Heart Vectors

How the pattern comes out of the wave

Sixteen short steps, from one cell to a bundle branch block. Each has something to try. The physics panels show the exact formulas the simulator uses.

Part I From a cell to a lead

1 One cell

A resting heart muscle cell keeps its inside about 85 mV more negative than the outside. When it is excited, sodium channels open and the inside swings positive within a couple of milliseconds: the upstroke. Calcium holds it there for a few hundred milliseconds, the plateau, and then potassium currents bring it back to rest. Every cell in the simulator follows this one curve, shifted to the moment the wave reaches it. Only the length of the plateau varies from place to place. Try this Shorten the action potential. Only the repolarisation moves.

Show the physics

Transmembrane potential s ms after activation, for a cell with duration D:

V_m(s) = V_\text{rest} + A\,\sigma\!\left(\tfrac{s}{k_\text{up}}\right)\left[1-\sigma\!\left(\tfrac{s-D}{k_\text{rep}}\right)\right]\left(1-c\,\min\!\left(\tfrac{s}{D},1\right)\right),\qquad \sigma(x)=\frac{1}{1+e^{-x}}

with V_rest = −85 mV, A = 105 mV, k_up = 4 ms, k_rep = 34 ms and a plateau that sinks by c = 45 % over D. A real upstroke is faster (1–2 ms); the simulator widens it so the wavefront spans a couple of its 2 mm cubes, otherwise the traces ripple. The slow recovery and the sinking plateau were fitted to the T waves of real ECGs; they stand in for the cell-to-cell spread of recovery that the model's few gradients do not have.

2 A row of cells

Put cells in a row and excite the left end. Each cell excites its neighbour, so a wave runs to the right at about half a metre per second. Where neighbouring cells are in the same state, no current flows between them. Current flows only where neighbours differ, at the front. The front acts like a tiny battery lying along the row, a dipole, pointing from the excited cells towards the resting ones. During activation it points the way the wave travels. Later the cells recover one by one and a second front appears. Its dipole still points from excited to resting tissue, which is now against the direction the recovery travels.

Try this Pause while the wave runs, then again while the cells recover, and compare the arrows.

Show the physics

The impressed current is $\mathbf J = -\sigma_i \nabla V_m$ . Between neighbouring cells a and b, h apart, the face carries the current dipole

\mathbf p_{ab} = \sigma_i h^2 \,(V_a - V_b)\,\hat{\mathbf n}_{ab}

so it vanishes wherever V_m is uniform and points from high to low V_m.

3 What an electrode sees

An electrode outside the tissue adds up the fields of all the dipoles, each weighted by distance and direction: nearer dipoles count more (1/r²), and a dipole side-on to the electrode counts less. A front coming towards the electrode turns its positive end to it, and the trace goes up. A front moving away shows its negative tail, and the trace goes down. A front passing underneath gives first up, then down. When every cell is in the same state, all resting or all on the plateau, there is no front and the trace is flat. That is why the ST segment sits near the baseline: the whole ventricle is on its plateau.

Try this Put the electrode at the right end, then the left end, then the middle. Then tick “right end recovers sooner” and watch the T wave flip. Step 12 explains why.

Show the physics

In an infinite uniform conductor of conductivity σ, the potential at an electrode at r is the sum over all faces f, with centres c_f:

\Phi(\mathbf r) = \frac{1}{4\pi\sigma}\sum_f \mathbf p_f\cdot\frac{\mathbf r-\mathbf c_f}{\lvert\mathbf r-\mathbf c_f\rvert^{3}}

4 Many sources, one vector

In a sheet of muscle the front is a line of small dipoles side by side. Seen from far away they add up like arrows, tip to tail, into one vector. In the heart this sum over every front at one instant is the heart vector. It turns and changes length during the beat, and its tip traces a loop.

Try this Start the wave in a corner, then in the middle. A wave spreading evenly in all directions cancels itself out. A wave running mostly one way makes a big vector.

Show the physics

\mathbf H(t) = \sum_f \mathbf p_f(t),\qquad \Phi(\mathbf r)\approx\frac{\mathbf H\cdot(\mathbf r-\mathbf c)}{4\pi\sigma\lvert\mathbf r-\mathbf c\rvert^{3}}\ \text{ when } \lvert\mathbf r-\mathbf c\rvert \gg \text{heart size}

5 A lead is a direction

Far from the heart, each limb lead reads the heart vector projected onto its own axis. Lead I is horizontal (0°), II points down and to the patient's left (right on the diagram) at +60°, III at +120°. The augmented leads aVL, aVF and aVR sit at −30°, +90° and −150°. A vector parallel to a lead's axis gives that lead its biggest deflection; a perpendicular one gives none. Because the axis of II is the sum of the axes of I and III, lead I + lead III = lead II at every instant.

