Mathematical Model Optimizes Shoe Midsoles for Peak Running Performance

By Lina Sorg

Multiple athletic brands are beginning to utilize three-dimensional (3D) printing to create the midsoles of running sneakers in order to optimize shoe performance. The midsole is the cushiony part on the bottom of a shoe that allows for energy absorption and return during the running process. In many standard types of running shoes, it comprises a slab of uniform foam that is thick and softer in the rearfoot and thin and stiff in the forefoot. However, novel 3D printing techniques now allow for more flexibility in midsole design. “We can keep the shape of the shoe the same but change the properties by altering the geometry of the lattice — either by how the unit cells are arranged, the particular shapes of the unit cells, or just by simply increasing or decreasing the beam’s thickness,” Sarah Fay of the Massachusetts Institute of Technology (MIT) Sports Lab said.

Figure 1. A simple two-dimensional (2D) model that uses two actuators/motors to represent muscle activity. The runner is a single-point mass. A second 2D model illustrates how the runner interacts with the shoe. Figure courtesy of Sarah Fay.

During a contributed presentation at the 2022 SIAM Conference on Mathematics of Data Science, which took place in a hybrid format in San Diego, Calif., last week, Fay discussed a recent partnership with Adidas to optimize shoe midsoles for peak running performance. Fay, who is currently a postdoctoral researcher at MIT, worked on this project while earning her Ph.D. under the direction of Anette "Peko" Hosoi. Given recent developments in 3D printing technology, Adidas approached the MIT Sports Lab and expressed an interest in understanding how certain midsole designs would ultimately contribute to the most optimal shoe.

Running performance is traditionally measured by running economy, which typically involves test subjects who run on a treadmill with oxygen masks that measure the required level of oxygen at a given speed. The highest-performing shoes require the lowest use of oxygen from the runners. Unfortunately, this process is time consuming, expensive, and often laden with paperwork. “It’s really important that we have a way of identifying shoes that could potentially be high performing before we put them through that test,” Fay said. “The topic of my work explores whether a model is able to predict how these new shoe designs will perform without having to make and test them so we can be more strategic about the ones that we do fabricate.”

Existing runner models range from very simple to quite complex. Though the complex models that account for each muscle, ligament, and tendon are more accurate, they require many parameters and a much longer solve time. As such, Fay developed a straightforward two-dimensional (2D) model—wherein the runner is a single-point mass—and used two actuators/motors to represent muscle activity (see Figure 1). One actuator works along the length of leg and the other produces a torque at the hip, which yields a reaction force with the ground. “We also have to model how this runner is interacting with the shoe, so we have a similar 2D model for the shoe that accounts for the fact that the stiffness is different from the front to back,” Fay said.

Figure 2. The most effective objective function was located somewhere between efforts to minimize the mean squared jerk and efforts to minimize the impulse from the ground. Figure courtesy of Sarah Fay.

The way in which one’s muscles react in certain situations varies based on context. “If we’re trying to get away from a lion, our muscles are going to do whatever they can to get us to move as fast as possible,” Fay said. “But if we’re trying to traverse a very long distance, we’re going to choose a gait and muscle activation pattern that allows us to travel far while conserving a lot of energy.” When it comes to recreational running, however, researchers must learn the factors for which they are optimizing from existing data. This begs the following question: How does the body decide what the muscles are doing, and what influences those decisions?

Figure 3. In July 2020, Adidas generated a sample shoe with properties that were influenced by Sarah Fay's research. Figure courtesy of Sarah Fay.

Fay used a simple mathematical system to address this query and chose the muscle activation patterns to optimize for an objective function that is both biological and intrinsic. She focused specifically on two objective functions that were based on physical quantities that are important to the bodies of runners: (i) The impulse that the runner feels from the ground and (ii) the mean squared jerk of the motion, which measures the rapidity of acceleration change. Next, she compared data from a public data set to the model in the ground reaction force space. This data set reflected 28 different runners—and all of their forces and trajectories—at three different speeds. These subjects ran on a treadmill with force plates that measured the vertical ground reaction force (the extent to which the ground pushes up on the runner) and the horizontal ground reaction force (which slows runners down and then speeds them up as they push off the ground). “Our model is 2D, so tracking the force and using that as a proxy for the center of mass is a lot better than trying to use the hip as a center of mass,” Fay said. “As you’re running, your center of mass moves from your hip to up and down your body in all kinds of ways."

When Fay optimized the model to minimize mean squared jerk, the model performed quite well in the vertical ground reaction force but was roughly one order of magnitude off in the horizontal direction. Minimizing the impulse from the ground generated much better results in the horizontal force but was significantly awry in the vertical ground reaction force. “Somewhere in between we’re doing a pretty good job with both of them,” Fay said (see Figure 2).

Figure 4. Sarah Fay (left) and her advisor Anette "Peko" Hosoi hold a pair of Adidas shoes with 3D-printed midsoles and properties that reflect their research. The shoes officially launched in August 2022. Photo courtesy of Sarah Fay.

Next, Fay examined the mean squared error of the ground reaction force vector for both the vertical and horizontal forces. By comparing the vectors for the model and data, she found the objective function that yielded the best match. “In this case, the best match arose when we cared about mean squared jerk only 25 percent and cared about the impulse felt with the ground 75 percent,” Fay said. This result is for the intermediate speed case (3.5 meters per second), though each of the three speeds begets slightly different results.

Fay then plugged her successful objective function into the model, ran the model with many different shoe types, and repeatedly solved the objective function to determine the gait and energy cost of each pair of shoes. “The gait that costs the least amount of energy is the shoe that we ended up making,” Fay said. Figure 3 depicts a sample shoe from July 2020 with properties that were influenced by her research. Furthermore, Adidas officially released a new shoe in August 2022 whose properties reflect Fay’s efforts (see Figure 4). “The exciting part of the work is that we were able to get all the way from a 2D model to actual shoe stuff,” Fay said. “The beauty of 3D printing is that we can customize the shoes to the runner and the kinds of speeds they’ll be running.”

Lina Sorg is the managing editor of SIAM News.