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Log Conversions and Decibels in Ultrasound

This video will talk about the logarithmic scale, how to convert to and from this. And also about the units of decibels. There will be a particular focus on its applications in ultrasound.

Firstly, the attenuation of an ultrasound as it passes through a medium such as muscle is given by this exponential decay expression.

Here, x is the distance or depth in the medium that the ultrasound has travelled.
I is the intensity of the ultrasound wave.
Two intensity values are used in this formula: the initial intensity at the start of the medium, and the intensity at a depth x.
The attenuation coefficient is alpha.

This is what the exponential decay looks like.

Now to convert this into the logarithmic scale, we can use a logarithmic unit called the decibel.

The decibel can be simply defined as the ratio of one value to another value, on a log scale.

For ultrasound, the ratio is of the intensity at depth x to the initial intensity.
That is, Ix over I0. Put in a log scale.

Here, is the formula of ultrasound attenuation where the left hand side is the definition of the decibel. Note that alpha subscript dB is a constant and is numerically different to alpha.
The plot shows a straight line with a negative gradient.

Let’s compare the two plots. The top plot is exponential decay, the bottom plot is using a log scale.

At a distance where the intensity has dropped to 50%, the corresponding value in the bottom plot is 3 decibels.
We can show this with a simple calculation. By substituting Ix = 0.5 I0 into the log expression, we arrive at -3.01 decibels, or an attenuation of 3 decibels.
If we double this distance, the attenuation in decibels is also doubled. And so on.
Therefore, there is a linear relationship between the distance the ultrasound has travelled, and its attenuation in decibels.

Another benefit is that the decibel can represent orders of magnitude, or smaller values, better.

What is the relationship between alpha dB and alpha?
If we take the log to base 10 on both sides of our original equation, and apply log rules, we arrive at this. Now comparing this with our other formula, we arrive at this relationship. Note that alpha dB is measured in decibels per centimetre, and alpha is measured in per centimeter.

Hopefully from this video, you will now be able to convert the attenuation of ultrasound intensity to and from the logarithmic scale using decibels and the relationship between the attenuation coefficients.

Log scales also have applications in pH for acidity and the Richter scale for earthquakes.
The units of decibel has uses in electronics and acoustics such as in ultrasound.