Forward Solution Modeling: An In Vivo Theoretical Simulator of the Knee

What Is Mathematical Modeling?

Mathematical modeling pertains to any simulation of a physical system using a computer. In practice, mathematical modeling is an extremely diverse field, which can be applied to any system, ranging from modeling weather patterns to analyzing artificial knees. In this chapter, mathematical modeling focuses on physics simulations of the human knee. Therefore total knee arthroplasty (TKA) mathematical models are used to analyze muscle forces, soft tissue forces, joint reaction forces, implant geometry, and in vivo kinematics to simulate a TKA patient that exists entirely on the computer (in silico). By modifying any one of these parameters, the results of the other parameters can derived and reviewed.

Therefore mathematical modeling can be used as a powerful design tool to assess or design both existing and future TKA implants. By evaluating various implant designs and analyzing the resulting mechanics, improved designs can be created. Furthermore, those designs can be evaluated with varying soft tissue constraints to determine under which conditions they perform best.

Why Mathematical Modeling?

The modern knee implant has a sophisticated design, which in general tries to restore the native kinematics and kinetics. Although current designs experience some success, new generations of TKA are constantly being developed. To develop more successful knee implants, it is necessary to first understand the forces that will be induced at the implant-bearing surfaces. Then the interplay between those forces and the geometry can be considered. These two factors play a major role in the determination of in vivo kinematics and the soft tissue forces.

There are two main ways of determining the loading in a post-TKA knee: telemetry and mathematical modeling. Telemetry produces the most accurate measurements of forces incurred by the TKA, which are directly measured in vivo using sensors placed within the implant. However, developing and manufacturing these implants is prohibitively expensive, so studies typically use small samples sizes of one to three subjects.

In addition to directly measuring the forces through telemetry, these bearing surface and soft tissue forces can also be calculated using mathematical models. Mathematical modeling of the knee incorporates that evaluation of all soft tissue constraints, interactive and independent motions, and muscle forces that are used to drive these motions, leading to the generation of equations of motion to solve for the desired results.

After a mathematical model has been developed and validated, it possesses many benefits that go beyond what telemetry can offer. For example, after a mathematical model has been created, conducting the theoretical analysis is extremely inexpensive. Furthermore, the model can be used to evaluate any number of patients, efficiently, after their input parameters are known. The model also allows for patient modifications to be made, in relatively short time periods that could not be accomplished using any other method. For example, the medial collateral ligament or lateral collateral ligament could be damaged or removed in a mathematical model, which could not be simulated using a living subject. These soft tissue modifications could be simulated using a cadaver, but then the test does not replicate in vivo conditions, leading to concerns that evaluation is being conducted in an altered environment not representing actual muscle forces found during in vivo conditions.

As a design tool, mathematical modeling offers unparalleled flexibility, leading to very powerful parametric analyses. The model can be used to run iterative simulations with only small design changes between each simulation, leading to a pathway design process that documents all changes that were evaluated. Alternatively, one could attempt to manufacture and test many implants modifications, but again this would be very expensive and not simulate the in vivo environment. Mathematical modeling can also provide detailed information about the internal stress on an implant, which can be very difficult to measure. This information can help to eliminate any potential stress shielding or stress concentrators on the implant or bone. Overall, mathematical modeling allows for the opportunity to conduct very efficient experiments that otherwise would be very expensive and impossible to perform in a nontheoretical environment.

Basic Concepts of Knee Modeling

All mathematical modeling of the dynamics of rigid body systems are derived using the principles of Newton's second law.

When modeling the knee, three rigid bodies—the femur, tibia, and patella—are the objects that can accelerate and decelerate within the system. The acceleration of the femur, tibia, and patella represent the right side of the above equation (mass [ M ] × acceleration [ A ]). Muscles, ligaments, tendons, and joint interactions are the forces within the system that generate or constrain this acceleration. Mathematical modeling is the process of accurately describing all of the forces and bodies within a system.

After the system is accurately modeled, the equations of motion are determined. Equations of motion are a series of differential equations that are more complicated than the simple force ( F ) = M × A because they incorporate rotations as well as translations. For example, one acceleration equation within the human body leg system could amount to five pages in length, which is very complicated. Each rigid body can be defined by six equations of motion because it can move in six different ways: three translations and three rotations. By applying known forces and simultaneously solving all of the equations, the resulting accelerations of the rigid body can be computed. When deriving a mathematical model, the reverse approach could be taken in which the accelerations of the bodies are known using an experimental measuring technique, leading to a solution that solves for the muscle and bearing surface forces. Typically these kinematic parameters that induce accelerations are computed using either motion capture systems or fluoroscopic techniques.