# Analyzing Uncertainty of an Ankle Joint Model with Genetic Algorithm

## Abstract

**:**

## 1. Introduction

#### Aim of the Study

## 2. Materials and Methods

#### 2.1. Overview of the Proposed Approach

#### 2.2. Ankle Model Assumed to Verify the Procedure

_{ext}= −5.00: 5.00 Nm in 11 steps. The negative values of M

_{ext}corresponded to the motion clinically referred to as dorsiflexion, while the positive value represented plantarflexion.

#### 2.3. Encoding the Adversarial Structures

#### 2.4. Objective Function

_{j}is the weight j (here: w

_{1i}= 1 for i = 1, …, (m − 1) and w

_{1i}= 2 for i = [0, m]), Δθ

_{k}(M

_{ext,i}) is the angular displacement of model k (k is either A or B) under the external moment load M

_{ext,i}(the considered loads were specified in the Section 2.2). In this case, the larger the value of Equation (1), the more different the structures were.

_{j}is the weight j (here: w

_{1i}= 1 for i = 1, …, (m − 1) and w

_{1i}= 2 for i = [0, m]), w

_{2}= 10.00), Δθ

_{k}(M

_{ext,i}) is the angular displacement of model k (k is either A or B) under the external moment load M

_{ext,i}(the considered loads were specified in the Section 2.2), not_passed is the number of loads for which the solver did not solve the models with desired accuracy (a sum for both of the adversarial structures; a similar approach was utilized in: [29,30,33]). The solutions, which contained models that were difficult to solve for all of the assumed loads were penalized with the second element of the objective function (2). The penalty was included with a relatively large weight in the objective function.

^{−10}(see [26] for further details).

#### 2.5. Optimization Procedure

#### 2.6. Generating the Initial Population for the Algorithm

## 3. Results

#### 3.1. Optimization Process

#### 3.1.1. Initial Runs of the Optimization

#### 3.1.2. An Extended Run with 2000 Generations

#### 3.1.3. Computing the Baseline

#### 3.2. Analyzing the Uncertainty of the Ankle Model

## 4. Discussion

#### 4.1. Optimization Procedure

#### 4.2. Effect of Uncertainties on the Ankle Model

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Assumed planar model of the ankle joint in the sagittal plane and a schematic representation of its spherical contact pairs, reproduced with permission from [26].

**Figure 2.**The proposed encoding of the adversarial structures—an example prepared based on an ankle joint model [26].

**Figure 6.**Two adversarial structures obtained with the algorithm for the ankle joint model (as seen in the prepared software). Only relevant parts of the contours of the bone segments were drawn. (

**a**) Model A; (

**b**) model B.

**Figure 7.**Angular displacement Δθ

_{i}versus the external moment M

_{ext}for the two most different adversarial structures A and B.

**Figure 8.**Values of the forces generated by the ligaments versus the external moment M

_{ext}for the two most different adversarial structures A and B, where: ‘con.’ is the magnitude of the contact force.

**Table 1.**Baseline obtained with one-at-a-time parameter modification compared to the results obtained with the proposed optimizational procedure.

Baseline | Optimization after 100 gen. | Optimization after 2000 gen. | |
---|---|---|---|

Value of the objective h(x) (obtained using Equation (2)) | −0.23 | −1.30 | −2.06 |

**Table 2.**Maximal angular displacement Δθ

_{i}in plantarflexion and dorsiflexion for the two most different adversarial structures A and B.

Δθ_{A} (deg) | Δθ_{B} (deg) | Avg^{1} (deg) | abs_diff^{1} (deg) | rel_diff^{1} (%) | |
---|---|---|---|---|---|

M_{ext} = 5.00 Nm | 41.21 | 31.35 | 36.28 | 9.86 | 27.18 |

M_{ext} = −5.00 Nm | −19.79 | −29.23 | −24.51 | 9.44 | 38.52 |

range of motion | 61.00 | 60.58 | 60.79 | 0.42 | 0.69 |

^{1}avg—average; abs_diff—absolute difference, rel_diff—relative difference, computed as: (abs_diff/|avg|) × 100%.

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**MDPI and ACS Style**

Ciszkiewicz, A.
Analyzing Uncertainty of an Ankle Joint Model with Genetic Algorithm. *Materials* **2020**, *13*, 1175.
https://doi.org/10.3390/ma13051175

**AMA Style**

Ciszkiewicz A.
Analyzing Uncertainty of an Ankle Joint Model with Genetic Algorithm. *Materials*. 2020; 13(5):1175.
https://doi.org/10.3390/ma13051175

**Chicago/Turabian Style**

Ciszkiewicz, Adam.
2020. "Analyzing Uncertainty of an Ankle Joint Model with Genetic Algorithm" *Materials* 13, no. 5: 1175.
https://doi.org/10.3390/ma13051175