Pedram Ramezani - Interactive Cabrera

Understanding the Cabrera System

The Cabrera system or Hexaxial reference system, often referred to as the Cabrera sequence or Cabrera circle, represents a logical sequence in which electrocardiogram (ECG) leads are observed. This circle offers a bird's-eye view of the heart's electrical activity in the frontal plane.

Historical Overview

Originating from the work of Dr. Enrique Cabrera, this system seeks to provide a more coherent representation of the heart's electrical vectors. Traditional ECG presentations often separate limb leads (Ⅰ, Ⅱ, Ⅲ) and augmented limb leads (aVL, aVR, aVF), but the Cabrera system presents them in a continuous sequence, offering a more comprehensive and intuitive understanding.

Visualization and Importance

Visualized as a circle, the Cabrera system starts with the lead I on the horizontal plane and moves clockwise, integrating the other leads and returning to the horizontal plane. This arrangement:

Offers a continuous transition from one lead to another.
Presents a more holistic view of the heart's electrical activity.
Facilitates easier recognition of the heart's electrical axis.

The Math Behind Projections

To comprehend the projections in the Cabrera system, a dive into vector mathematics is indispensable.

A Closer Look at Scalar Projection

Scalar projection, sometimes referred to as the "scalar component", is the length of the projection of one vector onto another. To understand this concept, we'll begin with the geometric interpretation of the dot product.

The Coordinate definition of the Dot Product

Given two vectors $\mathbf{a} = [a_1, \ldots, a_n]^T$ and $\mathbf{b} = [b_1, \ldots, b_n]^T$ , their dot product is defined as:

\mathbf{a} \cdot \mathbf{b} = \sum_{i = n}^{n} a_ib_i = a_1b_1 + \ldots + a_nb_n

If $\mathbf{a} = \mathbf{b}$ their dot product can also be defined as:

\mathbf{a} \cdot \mathbf{a} = a_1a_1 + \ldots + a_na_n = \sqrt{a_1^2 + \ldots + a_n^2}^2 = \vert \mathbf{a} \vert^2

The Geometric Interpretation of the Dot Product

In Euclidean space like the cabrera system in $\mathbb{R}^2$ with standard dot product we can also use the geometric definition:

\mathbf{a} \cdot \mathbf{b} = \vert \mathbf{a} \vert \vert \mathbf{b} \vert \cos(\theta)

Where:

$\vert \mathbf{a} \vert$ and $\vert \mathbf{b} \vert$ are the magnitudes (or lengths) of vectors $\mathbf{a}$ and $\mathbf{b}$ respectively.
$\theta$ is the angle between the two vectors.

The factor $\cos(\theta)$ is a critical component here. If we visualize this geometrically:

Imagine vector $a$ being "projected" onto vector $b$ .
This "projection" casts a shadow or a line segment onto vector $b$ .
The length of this shadow is the scalar projection of $a$ onto $b$ .

Now, recall the trigonometric definition:

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

In our case:

The "hypotenuse" is the magnitude of vector $a$ , $\vert \mathbf{a} \vert$ .
The "adjacent" side is the length of the shadow, i.e., the scalar projection.

So, rearranging the terms, the scalar projection of $a$ onto $b$ (often denoted as $\text{comp}_\mathbf{b}(\mathbf{a})$ ) is:

\text{comp}_\mathbf{b}(\mathbf{a}) = \vert \mathbf{a} \vert \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\vert \mathbf{b} \vert}

Basics of Vector Projection

Given our understanding of scalar projection, vector projection becomes a natural extension. Once we've found the scalar projection (or length) of one vector onto another, obtaining the vector projection is a matter of giving that length a direction.

In the context of scalar projection, the vector projection of $\mathbf{a}$ onto $\mathbf{b}$ is obtained by multiplying the scalar projection by the unit vector of $\mathbf{b}$ (because the unit vector of $\mathbf{b}$ provides the direction). Mathematically:

\text{proj}_\mathbf{b}(\mathbf{a}) = \text{comp}_\mathbf{b}(\mathbf{a}) \times \frac{\mathbf{b}}{\vert \mathbf{b}\vert }

Where:

$\text{comp}_\mathbf{b}(\mathbf{a})$ is the scalar projection of $\mathbf{a}$ onto $\mathbf{b}$ .
$\frac{\mathbf{b}}{\vert \mathbf{b}\vert }$ is the unit vector of $\mathbf{b}$ , which provides the direction.

Expanding upon our earlier definition of scalar projection:

\text{comp}_\mathbf{b}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\vert \mathbf{b} \vert}

Combining these two equations:

\text{proj}_\mathbf{b}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\vert \mathbf{b} \vert} \times \frac{\mathbf{b}}{\vert \mathbf{b}\vert } = \frac{\mathbf{a} \cdot \mathbf{b}}{\vert \mathbf{b} \vert^2} \times \mathbf{b}

Simplifying further by using the identity $\vert \mathbf{b}|^2 = \mathbf{b} \cdot \mathbf{b}$ :

\text{proj}_\mathbf{b}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}

This equation gives us the vector projection of $\mathbf{a}$ onto $\mathbf{b}$ , taking into account both magnitude (from the scalar projection) and direction (from $\mathbf{b}$ ).

Relevance to the Cabrera System

In ECG interpretation using the Cabrera system, the heart's primary vector is akin to vector $\mathbf{a}$ , and each lead acts as vector $\mathbf{b}$ . By projecting the heart's electrical activity onto each lead, we get a reading representative of the heart's activity relative to that lead's orientation.

Interactive Cabrera Circle