**1. Introduction**

Anterior cruciate ligament (ACL) deficiency (ACLD) is a common injury in people who play sports. The ACL plays an important role in maintaining the stability of the knee joint. However, because of the complexity of the knee joint and ACL, it is difficult to conduct kinematic and dynamic research on patients with ACLD. Studies have focused on building a mechanical model of ACL in vitro [1,2]. Since the establishment of a muscle model by Zajac et al. [3], musculoskeletal models have improved [4,5]. With the help of musculoskeletal models, many studies have investigated the kinematics and dynamics in ACLD-affected knees. Some studies have shown that patients with ACLD adopted quadricep avoidance [6,7] and a stiffening strategy [8], resulting in reductions in the knee flexion moment and peak knee flexion angle. ACLD affects a patient's gait patterns and further kinematics and dynamics [9]. Ren et al. [10] and Yin et al. [11], respectively, studied the kinematics and dynamics in patients with ACLD. Shelburne et al. [7] considered the role of muscles, explaining that in patients with ACLD, quadricep avoidance occurred to restore anterior tibial translation. Furthermore, increasing hamstring force was also sufficient, implying muscle compensation in the knee instability. Even after ACL reconstruction, patients still have a high risk of osteoarthritis [12,13] because of the loss of normal muscle compensation in patients with ACL reconstruction [14,15].

**Citation:** Li, H.; Huang, H.; Ren, S.; Rong, Q. Leveraging Multivariable Linear Regression Analysis to Identify Patients with Anterior Cruciate Ligament Deficiency Using a Composite Index of the Knee Flexion and Muscle Force. *Bioengineering* **2023**, *10*, 284. https://doi.org/10.3390/ bioengineering10030284

Academic Editors: Christina Zong-Hao Ma, Zhengrong Li, Chen He and Aurélien Courvoisier

Received: 9 January 2023 Revised: 10 February 2023 Accepted: 17 February 2023 Published: 22 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The clinical diagnosis of ACLD is complicated and expensive, and the diagnosis process requires the subjective judgment of clinicians. For auxiliary diagnosis, many studies have used statistics to study the gait of ACLD. Christian et al. [16] trained the gait trajectory points of patients with ACLD using support vector machines (SVMs) to extract trajectory features. Berruto et al. [17] counted the fluctuation range of the acceleration of the patient's legs with one ACL reconstruction in a pivot-shift test and demonstrated a significant difference between the ACLD-affected and contralateral sides. Zeng et al. [18] used kinematic data extracted by a motion capture system as features for neural network training. Kokkotics et al. [19] used different machine learning methods to identify patients with ACLD and ACL reconstruction from kinematics and dynamics data. However, only a few studies have used kinematics and dynamics data to diagnose patients with ACLD. There are even fewer studies that can be directly reproduced and can rapidly diagnose patients.

The feature choice is the most important variable, regardless of the statistical method. Reinbolt et al. [20] performed *t*-tests on the entire gait cycle and selected the peak points of the statistics as features to predict outcomes of rectus femoris transfer surgeries. Principal component analysis (PCA) has been widely used for dimensionality reduction and feature extraction [21]. Armstrong et al. [22] used PCA to extract the feature points of kinematics and reconstruct the kinematics process. Based on multiple parameters extracted from gait data, some indexes were developed to identify walking patterns of normal [23] and abnormal [24–27] gait. Schutte et al. [26] proposed a normalcy index to reflect gait deviations from the mean of normal gait. Liu et al. [28] assessed the abnormal gait in patients with ACLD using the normalcy index calculated by PCA based on kinematics and dynamics data. Similarly, Rozumalski et al. [29] combined a single muscle strength score using PCA to describe the overall lower body joint strength. Hicks et al. [30] used this score as a variable for multivariable regression to study crouch gait. Their regression model had 71% classification accuracy when the parameters were analyzed in detail. However, few studies have combined kinematics and muscle forces to extract features.

This study was performed to identify patients with ACLD using multivariable linear regression through a composite index that combined kinematics and muscle forces.

