Model and measurement studies on stages of prosthetic gait.


General thoughts

In contrast to Dym (2004), Kent and Franklyn-Miller (2011) suggest that it would be of benefit to develop a uniform, robust modelling strategy for both research and clinical rehabilitation practice. They reported that variation in modelling techniques limits the utility of findings reported in the literature. I do not agree with the idea of a uniform and robust modelling strategy. To my opinion specialised, custom-developed mathematical models can be used to model phenomena at various levels, and may serve different goals. Occam’s razer principle states that among competing hypothesis, the one that makes the fewest assumptions should be selected, meaning that one should proceed to simpler theories until simplicity can be traded for greater explanatory power. The simplest available theory need not to be the most accurate. Simple models, which are less accurate, can be used to investigate principles, whereas more complex models, which are more accurate, try to predict a more realistic outcome. The downside of these latter models is that many input parameters and unclear algorithms are required to make realistic predictions, which makes the model less convincing compared to the simple models. As uniform models require multi parameters settings that make the relations within the model unclear, both the research and clinical rehabilitation practice may benefit of a nonuniform modelling strategy, in which specialised, custom developed models are used, that are dedicated and tested to simulate and predict only limited aspects of the observed phenomena. This strategy makes it is easier to study and understand the relation between the identified and included parameters and the predictions made (figure 7). Clear relations improve the power of the model. The identified parameters and the clear relations will benefit the clinical practice, provided that the model is wel described.


Figure 7: A first-order view of mathematical modeling that shows how the questions asked in a principled approach to building a model relate to the development of that model (original figure from Dym, 2004), inspired by Carson and Cobelli 46.

I strongly feel that making acceptable and valid assertions is only possible if the predictions of a model based on (known) variables and parameters, and of which the assumptions are known, are tested according to the methodological modelling principles as described by Dym (2004). Knowing what is assumed improves the power of the model. Without knowing this, the model becomes a black box, of which the circumstances that apply cannot be identified. In our models we made several assumptions that can be debatable. Our models simulate situations in a thin slice of the 3D world, an euphemism for 2D modelling, which therefore has consequences for the predictions made by the models. Compared to our simplified 2D models, with a limited number of body segments and pinpoint joints, 3D models are far more complex, with more parameters and other algoritms. We are aware that the predictions we made with our 2D models might be different when using 3D models, and know that the predictions only apply in one plane. Knowing this, and being aware of it, helps us to understand better what the predictions mean for the strategies chosen by the TF amputees in the real world.
The whole process of methodological modelling is captured by the questions presented in figure 7. This flowchart is not an algorithm for building a good mathematical model, but the questions are key to problem formulation generally. The methodological modelling principles show that the predictions of the model have to be validated and verified. In this thesis the validation and verification of the models was done not only using data of TF subjects and AB subjects using a kneewalker in the conceptual world, but also with a physical mechanical Meccano○R model as the outcome of one of the mathematical models was a counterintuitive prediction.
When we studied what happened when a prosthetic limb with a flexible knee was moved forward by using a hip flexion moment, we found a remarkable result with the mathematical model. We hypothesized that the foot should move forward when applying hip torques because the whole limb swings forward. However, because of the flexible knee and the inertial properties, the foot moves relatively more in upward direction than in forward direction when applying large hip torques and not using ground friction. To verify this unexpected, counterintuitive prediction we used the mechanical model, to confirm these findings (figure 8).


Figure 8: Mechanical model used to confirm that the foot moves relatively more in upward direction than in forward direction when applying large hip torques (by pulling the rope with the hand) and not using ground friction (lower image sequence) compared to a low hip torques condition (upper image sequence). This strategy can be used to avoid a collision with an obstacle.

This in upward direction moving of the foot becomes more clear with a heavier foot. Our limb swing model shows the influence of the mass of the foot within only a few minutes of simulation time (figure 9). This example demonstrates that mathematical models allow us to gain first insights into changes in movement strategies and prosthetic design. Especially designing, creating and testing of prosthetic limb components can be a costly and lengthy process. Considering this, it is well worth to start this process with exploratory modeling studies.


Figure 9: Example of the influence of the mass of the foot. The foot in the right panel is twice as heavy as the foot in the left panel.

Although there are data of human subjects in this thesis, the focus of this thesis is to a large extent on the mathematical models. For people who feel that because of the use of these models instead of real patients, of whom we know use many compensation strategies, this thesis is of lesser value, I have the following consideration.
A funny incident that happened may illustrate that working with mathematical models, which are simplified representations of the real world, is not so far off from working with human subjects. Although these mathematical models consist of numerous equations and even more values, these models can show very human-like behaviour. During our search for the optimal trajectory of a prosthetic limb over an obstacle, based on the error values described in chapter 3, I used several computers that were assigned to the calculate all the simulated steps that were necessary to find the solution. This solution would produce a trajectory of the foot over the obstacle, in which the forefoot moved only a few centimeters over the top of the frontside of the obstacle. A fraction of a second later the heel had to pass the top of the backside of the obstacle at a similar distance. Of course the foot was not allowed to collide with the obstacle and the energy consumption should be as little as possible. This procedure, in which all the computers communicated via a server to share their best result, so that the next simulation trial of all the computers would be based on the best solution found until then, took about 2.5 days. I used the weekend so that I could use the computers of my colleagues. When the best solution in a very diverse mathematical landscape was found, the central server sent an e-mail to me, with the input parameters for the model. I entered these parameters in my simulation software at home, on a sunday afternoon, and waited for the outcome, which of course would be perfectly suited to the demands and criteria that were given. Please, have a look at figure 10 and imagine my surprise …


Figure 10: Result of 2.5 days simulation with a model that crosses an obstacle based on a optimization algorithm.

What have I learned from all of this? The obvious lesson learned was that I had to restrict the motions of the hip. The lesser obvious lesson I learned was that models behave like patients. Analogous to some patients, without the proper instruction, models will come up with solutions to a problem that can be very surprising, without being illogical from a certain perspective, comparable to compensation strategies. Many times I have seen this happening during my work as a physiotherapist. Patients who were asked to avoid an obstacle that was positioned in front of them on the ground decided to walk around it, instead of over it when the environment allowed it. From a certain, probably very functional, perspective they did it exactly as instructed, but not as intended. As a researcher and therapist, I dare to say that models can show very human like behaviour. Your instructions and the learning environment in which the patient moves are key in successful training, teaching and learning!


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