beyond all questions，the mechanical process of the heart working is no exception.It complies with the law of the general motion of matter.The law of the heart working is discussed as below.

1 Research method

According to the laws of physics（principles of mechanical work），the rules of the left and right ventricles working in systole and diastole are discussed with a model of the left ventricular systolic ejection of blood doing work，and then mathematically conclusions are drawn.

1.1 The heart working and local myocardium working in systole

1.1.1 The heart working

1.1.1.1 The left ventricle working

1.1.1.1.1 The left ventricle working in systole

Fig.1 Schematic diagram of the left ventricular systolic ejection of blood doing work

As shown in the Figure 1，suppose Ais the left ventricular end-diastolic position（previous contraction position）；Bis the end-systolic position；ΔSis a tiny area on the left ventricular muscle wall lining；Δhis the displacement of theΔSduring the systole；P1is the contractive pressure in the left ventricular cavity；ΔFis the pressure on theΔS；ΔVis the volume of theΔS ejection of blood in the contraction emptying.When the heart is contracting，suppose the work done of theΔS mascle wall isΔW.Based on the definition of mechanical work in physics，the ΔWbe expressed as follows：

And as well，

Thereupon，

Suppose the left ventricular systolic ejection of blood doing work is W1，its precise expression should be：

In Formula（3），under the P1each increment（dv）of the volume（V）is ejected，the Vsand Vdstand for the end-systolic and end-diastolic positions respectivly，i.e.，a range of the ventricular wall moving during the heart working.In practical application，the average value of the P1is regarded as a constant.Therefore，

1.1.1.1.2 The left ventricle working in diastole

In diastole，the left ventricular muscle wall reverts from Position Bto A.Usually，volume（V）of diastolic blood flow to the left ventricle is the same as the Vof its blood ejection.Suppose the left ventricle working to suction the blood into the left ventricular cavity in diastole is W2，and the ventricular pressure in diastole is P2.In principle，the computational method of the W2and that of the W1in Formula（3）are the same，merely in the opposite direction.Therefore，the W2should be：

Obviously，the left ventricle working in systole and diastole should be the sum of the W1and W2.

1.1.1.2 The right ventricle working

1.1.1.2.1 The right ventricle working in systole

Under normal circumstances，the right and left ventricular stroke volumes （V）are the same.Suppose W3is the right ventricle working in systole；P3is the right ventricular pressure.The mechanically，the right ventricle in systole works in the same way as the left ventricle working.Thus the right ventricle working W3should be：

1.1.1.2.2 The right ventricle working in diastole

Suppose the right ventricle working in diastole is W4，the ventricular diastolic pressure is P4.The mechanism of the right ventricle working in diastole is the same as the left ventricle working in diastole.Thus the W4should be：

1.2 The local myocardium working in systole

Suppose theΔS muscular wall moves from Position Ato B，its acceleration is a；the moving distance or displacement isΔh；the blood ejection quantity is ΔV；the blood’s density isρ，and it cannot be compressed.Because the bloodΔVis ejected by theΔS，the acceleration and displacement of theΔS muscular wall is the same as that of the bloodΔV.From Formula（1）the local myocardium ΔS workingΔWin systole is：

Additionally，

2 Results

①Suppose the heart working in systole and diastole is W，which should be the sum of the left and right ventricle working.The value of the Wshould be as follows：W ＝W1＋W2＋W3＋W4

Because each stroke volume Vis equal to the end-diastolic volume （EDV）minus the end-systolic volume（ESV），Formula（9）can be rewritten as：

W＝［P1＋（－P2）＋P3＋（－P4）］·（EDV－ESV） （10）

② As stated in Equation（8），the local myocardiumΔS workingΔWin the systole is：

3 Discussions

3.1 The mathematical relationship between the heart working Wand the parameters（EDV－ESV）and EDV

Over the past 100years，the law of the heart working is expressed by the Frank-Starling Law of the Heart，which deems that the heart working Wis a function of the EDV （length of the muscle fibers）［7］.

