ACL microtrauma: healing through nutrition, modified sports training, and increased recovery time.

PURPOSE: Sports injuries among youth and adolescent athletes are a growing concern, particularly at the knee. Based on our current understanding of microtrauma and anterior cruciate ligament (ACL) healing characteristics, this clinical commentary describes a comprehensive plan to better manage ACL microtrauma and mitigate the likelihood of progression to a non-contact macrotraumatic ACL rupture. METHODS: Medical literature related to non-contact ACL injuries among youth and adolescent athletes, collagen and ACL extracellular matrix metabolism, ACL microtrauma and sudden failure, and concerns related to current sports training were reviewed and synthesized into a comprehensive intervention plan. RESULTS: With consideration for biopsychosocial model health factors, proper nutrition and modified sports training with increased recovery time, a comprehensive primary ACL injury prevention plan is described for the purpose of better managing ACL microtrauma, thereby reducing the incidence of non-contact macrotraumatic ACL rupture among youth and adolescent athletes. CONCLUSION: Preventing non-contact ACL injuries may require greater consideration for reducing accumulated ACL microtrauma. Proper nutrition including glycine-rich collagen peptides, or gelatin-vitamin C supplementation in combination with healthy sleep, and adjusted sports training periodization with increased recovery time may improve ACL extracellular matrix collagen deposition homeostasis, decreasing sudden non-contact ACL rupture incidence likelihood in youth and adolescent athletes. Successful implementation will require compliance from athletes, parents, coaches, the sports medicine healthcare team, and event organizers. Studies are needed to confirm the efficacy of these concepts. LEVEL OF EVIDENCE: V.

Return to sports bridge program improves outcomes, decreases ipsilateral knee re-injury and contralateral knee injury rates post-ACL reconstruction.

PURPOSE: To present the results of a return to sports bridge program designed to reduce knee injuries following ACL reconstruction and physical therapy. METHODS: One hundred and fifty (male = 83, female = 67) patients participated in a whole body neuromuscular control, progressive resistance strength and agility training program. Post-program testing included functional movement form, dynamic knee stability, lower extremity power, agility, and sports skill assessments. Participants completed the Knee Outcome Survey-Sports Activity Scale (KOS-SAS) before and after program initiation. Pre-participation scores were re-estimated following program completion. RESULTS: Global rating KOS-SAS score at program entry was 75 ± 13. Post-program global rating and calculated KOS-SAS were 91.0 ± 9.8 and 90.9 ± 9.7, respectively (p < 0.0001). Pre-participation KOS-SAS score re-estimates at program completion were 54.5 ± 23.3 and 57.3 ± 18.5, respectively. The approximately 20% lower pre-program KOS-SAS score re-estimates (p < 0.0001) observed at program completion suggests that subjects had inaccurately high sports readiness perceptions at program entry. Perceived overall sports activity knee function ratings improved from 2.9 ± 0.6 (abnormal) at program entry to 1.3 ± 0.5 (normal) at completion (p < 0.0001). Most subjects returned back to sports at or above their pre-injury performance skill/performance level (84%, 126/150). By 6.8 ± 3.2 years (range = 2-13 years) post-surgery, ten subjects had sustained an ipsilateral knee re-injury or contralateral knee injury (6.7%). The 2.7% non-contact contralateral and 1.3% non-contact ipsilateral knee injury rates observed were significantly lower than those cited in previous reports. CONCLUSION: Supplementing primary ACL reconstruction and standard physical therapy with a return to sports bridge program prior to release to unrestricted sports performance was effective at improving patient outcomes and decreasing ipsilateral knee re-injury and contralateral knee injury rates. LEVEL OF EVIDENCE: II.

Self-diffusion in a quasi-two-dimensional gas of hard spheres.

