Distribution of particles which produces a "smart" material

If $A_q(\beta, \alpha, k)$ is the scattering amplitude, corresponding to a potential $q\in L^2(D)$, where $D\subset\R^3$ is a bounded domain, and $e^{ik\alpha \cdot x}$ is the incident plane wave, then we call the radiation pattern the function $A(\beta):=A_q(\beta, \alpha, k)$, where the unit vector $\alpha$, the incident direction, is fixed, and $k>0$, the wavenumber, is fixed. It is shown that any function $f(\beta)\in L^2(S^2)$, where $S^2$ is the unit sphere in $\R^3$, can be approximated with any desired accuracy by a radiation pattern: $||f(\beta)-A(\beta)||_{L^2(S^2)}<\epsilon$, where $\epsilon>0$ is an arbitrary small fixed number. The potential $q$, corresponding to $A(\beta)$, depends on $f$ and $\epsilon$, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles $D_m\subset D$, $1\leq m\leq M$, distributed in an a priori given bounded domain $D\subset\R^3$. The geometrical shape of a small particle $D_m$ is arbitrary, the boundary $S_m$ of $D_m$ is Lipschitz uniformly with respect to $m$. The wave number $k$ and the direction $\alpha$ of the incident upon $D$ plane wave are fixed.It is shown that a suitable distribution of the above particles in $D$ can produce the scattering amplitude $A(\alpha',\alpha)$, $\alpha',\alpha\in S^2$, at a fixed $k>0$, arbitrarily close in the norm of $L^2(S^2\times S^2)$ to an arbitrary given scattering amplitude $f(\alpha',\alpha)$, corresponding to a real-valued potential $q\in L^2(D)$.Comment: corrected typo

Dynamic spasticity of plantar flexor muscles in cerebral palsy gait

Objective: To quantify dynamic spasticity, i.e. the coupling between muscle-tendon stretch velocity and muscle activity during gait, of the gastrocnemius and soleus muscles in children with spastic cerebral palsy. Design: Prospective, cross-sectional study. Subjects: Seventeen ambulatory children with cerebral palsy with spastic calf muscles, and H matched typically developing children. Methods: The children walked at 3 different speeds. Threedimensional kinematic and electromyographic data were collected. Muscle-tendon velocities of the gastrocnemius medialis and soleus were calculated using musculoskeletal modelling. Results: In typically developing children, muscles were stretched fast in swing without subsequent muscle activity, while spastic muscles were stretched more slowly for the same walking speed, followed by an increase in muscle activity. The mean ratio between peak activity and peak stretch velocity in swing was approximately 4 times higher in spastic muscles, and increased with walking speed. In stance, the stretch of muscles in typically developing children was followed by an increase in muscle activity. Spastic muscles were stretched fast in loading response, but since muscle activity was already built up in swing, no clear dynamic spasticity effect was present. Conclusion: Spastic calf muscles showed increased coupling between muscle-tendon stretch velocity and muscle activity, especially during the swing phase of gait. © 2010 Foundation of Rehabilitation Information

How Crouch Gait Can Dynamically Induce Stiff-Knee Gait

Children with cerebral palsy frequently experience foot dragging and tripping during walking due to a lack of adequate knee flexion in swing (stiff-knee gait). Stiff-knee gait is often accompanied by an overly flexed knee during stance (crouch gait). Studies on stiff-knee gait have mostly focused on excessive knee muscle activity during (pre)swing, but the passive dynamics of the limbs may also have an important effect. To examine the effects of a crouched posture on swing knee flexion, we developed a forward-dynamic model of human walking with a passive swing knee, capable of stable cyclic walking for a range of stance knee crouch angles. As crouch angle during stance was increased, the knee naturally flexed much less during swing, resulting in a 'stiff-knee' gait pattern and reduced foot clearance. Reduced swing knee flexion was primarily due to altered gravitational moments around the joints during initial swing. We also considered the effects of increased push-off strength and swing hip flexion torque, which both increased swing knee flexion, but the effect of crouch angle was dominant. These findings demonstrate that decreased knee flexion during swing can occur purely as the dynamical result of crouch, rather than from altered muscle function or pathoneurological control alone. © 2010 The Author(s)

Muscle fatigue during repetitive voluntary contractions: a comparison between children with cerebral palsy, typically developing children and young healthy adults.

Aim: To combine peak torque and EMG analyses to investigate the hypothesis that 1) children with cerebral palsy (CP) have lower muscle fatigability than typically developing children (TD) and whether 2) muscle fatigue correlates with muscle strength. Methods: Seven CP children, eight TD children and ten young healthy adults (YHA) performed an all-out fatigue test of 35 maximal concentric knee extension and flexion contractions on an isokinetic dynamometer. Angular velocity was set at 60°/s. Peak torque (PT) was determined per repetition and either normalized to bodyweight or maximum voluntary torque. Surface-EMG of quadriceps and hamstring muscles was measured to obtain changes in median frequency (EMG- mf) and smooth rectified EMG amplitude per contraction. Results: Decline in PT differed between all groups for extensors and flexors, where YHA showed the largest decline and CP children the smallest decline over the course of the test. YHA showed a larger decline in EMG- mf of all quadriceps and hamstrings than TD and CP children, while TD children showed a larger decline in EMG- mf of m.rectus femoris and m.vastus lateralis than CP children. Interpretation: Results confirm that children with CP show lower fatigability than TD children and that the lower fatigability coincides with lower maximal muscle strength. © 2013 Elsevier B.V