#### **2. Materials and Methods**

#### *2.1. Participants*

Twenty-five patients with unilateral chronic ACLD (the contralateral side was intact) were recruited before ACL reconstruction (ACLD group). Their knees had been injured 6 months to 4 years before testing. Most injuries occurred during basketball. Exclusion criteria were that the patient had no prior ACL and concomitant meniscal and ligament rupture and no history of musculoskeletal disease of the hip or ankle. Their physical activity levels were assessed by the Tegner score, which is a reliable and valid tool for assessing the activity level of patients with ACLD [31]. Average activity level of all patients was normal before knee injuries (score range 3.0–6.0). A control group comprising nine healthy volunteers with no history of musculoskeletal injury or surgery in the lower extremities was selected (Control group). All participants were young males to rule out biomechanical differences between sexes [32]. Ethical approval was obtained from the university's ethics committee, and written informed consent was obtained from all participants. The morphological data are shown in Table 1, and the participants' characteristics of the groups were not significantly different.


**Table 1.** Characteristics of the participants in each group.

Data are presented as mean ± standard deviation. ACLD, anterior cruciate ligament deficiency; BMI, body mass index.

#### *2.2. Data Collection and Modeling Analysis*

From January 2014 to December 2016, the experimental 3D data were collected while the patients were jogging using an optical motion capture system (Vicon MX; Oxford Metrics, Yarnton, Oxfordshire, UK). The marker trajectory data were filtered at 12 Hz, and the force data were filtered at 100 Hz using a low-pass Butterworth filter. To track the segmental motion during jogging, all participants had a set of markers attached to their anatomical lower limbs at specific locations based on the plug-in-gait model. The participants were asked to run along a 10-m path at a self-selected speed, and the kinematic data were recorded by eight cameras. No participants reported pain during jogging. Ground reaction forces were collected using two embedded force plates at a sampling rate of 1000 Hz (AMTI, Advanced Mechanical Technology Inc., Watertown, MA, USA). Each participant stepped on the force plates at their self-selected speed. For each participant, five successful jogging trials were recorded, and these results were imported into multibody dynamics software, AnyBody Modeling System version 6.0.5 (AnyBody Technology, Aalborg, Denmark), to estimate the kinetics of the knee joint.

A lower extremity model [33] implemented in the AnyBody Modeling System was used for the analysis. The model comprised 12 body segments, and 11 joints were used to connect the segments. Six joint degrees of freedom were considered for each leg, with a spherical joint with three degrees of freedom for the hip joint and a universal joint with two degrees of freedom for the ankle joint. The knee joint was modeled as a hinge joint with one degree of freedom because of the soft tissue artifact error [34]. Based on the morphological parameters measured from each subject, each model was scaled with a mass–fat scaling algorithm to perform the subject-specific jogging simulation. The min/max recruitment principle solver based on the optimization of the objective function [35,36], which has good numerical convergence and physiological representation, was used to predict the muscle force during the inverse dynamics analysis. The objective function is generally formulated as follows [5]:

$$\text{Minimize } \max\left(\frac{f\_i^{(M)}}{N\_i}\right) \tag{1}$$

Subject to

$$\mathbf{C}\mathbf{f} = \mathbf{d}, \ 0 \le f\_i^{(M)} \le \mathcal{N}\_i \quad i \in \left\{1, \dots, n^{(M)}\right\} \tag{2}$$

where *n* (*M*) is the number of muscles, *f* (*M*) *i* is the respective muscle force, and *N<sup>i</sup>* is the strength of the muscle. **f** contains all unknown forces in the optimization problem. **C** is the coefficient-matrix for the unknown forces. **d** contains all known applied loads and inertia forces. Muscle parameters were obtained from a comprehensive musculoskeletal geometry dataset [37]. Some studies have validated the ability of computational muscle forces [38,39].

#### *2.3. Muscle Data Processing*

According to the characteristics of the model and anatomy related to the knee muscles, force data were output from 13 muscles: rectus femoris (RF), popliteus (POP), vastus (VAS), gastrocnemius lateralis (GL), gastrocnemius medialis (GM), soleus medialis (SOLm), soleus lateralis (SOLl), semitendinosus (ST), semimembranosus (SM), proximal sartorius (SAp), distal sartorius (SAd), biceps femoris long head (BFlh), and biceps femoris short head (BFsh). For each participant, separate simulations were performed based on the data from five different jogging trials. The average values of the five calculations were used to perform dynamics analysis using MATLAB version 2019b (MathWorks, Natick, MA, USA).

The acquired muscle force data was processed in a dimensionless manner, and the nondimensionalization of the force data was divided by the subject's gravity (mass × 9.8) [40]. To investigate one gait cycle, the kinematics and dynamics data were interpolated to a 0%–100% gait cycle. Additionally, to more intuitively study the patterns of knee flexion and muscle strength, all flexion and muscle data were normalized to their maximal muscle force within that cycle, leading to a normalized amplitude between 0 and 1 [41].