Try this Drag the arrow until III is flat. Where does it point? Then make I and aVF equal.

Show the physics

\mathrm{I}=\Phi_\text{LA}-\Phi_\text{RA},\quad \mathrm{II}=\Phi_\text{LL}-\Phi_\text{RA},\quad \mathrm{III}=\Phi_\text{LL}-\Phi_\text{LA},\quad \mathrm{aVR}=\Phi_\text{RA}-\tfrac12(\Phi_\text{LA}+\Phi_\text{LL})

With the three limb electrodes at the corners of an equilateral triangle far away, each lead is $\mathbf H\cdot\boldsymbol\ell$ for a fixed lead vector $\boldsymbol\ell$ : the hexaxial system. The simulator places them that way, 240 mm from the heart, on the surface of a spherical torso.

6 Chest leads are close

The six chest leads look at the heart in the horizontal plane, each measured against Wilson's central terminal, the average of the three limb electrodes. They sit only a few centimetres from the heart, so the far-away rule bends. A source near an electrode counts far more than the same source in the middle of the heart. V1 and V2 lie over the right ventricle and the septum, and are especially sensitive to what happens there.

Try this Drag the source towards V1. V1 grows well beyond its projection.

Show the physics

V_k = \Phi_{V_k} - \tfrac13(\Phi_\text{RA}+\Phi_\text{LA}+\Phi_\text{LL}),\qquad \Phi \propto \frac{\mathbf p\cdot(\mathbf r-\mathbf s)}{\lvert\mathbf r-\mathbf s\rvert^{3}}

The 1/r² falloff makes a source 3 cm from an electrode count four times as much as one 6 cm away. The playground computes every lead with the full sum, so this effect is always in the traces; the dashed projection in the slices is the far-field approximation.

Part II The normal beat

7 Atria and the P wave

The sinus node, high in the right atrium, fires first. The wave spreads over both atria from upper right to lower left at about a metre per second. The atrial walls are thin, so the vector is small. It points down and to the left: the P wave is upright in I and II and inverted in aVR, with an axis around +45° to +60°.

Try this Move the frontal slice's depth in the playground until the atria appear, then replay the P wave there.

8 The pause

A fibrous ring separates the atria from the ventricles and does not conduct. The only bridge is the AV node and the His bundle. The AV node holds the impulse back for about 80 ms, which gives the atria time to empty. Nothing large enough to see from the surface happens meanwhile, so the trace is flat. The P wave and this flat stretch together make the PR interval.

Try this Watch the heart vector shrink to nothing after the P wave, long before the QRS starts.

9 The septum goes first

The left bundle reaches the left side of the septum first, so the septum is activated from left to right. For its first 10 to 20 ms the heart vector points forward and to the right. Leads that look from the left, such as aVL and V6, see it moving away and write a small q wave. V1, in front and to the right, sees it coming and writes a small r wave.

Try this Pause at the first arrows in the septum and check which way they point.

10 Apex and free walls

The Purkinje network lines the inside of both ventricles and spreads the impulse at about 2 m/s, so the muscle is activated from the inside out. The two ventricles' fronts point in roughly opposite directions, but the left ventricular wall is much thicker. The left side wins: the vector swings to the left, down and back. That gives the tall R waves in I, II, V5 and V6 and the deep S waves in V1 and V2.

Try this Find the moment the right ventricle is fully activated but the left is not. Which way does the vector point?

11 The base comes last

The base of the heart, next to the valve rings, has no Purkinje fibres. It is reached last, by slow cell-to-cell spread from below, and its vector points up and back. Leads that look from below, II and aVF, end their QRS with a small s wave. When the base is done, the whole ventricle is on its plateau and the trace comes back to the baseline.

Try this Watch where the last activated tissue sits in the frontal slice.