A quasi-two-dimensional system of hard spheres strongly confined between two parallel plates is considered. The attention is focused on the macroscopic self-diffusion process observed when the system is seen from above or from below. The transport equation, and the associated self-diffusion coefficient, are derived from a Boltzmann-Lorentz kinetic equation, valid in the dilute limit. Since the equilibrium state of the system is inhomogeneous, this requires the use of a modified Chapman-Enskog expansion that distinguishes between equilibrium and nonequilibrium gradients of the density of labeled particles. The self-diffusion coefficient is obtained as a function of the separation between the two confining plates. The theoretical predictions are compared with molecular dynamics simulation results and a good agreement is found.

Inhomogeneous cooling state of a strongly confined granular gas at low density.

The inhomogeneous cooling state describing the hydrodynamic behavior of a freely evolving granular gas strongly confined between two parallel plates is studied, using a Boltzmann kinetic equation derived recently. By extending the idea of the homogeneous cooling state, we propose a scaling distribution in which all the time dependence occurs through the granular temperature of the system, while there is a dependence on the distance to the confining walls through the density. It is obtained that the velocity distribution is not isotropic, and it has two different granular temperature parameters associated to the motion perpendicular and parallel to the confining plates, respectively, although their cooling rates are the same. Moreover, when approaching the inhomogeneous cooling state, energy is sometimes transferred from degrees of freedom with lower granular temperature to those with a higher one, contrary to what happens in molecular systems. The cooling rate and the two partial granular temperatures are calculated by means of a Gaussian approximation. The theoretical predictions are compared with molecular dynamics simulation results and a good agreement is found.

Understanding an instability in vibrated granular monolayers.

We investigate the dynamics of an ensemble of smooth inelastic hard spheres confined between two horizontal plates separated by a distance smaller than twice the diameter of the particles, in such a way that the system is quasi-two-dimensional. The bottom wall is vibrating and, therefore, it injects energy into the system in the vertical direction and a stationary state is reached. It is found that if the size of the plates is small enough, the stationary state is homogeneous. Otherwise, a cluster of particles is developed. The instability is understood by using some effective hydrodynamic equations in the horizontal plane. Moreover, the theoretical prediction for the size of the system above which it becomes unstable agrees very well with molecular dynamics simulation results without any fitting parameter.

Boltzmann kinetic equation for a strongly confined gas of hard spheres.

A Boltzmann-like kinetic equation for a quasi-two-dimensional gas of hard spheres is derived. The system is confined between two parallel hard plates separated a distance between one and two particle diameters. An entropy Lyapunov function for the equation is identified. In addition to the usual Boltzmann expression, it contains a contribution associated to the confinement of the particles. The steady properties of the system agree with equilibrium statistical mechanics results. Equations describing the energy transfer between the degrees of freedom parallel and perpendicular to the confining plates are obtained for some simple initial configurations. The theoretical predictions are compared with molecular dynamics simulation data and a good agreement is found.

Kinetic equation and nonequilibrium entropy for a quasi-two-dimensional gas.

A kinetic equation for a dilute gas of hard spheres confined between two parallel plates separated a distance smaller than two particle diameters is derived. It is a Boltzmann-like equation, which incorporates the effect of the confinement on the particle collisions. A function S(t) is constructed by adding to the Boltzmann expression a confinement contribution. Then it is shown that for the solutions of the kinetic equation, S(t) increases monotonically in time, until the system reaches a stationary inhomogeneous state, when S becomes the equilibrium entropy of the confined system as derived from equilibrium statistical mechanics. From the entropy, other equilibrium properties are obtained, and molecular dynamics simulations are used to verify some of the theoretical predictions.

Stability analysis of the homogeneous hydrodynamics of a model for a confined granular gas.

The linear hydrodynamic stability of a model for confined quasi-two-dimensional granular gases is analyzed. The system exhibits homogeneous hydrodynamics, i.e., there are macroscopic evolution equations for homogeneous states. The stability analysis is carried out around all these states and not only the homogeneous steady state reached eventually by the system. It is shown that in some cases the linear analysis is not enough to reach a definite conclusion on the stability, and molecular dynamics simulation results are presented to elucidate these cases. The analysis shows the relevance of nonlinear hydrodynamic contributions to describe the behavior of spontaneous fluctuations occurring in the system, that lead even to the transitory formation of clusters of particles. The conclusion is that the system is always stable. The relevance of the results for describing the instabilities of confined granular gases observed experimentally is discussed.

