## Friday, May 21, 2010

### Kalin Lucas and his ruptured Achilles tendon

Basketball fans may recall that in this year's NCAA tournament, Michigan State University (located about a 90 minute drive from where I work here at Oakland University in Rochester Michigan) made it into the final four before losing to Butler. They might have won the entire tournament if they were not without their star, Kalin Lucus, who ruptured his left Achilles tendon in a second round game against the University of Maryland. Coach Tom Izzo, who is much beloved here in southeast Michigan, managed two more wins without Lucus, before losing in the semifinals.

Why did Lucus injure his Achilles tendon? There was no collision or accident, he just landed awkwardly. Readers of the 4th edition of Intermediate Physics for Medicine and Biology won’t be too surprised when they hear about sports injuries to the Achilles tendon. In Section 1.5 of our book, Russ Hobbie and I analyze the forces on the Achilles tendon and show that the tension in this tendon can be nearly twice the body weight.
The Achilles tendon connects the calf muscles (the gastrocnemius and soleus) to the calcaneus at the back of the heal (Fig. 1.9). To calculate the force exerted by this tendon on the calcaneus when a person is standing on the ball of one foot, assume that the entire foot can be regarded as a rigid body. This is our first example of creating a model of the actual situation. We try to simplify the real situation to make the calculation possible while keeping the features that are important to what is happening. In this model the internal forces within the foot are being ignored.

Figure 1.10 shows the force exerted by the tendon on the foot (FT), the force of the leg bones (tibia and fibula) on the foot (FB), and the force of the floor upward, which is equal to the weight of the body (W)...
We then go one to solve the equations of translational and rotational equilibrium to find that FT = 1.8 W and FB = 2.8 W, and conclude
The tension in the Achilles tendon is nearly twice the person’s weight, while the force exerted on the leg by the talus is nearly three times the body weight. Once can understand why the tendon might rupture.
Many sports injuries result from the laws of biomechanics. The problem is often that a tendon must exert a large force in order to create enough torque to maintain rotational equilibrium. For the Achilles tendon, the moment arm between the joint and the tendon is roughly half the moment arm between the joint and the ball of the foot, so the force on the tendon must be about twice the weight. Our bodies are often built this way: large forces are required to make up for small moment arms. I sometimes give a problem on one of my Biological Physics (PHY 325) exams that illustrates this by calculating the forces on the shoulder of a gymnast performing an “iron cross” on the rings. Here again, the torque exerted by the rings on the arm is huge because of the large moment arm (essentially the entire length of the arm itself), while the moment arm of the pectoral muscle is small because it connects to the humerus (the arm bone) only about 5 cm from the shoulder, at a small angle. The problem suggests that the pectoral muscle must supply a force of over twenty times the body weight! No wonder I was so poor at the rings in my high school physical education class. Readers interested in learning more about this topic might want to read Williams and Lissner's classic textbook Biomechanics of Human Motion. Russ and I cite the 1962 first edition, but I believe that the book has evolved into Biomechanics of Human Motion: Basics and Beyond for the Health Professions, by Barney LeVeau, due out later this year.

Hopefully these insights into biomechanics can help you appreciate how Kalin Lucus could suffer a season-ending injury so easily. I’m glad MSU was able to make it to the final four even without Lucus. However, to be honest, the Spartans were only my 4th favorite team in the tournament this year. Oakland University participated in March Madness for only the second time in the school’s history. Vanderbilt University (where I went for graduate school) also reached the Big Dance, and many predicted that the University of Kansas (where I attended college) would win the entire event. Unfortunately, all three schools lost in the first weekend of play. Russ didn’t fare much better, as the University of Minnesota lost in the first round. Congratulations to Duke University (home to an excellent Biomedical Engineering Department) for their ultimate victory.