Cardiology

Samuel Rajini / JPBMS, 2012, 23 (17)

Available online at www.jpbms.info

Research article JPBMS

ISSN NO- 2230 – 7885 CODEN JPBSCT NLM Title: J Pharm Biomed Sci.

JOURNAL OF PHARMACEUTICAL AND BIOMEDICAL SCIENCES

What to the world Einthoven talking about? Dr.T.Rajini Samuel* Cardiology Department, Venkataeswara Hospitals, 36-A Chamiers road, Nandanam, Chennai, India. *Presently doing M.D. Biochemistry in Sree Balaji Medical College and Hospital, Chromepet, Chennai. India.

Abstract: Aims: To make people understand what Einthoven was really talking about. Methods and results: In the hex axial reference system of ecg, plot the net voltages of bipolar limb leads and connect them. Similarly, plot the net voltages of unipolar limb leads and connect them. Each forms equilateral triangles. Convert the equilateral triangles into circles. Each circle have same origin, same orientation, but different radii because the bipolar and unipolar limb leads have different resistance. The ratio of their resistance is 4/3. Hence the ratio of the area of the two circles is always constant and equal to 4/3. The square root of 4/3 is 1.154. The ratio of their radii is 1.154 .Multiply each unipolar limb lead voltages by 1.154 and then plot. The two equilateral triangles are on the same circle. The correction factor 1.154 is used because bipolar and unipolar limb leads have different resistance and the ratio of their resistance is 4/3. Thus heart is situated in the centre of the electric field which it generates. The two upper limbs and the left lower limb are the extensions of its electrical field. Voltage recorded in a particular lead is the result of dot product between the lead vector (measured in metre) and the cardiac vector measured in volt/metre. Hence voltage is a scalar quantity. The angle (α) determination from the ecg: Tan α= (1.154*avF)/ Lead 1

Conclusion: Einthoven triangle is an absolute electrical equilateral triangle. Each cardiac wave can be represented in the form of circles.

Keywords: - Cardiac vector, Shifted Einthoven triangle. Introduction: It has been said that Einthoven triangle is not absolutely equilateral and it is not valid for analytical and vector procedures.[2-6] Also, it is said that cardiac vector is an imaginary quantity of dimensions of unknown physical quantity. [1] But it is not so. Einthoven triangle is an absolute electrical equilateral triangle. The cardiac vector is an electric field vector of dimension volt/meter. The voltage recorded in a particular lead is the result of the dot product (scalar product) between the lead vector and the cardiac vector. Many of the medical students and even most of the specialist doctors find difficult to understand the concept of the electrocardiogram. So it has been made a clear description of the concept of the electrocardiogram and the Einthoven’s triangle. The current research study will be very useful for teaching purposes and also it has suggested a new method of interpretation of ECG in the frontal plane. Most of the common cardiac diseases can be interpreted easily based on this research study. 1

Cardiac vector hypotheses

In triangle OLH: COS α = OL/OH (adjacent side/ hypotenuse) OL= OH COSα H: CARDIAC (HEART) VECTOR L: LEAD VECTOR (LEAD AXIS)

Journal of Pharmaceutical and Biomedical Sciences © (JPBMS), Vol. 23, Issue 23

Cardiology

Samuel Rajini / JPBMS, 2012, 23 (17)

α: The angle between the cardiac and lead vector OL: Voltage recorded in that lead The scalar or dot product between cardiac and lead vectors: H . L = (OH) (OL) COSα H . L^= (OH) COSα

Notations for the voltages recorded in that particular leads Lead 1 =a1 Lead 2= a2 Lead 3=a3 avR =b1 avL =b2 avF =b3

Hex axial reference figure Since OL= OH COSα, the voltage recorded in a particular lead is the result of dot product between cardiac and lead vector. H. L^= (OH) COSα From the above equation it is very clear that the voltage recorded in a particular lead depends on: 1. Both the magnitude and direction of the cardiac vector. 2. Only on the direction of the lead vector because here lead vector (L ^) is a unit vector of magnitude one and has only direction. Einthoven triangle:

