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In the present work, we adopt the idea of velocity averaging lemma to establish regularity for stationary linearized Boltzmann equations in a bounded convex domain. Considering the incoming data, with three iterations, we establish regularity in fractional Sobolev space in space variable up to order $1^-$.

Three-dimensional printed ankle-foot orthoses (AFO) have been used in stroke patients recently, but there was little evidence of gait improvement. Here, we designed a novel customized AFO with energy storage, named Energy-Storage 3D Printed Ankle-Foot Orthosis (ESP-AFO), and investigated its effects on gait improvement in stroke patients. 12 stroke survivors were recruited (9 females/3 males, age: 55.58 ± 5.9 y/o). Participants were instructed to walk (1) bare foot (BF), with (2) anterior AFO (AAFO), and with (3) ESP-AFO, respectively. Gait analysis by the motion capture system was performed bilaterally (SL: Sound limb; AL: Affected limb). Participant satisfaction and fatigue were also compared. We found that the ESP-AFO significantly increased bilateral gait velocity (SL: ESP-AFO 52.62 ± 19.18 cm/s, BF 47.61 ± 22.78 cm/s; AL: ESP-AFO 52.78 ± 19.57 cm/s, BF 47.31 ± 23.27 cm/s), stride length (SL: ESP-AFO 78.96 ± 19.16 cm, BF 74.89 ± 21.92 cm; AL: ESP-AFO 79.06 ± 17.82 cm, BF 74.08 ± 21.31 cm). ESP-AFO reversed drop-foot in swing phase and created a higher ankle moment at terminal stance. Patients also reported more satisfaction with the ESP-AFO (total satisfaction: ESP-AFO 4.3 ± 0.47, AAFO 3.3 ± 0.62). We demonstrated that the novel custom-made ESP-AFO improved the gait performance with better satisfaction than traditional AFOs did. (Registry number of clinical trial: A-BR-106-005)

Ankle-foot orthoses (AFO) were well-used for stroke patients. Our study developed a new 3D printed AFO with the function of Energy Storage. It would be expected to improve the gait of the stroke patients. This study made a 3D printed joint part fixed between the foot plate and shank structure of AFO. The joint part was made of Nylon and inserted a steel plate to increase the strength against the inversion motion of foot. The potential energy from the dorsiflexion of ankle could be saved in the joint part and released in the pre-swing phase of the gait cycle to help walking. Due to the local health insurance policy patients can replace a new AFO every three months. The life of 3D printed AFO which was used in the condition of 2000 steps one day should last at least 3 months. Therefore this study tested the life of 3D printed AFO using the material test machine. The reciprocating motion of 3D printed AFO was limited in the range of 3 degree of plantarflexion and 10 degree of dorsiflexion. The results showed the shank structure broke in the clamping site of the jig after 144000 times of bending. The broken part showed the separating direction was not in the depositing direction of 3D printing path. It means the printing pattern of every layer is also a key issue for the strength of 3D printed object. In the future, this study will check the strength of 3D printed samples with different printing patterns and redesign the shank structure for the better life.