Analysis of M-mode Echocardiogram

About this Notebook

This demonstration notebook is based on the work of Dr. Andrew J. Maxwell ([1]), who has used  Mathematica  for analysis of M-mode echocardiosignals. In this example, basic signal processing is applied to an echocardiosignal, recorded from a 3-year-old patient. Signals of the left ventricle, the left ventricular posterior wall (LVPW) , and the septum are presented and, velocities and accelerations of the walls are calculated and displayed. The calculated velocities and accelerations carry potentially useful clinical information, in particular, for non-invasive heart transplant rejection surveillance. As a side note, interpolation of the signals is made with quartic polynomials.

Introduction to Physiology of the Human Heart

Before we begin the signal analysis, we briefly review the physiology of the human heart and  echocardiography as well. The heart is a four-chamber blood pump, consisting of two ventricles (lower chambers) and two atria (upper chambers). The right ventricle sends blood to the lungs through the pulmonary artery (Fig. 1, [2]). The left ventricle sends blood to the rest of the body though the aorta. The left atrium receives oxygenated blood from the lungs to be sent to the body, whereas the right atrium receives blood from the veins to be sent to the lungs through the right ventricle. A number of  cardiostructures are visible with echocardiography: myocardium, the main muscle of the heart; the endocardium, a membrane lining the chambers; the septum, a muscular wall that divides the heart in half; the epicardium, a membrane covering the heart.

Introduction to M-mode Echocardiography

Echocardiography is a visualization technique that uses ultrasound as the information carrier. Ultrasound is an acoustic wave with a frequency higher than 20 kHz. However, for diagnostics, the ultrasound used is typically in a higher frequency range of about 2 MHz, which allows improved accuracy with somewhat reduced penetration.
As ultrasound penetrates through a non-homogeneous media, it reflects from the boundaries of the regions with different acoustic impedance. An echocardiogram is a recording of the reflected ultrasonic beam.
There are two major types of echocardiography: M-mode (below left) and two-dimensional (right, Fig. 2,[3]).

In the M-mode a single beam of ultrasound is used (Fig. 3,[3]).

The reflections of the signal are recorded and displayed as monochronic dots. The location of which is proportional to the distance from the reflective region and the intensity contains the information about acoustic impedance of the region. The M-mode has excellent axial resolution, but it carries essentially one-dimensional information. The two-dimensional echocardiograph typically uses a multi-element transducer, which generates a single ultrasonic beam with changing direction. This technique allows one to create a two-dimensional image, which has good lateral resolution but lacks accuracy in the axial dimension.

Analysis of M-mode Echocardiogram

Import of the Original Data

In this section, the original data is downloaded from the data files, and scaling functions are defined and applied to data. Then, a common list of data is formed and displayed. Finally, the heartbeat is calculated.

Interpolation of a Single Cycle with Quartic Polynomials

In this section, a composite cycle is created and displayed. The standard kernel function Fit is used for the data interpolation. The results of the interpolation are then displayed together with the composite cycle.

Calculations and Graphics for the Wall and Chamber: Sizes, Velocities, and Acceleration

In this section, the signals for the left ventricle, the left ventricular posterior wall, and the septum are calculated and displayed. First and second derivatives are calculated with the standard kernel function D, and displayed.

Ventricle, Left Ventricle Posterior Wall, and Septum Sizes

Left Ventricle: Size,Velocity, and Acceleration

Note: Due to the use of quartic polynomials for interpolation, the result of the second derivative will be a quadratic polynomial. It is sufficient for our goals of calculating the difference between the maximum and the minimum values  of the derivatives.

Left Ventricle Posterior Wall

Septum

Conclusion

In this notebook,  Mathematica was applied to a set of  problems in biomedical signal processing. Echocardiosignals of the left ventricle, left ventricle posterior wall, and septum were interpolated by means of  the Mathematica function Fit. In addition, velocities and accelerations of the heart chamber's walls were calculated and graphed. These velocities and accelerations carry potentially useful clinical information, in particular, the up-stroke of the ventricular diameter curve and the minimum diameter achieved are important for non-invasive heart transplant rejection surveillance.

Bibliography

1. Dr. Andrew J. Maxwell. A set of Mathematica Notebooks (in code).

2. The American Heart Association Homepage: http://www.amhrt.org/Heart_and_Stroke_A_Z_Guide/hworks.html.
3. College of Veterinary Medicine, the University of Tennessee. Homepage: http://www.vet.utk.edu/m-mode.html.
4.  Dr. Daniel Shindler. Echocardiography Journal. Homepage: http://www2.umdnj.edu/~shindler/index.html.