First triangle (formed by the three bipolar limb lead voltages): Let A, B, C be the vertices of the triangle (bigger one red color) formed by the voltages of the bipolar limb leads 1, 2 and 3 in the hexaxial reference figure. Cosine formulae a² = b² + c² -2bcCOSA cos60° = 1/2 cos120°= - 1/2 (negative sign is there)

Since we have taken lead vectors as unit vectors of magnitude one and having only direction, all the three limb lead vectors are of magnitude one and thus they form an equilateral triangle.

Shifted Einthoven triangles (A new discovery) Proof: Formula used: Lead 1+ lead 3 = lead 2 (Einthoven equation) avR + avL + avF = 0 The above two equations are based on Kirchoff’s law which is widely accepted. Cosine formulae (for a triangle with a, b, c as the sides and A as the angle) a² = b² + c² -2bcCOSA

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(AB) ² = a1² + a2² – a1a2 (Because the angle is 60° between lead 1 and 2 and cos60°= ½) (BC) ²= a2²+ a3² + a2a3 (here the angle between lead 2 and lead 3 is 120°) (AC) ²= a1²+ a3² – a1a3 (angle between lead 1 and lead 3 is 60°) Note that the angle between each limb leads will be different in each quadrant of the hexaxial reference system. In some it will be 60° and in others will be 120°. Lead 1+ lead 3 = lead 2 (Einthoven equation) Thus a1+ a3=a2 But here a3 is negative (lead 3 is negative here) Hence a1-a3=a2 Using the above formula and substituting it in (BC)² AND (AC)² the following results was found. (BC) ²= a2² + a3² + a2a3 =a2² + (a1-a2)² + a2 (a1-a2) = a2² + a1² + a2² – 2a1a2 + a1a2 –a2² =a1² + a2² –a1a2 = (AB) ² (AC) ²= a1² + a3² – a1a3 =a1²+ (a1-a2)² - a1(a1-a2)

Journal of Pharmaceutical and Biomedical Sciences © (JPBMS), Vol. 23, Issue 23

Cardiology

Samuel Rajini / JPBMS, 2012, 23 (17)

=a1²+ a1² + a2² - 2a1a2 – a1² +a1a2 =a1² + a2² -a1a2= (AB)² Thus (AB) ²= (AC) ²= (BC)²

The first equilateral triangle (bigger one in red color) is formed by the three bipolar limb leads 1, 2, and 3. The second equilateral triangle (green color) is formed by the three unipolar limb leads avR, avL and avF.

So, all the sides of the triangle are equal and hence it is an absolute electrical equilateral triangle. Second triangle (formed by the three unipolar limb leads) Let R, L, F be the vertices of the triangle( in green color in the fig) formed by the voltages of the unipolar limb leads avR, avL, and avF in the hexaxial reference system. Similarly by using the cosine formula we can see that:(RL)²= b1² + b2² –b1b2 (RF)² =b1² + b3² – b1b3 (FL)²= b2² + b3² + b2b3 As it is evident that:avR + avL + avF = 0 Thus b1+ b2 + b3 =0 But here b1 is negative (avR is negative here). Hence b2+ b3=b1 Using the above formula and substituting it in the (RF)² and (FL)² the following results was found. (RF)²= b1² + b3² – b1b3 =b1² + (b1-b2)² –b1(b1-b2) =b1² + b1² + b2² -2b1b2 –b1² + b1b2 = b1² + b2² –b1b2 = (RL)² (FL)² = b2²+ b3²+ b2b3 =b2²+ (b1-b2)² + b2( b1-b2) =b2² + b1² + b2² -2b1b2 + b1b2 –b2² = b1² + b2² –b1b2 =( RL)²

We already know the following relationship between the bipolar and unipolar limb lead voltages. avR= -(lead 1+ lead 2)/2 avL= lead1 - (lead2) /2 avF= lead 2- (lead 1) /2 Hence in this case: b1= -(a1 + a2) / 2 b2= a1 – a2/2 = (2a1 - a2)/2 b3= a2 – a1/2 = (2a2-a1)/2

Lead 1+ lead 3 = lead 2 (Einthoven equation) avR + avL + avF = 0 The above two equations are based on Kirchoff’s law which is widely accepted.