Anomalous self-diffusion in a freely evolving granular gas near the shearing instability.

The self-diffusion coefficient of a granular gas in the homogeneous cooling state is analyzed near the shearing instability. Using mode-coupling theory, it is shown that the coefficient diverges logarithmically as the instability is approached, due to the coupling of the diffusion process with the shear modes. The divergent behavior, which is peculiar in granular gases and disappears in the elastic limit, does not depend on any other transport coefficient. The theoretical prediction is confirmed by molecular dynamics simulation results for two-dimensional systems.

Hydrodynamics for a model of a confined quasi-two-dimensional granular gas.

The hydrodynamic equations for a model of a confined quasi-two-dimensional gas of smooth inelastic hard spheres are derived from the Boltzmann equation for the model, using a generalization of the Chapman-Enskog method. The heat and momentum fluxes are calculated to Navier-Stokes order, and the associated transport coefficients are explicitly determined as functions of the coefficient of normal restitution and the velocity parameter involved in the definition of the model. Also an Euler transport term contributing to the energy transport equation is considered. This term arises from the gradient expansion of the rate of change of the temperature due to the inelasticity of collisions, and it vanishes for elastic systems. The hydrodynamic equations are particularized for the relevant case of a system in the homogeneous steady state. The relationship with previous works is analyzed.

Power-law decay of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state.

The hydrodynamic part of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state has been calculated by using mode-coupling theory for a finite system with periodic boundary conditions. The existence of the shearing instability, leading to a divergent behavior of the velocity flow fluctuations, is taken into account. A time region in which the velocity autocorrelation function exhibits a power-law decay, when time is measured by the number of collisions per particle, has been been identified. Also the explicit form of the exponential asymptotic long time decay has been obtained. The theoretical prediction for the power-law decay is compared with molecular dynamics simulation results, and a good agreement is found, after taking into account finite size corrections. The effects of approaching the shearing instability are also explored.

Homogeneous hydrodynamics of a collisional model of confined granular gases.

The hydrodynamic equation governing the homogeneous time evolution of the temperature in a model of confined granular gas is studied by means of the Enskog equation. The existence of a normal solution of the kinetic equation is assumed as a condition for hydrodynamics. Dimensional analysis implies a scaling of the distribution function that is used to determine it in the first Sonine approximation, with a coefficient that evolves in time through its dependence on the temperature. The theoretical predictions are compared to numerical results obtained by the direct simulation Monte Carlo method and a good agreement is found. The relevance of the normal homogeneous distribution function to derive inhomogeneous hydrodynamic equations, for instance using the Champan-Enskog algorithm, is indicated.

Memory effects in the relaxation of a confined granular gas.

The accuracy of a model to describe the horizontal dynamics of a confined quasi-two-dimensional system of inelastic hard spheres is discussed by comparing its predictions for the relaxation of the temperature in a homogenous system with molecular dynamics simulation results for the original system. A reasonably good agreement is found. Next the model is used to investigate the peculiarities of the nonlinear evolution of the temperature when the parameter controlling the energy injection is instantaneously changed while the system was relaxing. This can be considered as a nonequilibrium generalization of the Kovacs effect. It is shown that, in the low-density limit, the effect can be accurately described by using a simple kinetic theory based on the first Sonine approximation for the one-particle distribution function. Some possible experimental implications are indicated.

Shearing instability of a dilute granular mixture.

The shearing instability of a dilute granular mixture composed of smooth inelastic hard spheres or disks is investigated. By using the Navier-Stokes hydrodynamic equations, it is shown that the scaled transversal velocity mode exhibits a divergent behavior, similarly to what happens in one-component systems. The theoretical prediction for the critical size is compared with direct Monte Carlo simulations of the Boltzmann equations describing the system, and a good agreement is found. The total energy fluctuations in the vicinity of the transition are shown to scale with the second moment of the distribution. The scaling distribution function is the same as found in other equilibrium and nonequilibrium phase transitions, suggesting the existence of some kind of universality.