The effect of walking speed on hamstrings length and lengthening velocity in children with spastic cerebral palsy

Children with cerebral palsy often walk with reduced knee extension in terminal swing, which can be associated with short length or slow lengthening velocity of hamstrings muscles during gait. This study investigated the role of two factors that may contribute to such short and slow hamstrings: walking speed and spasticity. 17 children with spastic cerebral palsy and 11 matched typically developing children walked at comfortable, slow, and fast walking speed. Semitendinosus muscle-tendon length and velocity during gait were calculated using musculoskeletal modeling. Spasticity of the hamstrings was tested in physical examination. Peak hamstrings length increased only slightly with walking speed, while peak hamstrings lengthening velocity increased strongly. After controlling for these effects of walking speed, spastic hamstrings acted at considerably shorter length and slower velocity during gait than normal, while non-spastic hamstrings did not (all

Validation of hamstrings musculoskeletal modeling by calculating peak hamstrings length at different hip angles

Accurate estimates of hamstrings lengths are useful, for example, to facilitate planning for surgical lengthening of the hamstrings in patients with cerebral palsy. In this study, three models used to estimate hamstrings length (M1: Delp, M2: Klein Horsman, M3: Hawkins and Hull) were evaluated. This was done by determining whether the estimated peak semitendinosus, semimembranosus and biceps femoris long head lengths, as measured in eight healthy subjects, were constant over a range of hip and knee angles. The estimated peak hamstrings length depended on the model that was used, even with length normalized to length in anatomical position. M3 estimated shorter peak lengths than M1 and M2, showing that more advanced models (M1 and M2) are more similar. Peak hamstrings length showed a systematic dependence on hip angle for biceps femoris in M2 and for semitendinosus in M3, indicating that either the length was not correctly estimated, or that the specific muscle did not limit the movement. Considerable differences were found between subjects. Large inter-individual differences indicate that modeling results for individual subjects should be interpreted with caution. Testing the accuracy of modeling techniques using in vivo data, as performed in this study, can provide important insights into the value and limitations of musculoskeletal models. © 2008 Elsevier Ltd. All rights reserved

Assessment of muscle endurance of the knee extensor muscles in adolescents with spastic cerebral palsy using a submaximal repetitions-to-fatigue protocol

Objective To compare muscle endurance in adolescents with spastic cerebral palsy (CP) with typically developing (TD) peers using a submaximal repetitions-to-fatigue (RTF) protocol

Reproducibility and validity of video screen measurements of gait in children with spastic cerebral palsy.

PURPOSE: To determine the reproducibility and validity of video screen measurement (VSM) of sagittal plane joint angles during gait. METHODS: 17 children with spastic cerebral palsy walked on a 10m walkway. Videos were recorded and 3d-instrumented gait analysis was performed. Two investigators measured six sagittal joint/segment angles (shank, ankle, knee, hip, pelvis, and trunk) using a custom-made software package. The intra- and interrater reproducibility were expressed by the intraclass correlation coefficient (ICC), standard error of measurements (SEM) and smallest detectable difference (SDD). The agreement between VSM and 3d joint angles was illustrated by Bland-Altman plots and limits of agreement (LoA). RESULTS: Regarding the intrarater reproducibility of VSM, the ICC ranged from 0.99 (shank) to 0.58 (trunk), the SEM from 0.81 degrees (shank) to 5.97 degrees (trunk) and the SDD from 1.80 degrees (shank) to 16.55 degrees (trunk). Regarding the interrater reproducibility, the ICC ranged from 0.99 (shank) to 0.48 (trunk), the SEM from 0.70 degrees (shank) to 6.78 degrees (trunk) and the SDD from 1.95 degrees (shank) to 18.8 degrees (trunk). The LoA between VSM and 3d data ranged from 0.4+/-13.4 degrees (knee extension stance) to 12.0+/-14.6 degrees (ankle dorsiflexion swing). CONCLUSION: When performed by the same observer, VSM mostly allows the detection of relevant changes after an intervention. However, VSM angles differ from 3d-IGA and do not reflect the real sagittal joint position, probably due to the additional movements in the other planes

How Crouch Gait Can Dynamically Induce Stiff-Knee Gait

Children with cerebral palsy frequently experience foot dragging and tripping during walking due to a lack of adequate knee flexion in swing (stiff-knee gait). Stiff-knee gait is often accompanied by an overly flexed knee during stance (crouch gait). Studies on stiff-knee gait have mostly focused on excessive knee muscle activity during (pre)swing, but the passive dynamics of the limbs may also have an important effect. To examine the effects of a crouched posture on swing knee flexion, we developed a forwarddynamic model of human walking with a passive swing knee, capable of stable cyclic walking for a range of stance knee crouch angles. As crouch angle during stance was increased, the knee naturally flexed much less during swing, resulting in a ‘stiff-knee’ gait pattern and reduced foot clearance. Reduced swing knee flexion was primarily due to altered gravitational moments around the joints during initial swing. We also considered the effects of increased push-off strength and swing hip flexion torque, which both increased swing knee flexion, but the effect of crouch angle was dominant. These findings demonstrate that decreased knee flexion during swing can occur purely as the dynamical result of crouch, rather than from altered muscle function or pathoneurological control alone.Biomechanical EngineeringMechanical, Maritime and Materials Engineerin