How is adding up action potentials equivalent to measuring the direction of depolarization vectors?

Question

An ECG of the heart measured from lead II looks like this:

I have seen two ways of arriving at this image, the first is by considering the depolarization and the repolarization vectors and their direction's relative to lead II, like in the image below:

The other way was by adding up all the action potentials produced by myocariocytes

My question is how do these two methods arrive at the same ECG image, how is adding up action potentials equivalent to measuring the direction of depolarization(repolarization) vectors?

Answer 1

Electrocardiography (ECG)

ECG Leads

Firstly, it is important to clarify that a lead (like lead 2) does not refer to one of the electrodes on the patient. Each lead represents the potential difference between two of the electrodes. In the case of Lead 2, it is the potential difference between the electrodes on the left leg and the right arm.

The vectors of the standard leads are approximately in the coronal plane, while the vectors of the chest leads are in the transverse plane (the chest leads V1-V6 represent the potential difference between the left leg electrode and the corresponding chest electrode).

The augmented leads (aVR and aVF) are mathematical derivations based on the others.

Thus, the leads taken together provide a three-dimensional representation of the average electrical potential through the body (and by minimising other muscle movement, what is left is electrical activity due to the heart).

Fig 1: Electrical axis of ECG leads (from ECGpedia.org)

ECG Waveform

As regards why we arrive at the same approximate ECG tracing, remember each lead is unaware (as it were) of any individual electrical potentials within different components of the heart; they only measure the average depolarisation along their vector over time, through every part of the body between the two electrodes.

Each lead assesses the depolarisation from a different perspective, so the leads vary in the relative positive or negative deflection.

For example, a wave of depolarisation moving perpendicular (orthogonal) to the vector of the electrodes (on average) will not be detected. Lead 2 is often used to detect the rhythm as the depolarisation in the atria is propagated directly along the vector of lead 2, leading to the positive deflection of a P wave. Conversely, P waves are often quite flat in aVL, as the vector of average depolarisation over the atria is approximately orthogonal to aVL (see diagram above).

Because all the leads are measuring the same thing at the same time, and because the heart has a set sequence (thanks to the SA and AV nodes), the tracings of each lead look broadly similar in terms of timing.

Vectors

As to why summing the overall vectors works, this is just how vectors work mathematically. They represent quantities where the direction is important, as opposed to scalar quantities, which only have a value.

Fig 2: Vector addition (from grc.nasa.gov)

It is useful to think of the value you see in a given lead at any point in time representing the total addition and subtraction of all electrical vectors (relative to that lead).

See more about vectors here and vector addition at this NASA page.

Answer 2

As far as I understand, in the vector-cardiogram, the Heart Vector H(t) changes from time to time, and its projection to each induction direction is induction I, II and III. In other words, we have first the following time-dependent Vector.

Here, θ(t) is the declination angle between H(t) and I induction measured in counterclockwise direction.

The Lissajous curve plotting the trajectory (green dotted line) of the Heart Vector (yellow arrow) is called a Vectorcardiogram.

https://i.stack.imgur.com/AydpU.png

And according to the same manner of thinking - which is a bit virtual - if the change in potential corresponding to the curve in each of your graphs could be measured independently in at least two inductions, you could create a vector for each component. Let it be H1, H2, H3, ...　then the vector H, which is the basis of the entire ECG, would be decomposed as follows

H=H1+H2+...

---

Let us now review the relationship between each induction and the vector ECG.

Since the time variable t is considered fixed thereafter, it is omitted from the equation. In the figure below, and,

If we simply write I, II, and III induction readings (V I, V II, and V III) as I, II, and III, respectively.

The following relationship is obtained from the assumption that the projection of H in each induction direction is I, II, and III induction, respectively;

Here, from the following calculations, we find that the Einthoven's relation formula holds as follows;

That means,

Then, eliminating the term cos from equations (1-2-II) and (1-2-III), we obtain;

So,

And by combining (1-6) with the Einth-Haens equation, we obtain

So finery we get,

So much for the essential argument.

ーーーーーーーーーーーーーーーーーーーー

From now on, we will look at the relationship between the potentials actually measured by the electrocardiograph, namely aVL, aVR, aVF, VL VR, and VL, and the Heart Vector.

Now, two types of circuits are often used in ECGs: Wilson-type circuits that measure VL, VR, and VF, and Coldberger-type circuits that measure aVL, aVR, and aVF. Since both circuits are grounded, the absolute reference of the circuit (i.e., the ground electrode) is the potential of the earth's surface (this is how it is expressed in circuit books, but frankly it can be anywhere as long as the potential is not indefinite), and the potentials of the L, R, and F electrodes from the absolute reference of the circuit can be calculated respectively as　.

Since the potential difference does not depend on the way the reference is taken, we get

In the Coldberger-type circuit, aVL, aVR,aVF are recorded, and the circuit configuration is shown in the figure below, respectively (actually, all are measured simulteniously)

If we follow the mannerisms of a high school physics textbook, we get

This relational equation is called the Coldberger's equation.

means

So,

Similarly, in the Wilson-style circuit configuration,

Here,

So, we get

and,

In light of the above discussion；

So,

And finally we get,

Here, I will write it down to avoid confusion, the conclusion of the discussion of above two types of circuit　configurations can be summarized as follows, although this is also valid for ordinary

See also

• How to determine the "argument" of the electrical axis of the ECG from aVF ＆ VI
• Is the definition of aVF consistent with Novosel’s formula?

◆References
https://www.physoc.org/magazine-articles/trigonometry-of-the-ecg/ https://www.bem.fi/book/15/15.htm