Homogeneous steady state of a confined granular gas.

The nonequilibrium statistical mechanics and kinetic theory for a model of a confined quasi-two-dimensional gas of inelastic hard spheres is presented. The dynamics of the particles includes an effective mechanism to transfer the energy injected in the vertical direction to the horizontal degrees of freedom. The Enskog approximation is formulated and used as the basis to investigate the temperature and the distribution function of the steady state eventually reached by the system. An exact scaling of the distribution function of the system having implications on the form of its moments is pointed out. The theoretical predictions are compared with numerical results obtained by a particle simulation method, and a good agreement is found.

Internal energy fluctuations of a granular gas under steady uniform shear flow.

The stochastic properties of the total internal energy of a dilute granular gas in the steady uniform shear flow state are investigated. A recent theory formulated for fluctuations about the homogeneous cooling state is extended by analogy with molecular systems. The theoretical predictions are compared with molecular dynamics simulation results. Good agreement is found in the limit of weak inelasticity, while systematic and relevant discrepancies are observed when the inelasticity increases. The origin of this behavior is discussed.

Thermal segregation of intruders in the Fourier state of a granular gas.

A low density binary mixture of granular gases is considered within the Boltzmann kinetic theory. One component, the intruders, is taken to be dilute with respect to the other, and thermal segregation of the two species is described for a special solution to the Boltzmann equation. This solution has a macroscopic hydrodynamic representation with a constant temperature gradient and is referred to as the Fourier state. The thermal diffusion factor characterizing conditions for segregation is calculated without the usual restriction to Navier-Stokes hydrodynamics. Integral equations for the coefficients in this hydrodynamic description are calculated approximately within a Sonine polynomial expansion. Molecular dynamics simulations are reported, confirming the existence of this idealized Fourier state. Good agreement is found for the predicted and simulated thermal diffusion coefficient, while only qualitative agreement is found for the temperature ratio.

Rheological effects in the linear response and spontaneous fluctuations of a sheared granular gas.

The decay of a small homogeneous perturbation in the temperature of a dilute granular gas in the steady uniform shear flow state is investigated. Using kinetic theory based on the inelastic Boltzmann equation, a closed equation for the decay of the perturbation is derived. The equation involves the generalized shear viscosity of the gas in the time-dependent shear flow state, and therefore, it predicts relevant rheological effects beyond the quasielastic limit. Good agreement is found when comparing the theory with molecular dynamics simulation results. Moreover, the Onsager postulate on the regression of fluctuations is fulfilled.

Cooling rates and energy partition in inhomogeneous fluidized granular mixtures.

The local cooling rates of the components of a vibrated binary granular mixture in a steady state are investigated. The accuracy of the expression obtained by assuming a local homogeneous cooling state distribution of the gas is analyzed by comparing it with molecular dynamics simulation results. A good agreement is observed. Also, the profiles of the partial temperatures are compared with the theoretical prediction following from the application of the Chapman-Enskog method to solve the kinetic Enskog equations of the mixture. In this case, the agreement is satisfactory if the boundary layers near the walls are excluded. The implications of the results are discussed.

Fluctuating Navier-Stokes equations for inelastic hard spheres or disks.

Starting from the fluctuating Boltzmann equation for smooth inelastic hard spheres or disks, closed equations for the fluctuating hydrodynamic fields to Navier-Stokes order are derived. This requires deriving constitutive relations for both the fluctuating fluxes and the correlations of the random forces. The former are identified as having the same form as the macroscopic average fluxes and involving the same transport coefficients. On the other hand, the random force terms exhibit two peculiarities as compared with their elastic limit for molecular systems. First, they are not white but have some finite relaxation time. Second, their amplitude is not determined by the macroscopic transport coefficients but involves new coefficients.