Computational and Mathematical Methods in Medicine

Volume 2017, Article ID 3602928, 8 pages

https://doi.org/10.1155/2017/3602928

## Predictive Behavior of a Computational Foot/Ankle Model through Artificial Neural Networks

^{1}Department of Biomedical Engineering, Virginia Commonwealth University, 401 West Main Street, P.O. Box 843067, Richmond, VA 23284-3067, USA^{2}Department of Electrical Engineering, Virginia Commonwealth University, 601 West Main Street, P.O. Box 843072, Richmond, VA 23284-3072, USA^{3}Department of Mathematics & Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, P.O. Box 842014, Richmond, VA 23284-2014, USA

Correspondence should be addressed to Jennifer S. Wayne; ude.ucv@enyawj

Received 15 September 2016; Revised 15 December 2016; Accepted 20 December 2016; Published 30 January 2017

Academic Editor: Kaan Yetilmezsoy

Copyright © 2017 Ruchi D. Chande et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Computational models are useful tools to study the biomechanics of human joints. Their predictive performance is heavily dependent on bony anatomy and soft tissue properties. Imaging data provides anatomical requirements while approximate tissue properties are implemented from literature data, when available. We sought to improve the predictive capability of a computational foot/ankle model by optimizing its ligament stiffness inputs using feedforward and radial basis function neural networks. While the former demonstrated better performance than the latter per mean square error, both networks provided reasonable stiffness predictions for implementation into the computational model.

#### 1. Introduction

Computational models of diarthrodial joint function depend on accurate reproduction of bony and soft tissue characteristics. Certain characteristics may be readily acquired from imaging modalities while others require experimentation. This is particularly challenging when developing patient-specific models for which soft tissue material properties, for example, cannot be easily obtained. In such cases, existing literature is referenced and best estimates serve as input into the computer model. Further complicating model inputs are soft tissue properties that have not or cannot be determined experimentally and for which the properties of known tissues with similar functions are applied. In either situation, data from the literature usually includes a wide range of values or large standard deviations, impacting the efficacy of the model. Because such tissue inputs directly affect the function of the computational model, the model’s predictive capability is only as successful as the information provided to the model. Thus by improving the accuracy of the inputs, the performance of the model will be improved. In the current work, a means of optimizing a model’s inputs, specifically ligament stiffness, was sought for the greater purpose of enhancing the predictive ability of the computational model.

To optimize the ligament stiffness, artificial neural networks (ANNs) were considered. ANNs are mathematical models in which interconnected computational units or neurons [1, 2] are utilized to “learn” relationships among data [2, 3]. To learn this relationship, ANNs attempt to minimize a given cost function by using an iterative process, that is, learning rule [3, 4], to continually adjust system weights until a target is achieved [1]. Once it learns the relationship between known input-output data, the ANN can then apply this knowledge to similar, never-before-seen data to predict an output [1, 3]. Therefore, ANNs are capable of generalizing, meaning they are able to determine a reasonable output based on learned knowledge [1, 5]. Also, they can be utilized without knowing much about the input-output relationship a priori [1], an advantage over statistical regression in which a mathematical formulation for the problem is known or assumed to be known [1, 5]. Additionally, ANNs are applicable to nonlinear problems [1]. Generally, neural networks are useful in several applications including pattern and image recognition, classification, and curve-fitting, and various examples of these applications can be found within the biomedical field, including musculoskeletal modelling [1, 3–12].

Different types of ANNs exist and determination of which type to use is usually dependent on a given project’s application [5]. Because the current work falls under function approximation, the following descriptions will focus on feedforward (FFN) and radial basis function (RBFN) networks (Figure 1). Structurally, the basic unit of any ANN is the neuron or node. In the case of a FFN, each input is first multiplied by a weight factor, and then all weighted inputs are summed together along with a bias prior to passing through the activation function. The activation function, also referred to as a transfer function, can take various forms (e.g., linear, piece-wise, or sigmoidal [1, 6, 13]), but in each case its purpose is to limit the output of a given neuron within a certain range [1]. As for RBF networks, source inputs are not weighted; rather, these inputs are passed through a distance function essentially calculating how far the vector formed by the input data is from a predetermined vector of the same dimensionality, known as a “center.” The result of this calculation is passed through a radial basis function, such as a Gaussian function, and the produced value is the output of the neuron [1, 14].