Conclusion: The two upper limbs and left lower limb are the vertices of an electrical equilateral triangle. Thus Einthoven triangle is an electrical equilateral triangle and not an anatomical triangle. These shifted triangles are also valid for vector and analytical procedures.

Hexaxial reference figure

Notations for the voltages recorded in that particular leads Lead 1 =a1 Lead 2= a2 Lead 3=a3 avR =b1 avL =b2 avF =b3

We also know the following.

Hence proved that (RL)², (RF)² and ( FL)² are equal. All the sides of the triangle are equal and hence it is an equilateral triangle.

Ratio of the two triangles

Let the voltages recorded in a particular limb leads be the following.

Proof: Let A, B, C be the vertices of the equilateral triangle formed by the voltages of the bipolar limb leads 1, 2 and 3 in the hexaxial reference system. Let R, L, F be the vertices of the equilateral triangle formed by the voltages of the unipolar limb leads avR, avL, and avF in the hexaxial reference system. Formula used: Cosine formulae (for a triangle with a, b, c as the sides and A as the angle) a² = b² + c² -2bcCOSA Hence in this case we have, (AB)² = a1² + a2² -2a1a2COS60° (we know cos60° is ½) (AB) ² =a1² + a2² - a1a2 (equation 1) (RL)² = b1² + b2² -2b1b2COS60° =b1² + b2² - b1b2 But in the above diagram avR is in negative quadrant. Hence b1 is negative.

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Journal of Pharmaceutical and Biomedical Sciences © (JPBMS), Vol. 23, Issue 23

Cardiology

Samuel Rajini / JPBMS, 2012, 23 (17)

So, (RL)² = b1² + b2² +b1b2 ( equation 2) It’s already stated that,

Hence it is suggested that combination of the 12-lead ECG and this special type of frontal plane electrocardiogram provide the optimum approach to ECG interpretation as compared to either method alone.

b1 = - (a1+a2)/2 b2 = (2a1-a2) /2 Hence (RL)² can be rewritten as follows: (RL)² = (a1² + a2² + 2a1a2) / 4 + (4a1² + a2² -4a1a2) / 4 1/4( 2a1² +2a1a2 –a1a2 –a2²)

Why most of them don’t accept the Einthoven’s hypothesis?

=( 5a1² + 2a2² -2a1a2 )/4 – 1/4 ( 2a1² - a2² + a1a2 )

What are the questions arises which doubted the Einthoven’s hypothesis?

=1/4( 3a1² +3a2² - 3a1a2 )

Explanations for each of the questions are given below.

(RL)²= 3/4( a1² + a2² -a1a2 ) ( equation 3) Ratio of the two equilateral triangles: From the above equations we see that, (AB)² / (RL)² = 4/3 An equilateral triangle can be converted into a circle with the three vertices of that triangle lying on the circle with the centroid of the triangle as the centre of the circle. Here : (Radius)² = ( side)² / 3 ( in mathematics ) Since the ratio of the area of the two equilateral triangles is constant the ratio of the area of the circles is also constant and equal to 4/3. Hence the ratio of the radii of the two circles is square root of 4/3 and is equal to 1.154. Hence if we multiply each unipolar limb lead voltages by 1.154 ( the correction factor ) and then plot it in the hexaxial reference system the two equilateral triangles lie on the same circle with same origin, same centre and same orientation.