But in Formula（10）deduced by the authors，the mathematical relationship between the W，（EDV－ESV）and EDVshows that the Wis a function of the（EDV－ESV），and there is no functional relation between the Wand EDVthat are isolated from the（EDV－ESV）.This indicates that the Frank-Starling Law of the Heart is negated by the laws of physics and mathematics in Formula（10），and it is not a true law［1－4，9－11］.

On the other hand，perhaps because the Frank-Starling Law of the Heart possesses a high authority，people tend to forget to use a most simple physical principle about the definition of mechanical work to explore its truthfulness or to doubt its rightfulness.The energy of cardiac contraction （cardiac work）is a function of the length of the muscle fibers （EDV）［7］，the Frank-Starling Law of the Heart，which notices the initial length of the cardiac muscle fibers（EDV，the starting point of the ventricular wall movement during the contraction），but it neglects the change in the length of muscle fibers（ignoring the end-point of the ventricular wall movement during the contraction，ESV）and its potency dimension.Apparently，the principle of the heart working stated by the Frank-Starling Law of the Heart does not comply with the basic concept of the“mechanical work”in physics：Lacking the end-point of the ventricular wall movement during the contraction，the cardiac work cannot be calculated.How could it be a right expression for the law of the heart working activities？

However，quite a few physiologists think that Frank-Starling Law of the Heart is summarized from experiments，and a large amount of clinical practice shows that the preload and after-load of the heart can regulate the heart working，which supports the Frank-Starling Law of the Heart in terms of clinical examples.And so，they hold that it is the correct theory，taking an unshakable position in physiology.

Here，we need to rethink profoundly that just because the Frank-Starling Law of the Heart is from vitro animal experiments，even if it does not conform to the accepted laws of physics，could the experimental results conseguentily become a correct theory？There is no example that increased cardiac preload method is adopted to strengthen the function of the heart in the treatment of heart failure cases.Why don’t you think it to be vital evidence indicating that the Frank-Starling Law of the Heart lacks clinical practice as supportive instances？

3.2 Local myocardium working in systole

In Equation（8）ΔW＝ρΔV·a·Δh，whenΔV is equal to 1cubic centimeter，Equation（8）can be simplified into：

In this condition，suppose the work done of the ΔSduring the local myocardium contracting is W5.The W5should be：

The average value of the“a”is regaded as a constant，therefore，

Obviously，the amount of local myocardium working is in direct proportion to the“ah”value of the ventricular muscle wall moving at one point during systole.

According to the epistemology principle that the universality of matter is contained in its particularity，the general law of the muscle wallΔS working should include a special condition，in whichΔVis equal to 1cubic centimeter.This prompts that the“ρah”can be used as an evaluation index of the systolic performance of the local myocardium.If the difference of the“ρ”（density of blood）in most people＇s bodies is not big，the“ah”value can be used as an evaluation index of the systolic performance of the local myocardium.The same reason，the“ah”can be used as an evaluation index of the diastolic performance of the local myocardium.

As we all know，the performance of the local myocardium may represent the cardiac function.And so，the“ah”value in ultrasonic detection of the ventricular muscle wall moving at one point during systole can be regarded as an evaluation index of the systolic function，and the“ah”value in diastole can be regarded as an evaluation index of the diastolic function.

4 Conclusion

①The law of the heart working in systole and diastole or Formula for calculating the heart working should be：

②Formula deduced by the authors of this paper proved that there is no functional relation between the heart working Wand the EDV（the initial length of the myocardium fibers）——the Frank-Starling Law of the Heart has no theoretical necessity and it is not a true law.This theoretical research shows that the so-called convincing experimental evidence to support the Frank-Starling Law of the Heart is nonexistent，even at the molecular level.

③ The“ah”value in ultrasonic detection of the ventricular muscle wall moving at one point during systole and diastole can be regarded as an evaluation index of the systolic and diastolic function respectively.

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