If you are a sonographer, whether you recognize it or not, you are in the detective business. As much as any of the colorful characters featured on CSI, Bones, or (for you folks a few years older) Columbo, you have to make sense of what is oftentimes opaque or confusing data. It is your ability to deductively reason that often makes the difference between a correct diagnosis and misinterpretation.
A misconception held by many new and developing ultrasound professional is that math equates to memorization of values and equations. This is generally followed with statements such as “Why do I need to know equations when the system does all the calculations for me?” Although the statement regarding the system is true, this thought process does not take into consideration some very important aspects of imaging.
I have seen many imaging situations in which calipers were carefully placed to make a measurement, and then the user clicked on the wrong calculation, not recognizing the error. In these cases, the numerical results were far from reasonable, but unfortunately the assumption was made that the system would always produce the right results. The mistaken belief was that once the calipers were set, errors were not possible. These instances may be few and far between, but that is little consolation to a specific patient now attached to inaccurate data for their diagnosis.
Much more importantly, there are two very compelling reasons why you should know the equations and understand the relationships between variables you measure and the resulting calculations: 1) understanding error sources and 2) the ability to predict the body’s compensatory mechanisms for disease.
The ability to assess errors and error sources derives directly from understanding the mathematical relationships used in the calculations. Consider what happens if you measure the radius of a vessel to determine cross-sectional area. If your measurement is off axis, or if you are unable to place the cursors accurately, the measured radius will have some degree of error. Since the area calculation squares the radius, the error increases non-linearly. Of course going from one-dimensional measure to three-dimensional calculations (such as radius to volume) will result in an even more rapid increase in the calculated error. For example, a 10% error in radius would result in approximately 30% error in the volume calculation. The same problem exists when calculating a pressure gradient from a measured Doppler peak velocity. In this case, the error comes first from non-linear angle effects on the measured velocity, and then from the squaring that occurs by employing the modified Bernoulli Equation. In these cases, a little bit of angular error can go a long way!
When the body is stressed, whether through exercise or disease, it tries to compensate. Most of the body’s responses to these stresses are non-linear. For example, when there is a reduction in flow cross-sectional area, the local resistance increases by the change in radius to the fourth power. To compensate for this increase in resistance, there is generally dilatation of the distal vascular beds.
Consider then, the significance of the radius of a mass increasing by a factor of two over a six month period. Because of the cubed relationship between radius and volume, a factor of two increase in radius implies a volume increases by a factor of eight. The ramifications are clear, this degree of increase implies a change in flow volume to the mass, a likely change in vasculature to the mass, and high probability of cancer. By understanding the equations that govern the assessed parameters, a “small sign” can give strong warnings to look for other secondary changes in the body.
Which brings us back to the detective aspect of your profession. As the television characters so dramatically review the crime scenes and make seemingly amazing observations from visual evidence, you too will be called upon to exercise similar deductive skills. What you may not fully appreciate today is that those “skills” are informed by an implicit comprehension of underlying mathematical correlations.
Oh, there is another important reason why you need to understand mathematical relationships. This ability it tested heavily in physics questions on the various credentialing exams.