Discussion: In Einthoven’s triangle, heart (in zero potential) is situated in the centre of the hexaxial reference system. When the heart acquires certain potential (during depolarization and depolarization) the triangle gets shifted but maintains its equilaterality. Hence Einthoven triangle is not an anatomical triangle. The right arm, left arm, and left leg are the vertices of an electrical equilateral triangle. Thus each cardiac wave (p, qrs, t) can be represented in the form of circles. All circles (see the diameter) should be formed in the left lower quadrant except QRS which can go up to -30 degree. When the angle between the ‘QRS’ and ‘T’ circles widen it usually denotes ischemia. Larger the size of a circle, higher will be the voltage. No circle should be formed during STsegment since it is an isoelectric period. Formation of circle and it’s magnitude during the ST-segment indicate the amount of myocardial injury. 4

Thus by seeing the size and location of the circles in the hexaxial reference system, interpretation can be done easily.

1. However vector addition rules are valid for any shape of the vector triangle. (lead 1 + lead 3 = lead 2 ) Here many of them thought that this equation is referring to the lead vectors. But it is not so. Here it refers to the voltage recorded in a particular lead. Voltage is a scalar quantity in physics and not a vector. 2. In ecg, most of them think that voltage is a vector quantity and they are using voltage as a vector for practical purposes. We should not change the physics principle. It is already accepted one in physics that voltage is a scalar quantity. 3. Einthoven triangle is not valid for analytical and vector procedures. In mathematical proof it has been have clearly shown that by any method the shifted Einthoven triangles can be shown to be an absolute equilateral triangle. 4. There is no evidence that the triangle depends on geometrical properties of equilaterality to depict. The triangle which has been constructed in the hexaxial reference system is a perfect and absolute equilateral triangle with equal sides and each angle being 60. 5. Equilateral triangle is only one of an unlimited number of possibilities. No. there is only one possible equilateral triangle. No other shape of the triangle is possible. Most of the researchers got wrong answer and not able to find the correct answer because they thought that Einthoven triangle is a vector triangle. In the proof it has been clearly shown that the vector triangle concept would be simply explained that the triangle gets shifted. We need not to prove the shifted vector triangle. It is understood. 6. We cannot consider the human body to be a homogeneous spherical volume conductor, as further studies by other researchers have shown that the human body to be an in homogeneous cylindrical volume conductor.

Journal of Pharmaceutical and Biomedical Sciences © (JPBMS), Vol. 23, Issue 23

Cardiology

Samuel Rajini / JPBMS, 2012, 23 (17)

Einthoven was a doctor, physiologist, and he was also well versed in physics. In the proof it has been shown the electrical field of the heart as a circle with heart at the centre of the circle. Present research work clearly explained that the circle forms only when the heart acquires certain potential (depolarization and repolarisation). When the heart has zero potential there is no circle since there’s a no electrical field. In physics, there is one principle about homogeneous volume spherical conductor (about the charge, potential, and of its electrical field) and what the present research shows is about the electrical field of the heart as circle, looking similar to that principle and that diagram. For example, we are comparing our human eye to camera and human brain to a computer. They are not exactly similar but they can be compared. Similarly, Einthoven compared this principle to a homogeneous volume spherical conductor to make others understand while he was teaching. But that comparison itself confused many of them because many of them don’t know the physics principle of homogeneous volume spherical conductor.

Table 2.Patient 90/f (From the figure1, the bipolar voltages of the qrs( in mm) is taken and the voltages of the unipolar limb leads is multiplied by 1.154 and then taken). Leads QRS 2ND BEAT Lead 1 9.0 Lead 2 - 4.0 Lead 3 -13.0 avR - 2.885 avL 12.694 avF -9.809° Angle -47.46° From the figure 1, the bipolar voltages of the qrs( in mm) is taken and the voltages of the unipolar limb leads is multiplied by 1.154 and then taken. This is to show that the two equilateral triangles lie on the same circle with heart at the centre. The constant 1.154(correction factor) is taken because the bipolar and unipolar limb leads have different resistance. Table 3.Patient 35yrs/female (from figure 5, the net voltages (in mm) of the qrs and t vectors are taken). 3rD BEAT QRS T Lead 1 8.0 2.0 Lead 2 8.0 2.0 Lead 3 0 0 avR -8.0 -2.0 avL 4.0 1.0 avF 4.0 1.0 Angle 29.98° 29.98° From figure 5, the net voltages ( in mm) of the qrs and t vectors are taken. Here qrs/t angle is normal.