12 Recovery and the T wave

Repolarisation is the opposite charge movement, so you might expect the T wave to point against the QRS. In a healthy heart it does not, because recovery does not run in the order of activation. Action potentials are shorter in the outer wall than in the inner wall, enough that the outer wall recovers first even though it was activated last. The base, activated last, also recovers last. During the T wave the still-excited tissue lies inside and at the base, and the vector points from there towards the recovered tissue: outwards and towards the apex, much as during the QRS. So the T wave points the same way as the QRS. It is concordant. In the horizontal plane, the T vector sits further forward than the QRS vector, so V1 can show an upright T after its negative QRS, or, as here, an almost flat one. That is a normal QRS–T angle, not a discordance.

Try this Compare with step 3: with uniform action potentials the T wave flips.

Part III Bundle branch blocks

13 Right bundle branch block

Cut the right bundle. The left bundle still activates the septum from left to right, so the small r in V1 and the q in V6 stay, and the left ventricle is activated on time. The right ventricle now has to be reached from the left, through the septum, cell by cell, and its own Purkinje network, entered from the muscle instead of from its bundle, carries the activation on at less than half its usual speed. By the time the left ventricle has finished, the right ventricle is still being activated, and nothing opposes its forces any more. The vector makes a late, slow turn to the right and forward. V1 sees it coming and writes a second positive wave, R′. I and V6 see it leaving and write a wide, slurred S. The QRS lasts 120 ms or more.

Try this Pause on the R′ in V1 and look at which ventricle is still activating.

14 From incomplete to complete

A bundle does not have to be cut to matter. Delay the right bundle, and activation from the left starts to invade the right ventricle before the right bundle gets there. Step by step the R′ in V1 grows and the QRS widens, from normal through incomplete block (110 to 119 ms) to complete block.

Try this Move the slider one stop at a time and read the QRS width under the leads.

15 Left bundle branch block

Cut the left bundle instead. Now the right bundle starts the ventricles, and the septum is activated from its right side, right to left. The early septal vector points left, so the q waves in aVL and V6 disappear, and so does the small r in V1. The left ventricle is reached only through the septum and fills slowly through working muscle, its lateral and posterior base last. The vector points left and back for a long time: broad, notched or slurred R waves in I, aVL and V6, a deep QS or rS in V1, and a QRS of about 140 ms or more.

Try this Pause where the R in V6 stalls on its way up and find the front in the slice.

16 Why the T wave flips in blocks

In a block the order of activation changes, and recovery follows the new order: the part activated last also recovers last. During the T wave the late ventricle is still excited and the early one has recovered, so the vector points away from the late ventricle. The late QRS forces pointed into it. The ST segment and T wave therefore point against the main or terminal QRS deflection: an inverted T in V1 in right bundle branch block, inverted T waves in I and V6 in left bundle branch block. These are secondary changes. They follow from the conduction and do not mean ischaemia by themselves.

Try this Compare the T wave with the last part of the QRS in each lead.

Model What this simulator leaves out

The heart is built from ellipsoids, 2 mm cubes, about 26 000 of them. Real hearts are not this smooth.
Muscle conducts nearly three times faster within the wall (0.7 m/s) than across it (0.26 m/s), the same in every direction within the wall. Real fibres turn through the wall and conduct fastest along their own direction; the model averages that.
The Purkinje network is a layer on the inner wall, not a branching tree. It conducts fast only when its own bundle drives it; reached from the muscle (beyond a block, or before a delayed bundle arrives) it conducts at less than half that speed. So a long enough delay looks exactly like a block.
The body is a uniform sphere with an insulating surface, with no lungs, blood or real chest shape. This changes the amplitudes, not the shape of the story.
The heart's orientation is fixed, its long axis about 65° in the frontal plane, so the normal QRS axis comes out near +30°. In right bundle branch block the measured axis (from the whole QRS) swings to about +90°, because the slow terminal forces point right; machines often quote the axis of the first 60 ms instead.
In left bundle branch block V4 shows a notch on its S wave. In a real tracing that notch (Cabrera's sign) suggests an old anterior infarct; here it comes from where V4 sits over the model heart, not from any scar.
In left bundle branch block the S in V1–V3 reaches its deepest point earlier and more sharply than in most real tracings, and II and aVF end their QRS positive where real ones mostly end negative (in real left bundle branch block the inferior leads vary a lot).
One beat at 60/min repeats. Rate, rhythm and escape rhythms are not modelled.
The numbers that govern the beat (speeds, delays, action potential durations, electrode positions) start from human measurements. They were then tuned so the normal ECG and the blocks meet the published criteria and look like real tracings: each was compared lead by lead with the median beats of 197 clinical ECGs from the PTB-XL database (normal, complete and incomplete right, and complete left bundle branch block).

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