ACL deficiency affects stride-to-stride variability as measured using nonlinear methodology

Abstract

Previous studies suggested that the small fluctuations present in movement patterns from one stride to the next during walking can be useful in the investigation of various pathological conditions. Previous studies using nonlinear measures have resulted in the development of the “loss of complexity hypothesis” which states that disease can affect the variability and decrease the complexity of a system, rendering it less able to adjust to the ever changing environmental demands. The nonlinear measure of the Lyapunov Exponent (LyE) has already been used for the assessment of stride-to-stride variability in the anterior cruciate ligament (ACL) deficient knee in comparison to the contralateral intact knee. However, there is biomechanical evidence that after ACL rupture, adaptations are also present in the contralateral intact knee. Thus, our goal was to investigate stride-to-stride variability in the ACL deficient knee as compared to a healthy control knee. Seven subjects with unilateral ACL deficiency and seven healthy controls walked at their self-selected speed on a treadmill, while three-dimensional knee kinematics was collected for 80 consecutive strides. A nonlinear measure, the largest LyE was calculated from the resulted knee joint flexion-extension data of both groups. Larger LyE values signify increased variability and increased sensitivity to initial conditions. Our results showed that the ACL deficient group exhibited significantly less variable walking patterns than the healthy control. These changes are not desirable because they reflect decreases in system’s complexity, which indicates narrowed functional responsiveness, according to the “loss of complexity hypothesis.” This may be related with the increased future pathology found in ACL deficient patients. The methods used in the present paper showed great promise to assess the gait handicap in knee injured patients.

Appendices

Appendix 1

The behavior of a continuously changing system over time can be periodic, random or chaotic.

Periodic systems are organized. They are repeatable and predictable (Fig. 3).

Fig. 3

Graphic representation of a periodic system [sin(1/10)] (a) and the corresponding phase plane plot (b), where the time series data is plotted versus the first derivative

Random systems, on the other hand, contain no order. They are unpredictable and their behavior is never repeated (Fig. 4).

Fig. 4

Graphic representation of a random system (Gaussian noise centered on 0 and a standard deviation of 1.0) (a) and the corresponding phase plane plot (b)

Chaotic systems have characteristics of both. They seem to be random and unpredictable but they contain order and are deterministic in nature. They are very flexible and can operate under various conditions (Fig. 5).

Fig. 5

Graphic representation of a chaotic system (the Lorentz attractor) (a) and the corresponding phase plane plot (b)

Appendix 2

To properly reconstruct a state space, it is essential to quantify an appropriate time delay and embedding dimension for the investigated data set.

Τo reconstruct the state space, a state vector was created from the data set. This vector was composed of mutually exclusive information about the dynamics of the system (Eq. 1).

$$ {\mathbf{y}}(t) = [x(t),{\text{ }}x(t - T_{1} ),{\text{ }}x(t - T_{2} ), \ldots ], $$

(1)

where y(t) is the reconstructed state vector, x(t) is the original data and x(t − T _i ) is time delay copies of x(t). The time delay (T _i ) for creating the state vector is determined by estimating when information about the state of the dynamic system at x(t) was different from the information contained in its time-delayed copy. If the time delay is too small then no additional information about the dynamics of the system will be contained in the state vector. Conversely, if the time delay is too large then information about the dynamics of the system may be lost and can result in random information. Selection of the appropriate time delay is performed by using an average mutual information algorithm 1 (Eq. 2).

$$ I_{{x(t),x(t + T)}} = {\sum {P{\left( {x(t)),x(t + T)} \right)}} }\log _{2} {\left[ {\frac{{P{\left( {x(t),x(t + T)} \right)}}} {{P{\left( {x(t)} \right)}P{\left( {x(t + T)} \right)}}}} \right]}, $$

(2)

where T is the time delay, x(t) is the original data, x(t + T) is the time delay data, P(x(t), x(t + T)) is the joint probability for measurement of x(t) and x(t + T), P(x(t)) is the probability for measurement of x(t), P(x(t + T)) is the probability for measurement of x(t + T). The probabilities are constructed from the frequency of x(t) occurring in the time series. Average mutual information is iteratively calculated for various time delays and the selected time delay is at the first local minimum of the iterative process. This selection is based on previous investigations that have determined that the time delay at the first local minimum contains sufficient information about the dynamics of the system to reconstruct the state vector.

It is additionally necessary to determine the number of embedding dimensions to unfold the dynamics of the system in an appropriate state space. An inappropriate number of embedding dimensions may result in a projection of the dynamics of the system that has orbital crossings in the state space that are due to false neighbors and not the actual dynamics of the system. To unfold the state space we systematically inspect x(t) and its neighbors in various dimensions (e.g. dimension = 1, 2, 3,…). The appropriate embedding dimension occurs when neighbors of the x(t) stop being unprojected by the addition of further dimensions of the state vector (Eq. 3).

$$ {\mathbf{y}}(t) = [x(t),{\text{ }}x(t + T),{\text{ }}x(t + 2T), \ldots x(t + (d_{E} - 1)T)], $$

(3)

where d _E is number of embedding dimensions, y(t) is the d _E -dimensional state vector, x(t) is the original data, and T is the time delay. A global false nearest neighbors algorithm with the time delay determined from the local minimum of the average mutual information is used to determine the number of necessary embedding dimensions to reconstruct the step time interval data series. The calculated embedding dimension indicates the number of governing equations that were necessary to appropriately reconstruct the dynamics of the system. The Tools for Dynamics (Applied Chaos LLC, Randle Inc., San Diego, CA, USA) software was used to calculate the embedding dimension for our data sets, and it was found to five.