The word spherical is a clue here because human body is not spherical is a known fact (Einthoven himself would have known that) 7. Einthoven never published a detailed description of the reasoning behind the Einthoven’s triangle. It has been discussed in a detailed description. 8. Cardiac vector is an imaginary quantity of dimensions of unknown physical quantity. Cardiac vector is an electrical field vector of dimensions volt/metre.

Results:

Figure 1.ECG of the patient 90yrs/female.

The new discovery of two equilateral triangles (that can be converted into circles) in the hexaxial reference system is constructed for the following ECG’s. We also see that the heart is situated in the centre of the electric field which it generates. Table 1.From the figure 1, the bipolar and unipolar limb lead voltages of the qrs vector (in mm) is taken Leads QRS 2ND BEAT Lead 1 9.0 Lead 2 - 4.0 Lead 3 -13.0 avR - 2.5 avL 11.0 avF -8.5 Angle -47.46° From the figure 1, the bipolar and unipolar limb lead voltages of the qrs vector(in mm) is taken. This is to show that two circles with same origin, same orientation but with different radii is seen.

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Figure 2. 2 circles (without triangles) of the ‘qrs’ voltages of the patient 90/female

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Cardiology

Samuel Rajini / JPBMS, 2012, 23 (17)

Figure 3. 2 circles (with 2 triangles) of the ‘qrs’ voltages of the patient 90/female

Figure 7. 2 circles (without triangles) of the‘t’ voltages of the patient 35yrs/female

Figure 4. 1 circle (with 2 triangles) of the ‘qrs’ voltages of the patient 90/female Figure 8. Einthoven triangle

Thus we see that in the hexaxial reference system net voltages of the bipolar limb leads and the net voltages of the unipolar limb leads each form equilateral triangles. By applying the correction factor (1.154) the two equilateral triangles can be seen to lie on the same circle with heart at the centre.

References:

Figure 5. ECG of the patient 35yrs/female.

VCG

Figure 6. 2 circles (without triangles) of the ‘qrs’ voltages of the patient 35yrs/female

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1. Katzeff IE, Gathiram P, Edwards H, Booyens J Dynamic electrocardiography. 111. The imaginary cardiac vector hypothesis: theoretical basis. Med hypotheses 1981 mar;7(3):373-92 2. Pentti rautaharju, farida rautaharju Investigative electrocardiography in epidemiological studies and – google books result Books.google.co.in/ books? isbn=1846284651-2006-294 pages. 3. Zang z.zao, M.D. An experimental study of burger triangles constructed from toad hearts in situ American heart association journals Circulation research.1956;4:211-216 doi: 4. Guyton and Hall. Textbook of medical physiology. tenth edition. published by Elsevier.publication year 2003. 5. William F. Ganong. Review of medical physiology. twenty first edition. published by the Mcgraw-hill companies. publication year 2003.

Journal of Pharmaceutical and Biomedical Sciences © (JPBMS), Vol. 23, Issue 23

Cardiology

Samuel Rajini / JPBMS, 2012, 23 (17)

6. Leo Shamroth. An introduction to electro cardiography.7TH edition. Published by Blackwell science ltd. publication year 2010.

Conflict of interest: - Not declared Source of funding:-None

Corresponding Author:Dr.T.Rajini Samuel., 276 third main road TNHB Colony Velachery, Chennai, 600 042, India Contact no: +91 9884971317. Email: [email protected], [email protected], [email protected].

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