In this paper, we consider the periodic solutions of a variable coefficient wave equation which models the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. Under Dirichlet–Neumann boundary conditions, we find some important properties for the variable coefficient wave operator. Then, based on these properties, we obtain the existence and multiplicity of periodic solutions by using the Leray–Schauder degree theory.
Rationale:
Clavicle fractures are common, accounting for 2.6 to 4% of all fractures, which typically result from direct injuries, including direct force on the shoulder after falling. However, bipolar clavicle fractures are rare, accounting for only 2.8% of all clavicle fractures, and their injury mechanism is speculated to evolve from two independent and continuous forces affecting the clavicle. Due to its low incidence, there is great controversy regarding the treatment of this fracture, as there is no relevant treatment standard or guideline to date.
Patient concerns:
In this case report, we describe a rare case of bipolar clavicle fracture in a 76-year-old man with multiple systemic fracture complications due to a traffic injury. He presented with limited shoulder function and movement upon arrival in the emergency room.
Diagnosis:
Bipolar clavicle fracture in the right shoulder (Robinson type 1B2, Robinson type 3B2)
Interventions:
We performed trans-sternoclavicular locking plate and lateral clavicular hook plate treatments and instructed patients to perform reasonable postoperative functional exercises.
Outcomes:
Three months postoperatively, the pain was almost completely relieved with a DASH score of 40.0. Furthermore, radiographic examination of the clavicle showed satisfactory fracture healing. The patient had no further demands for shoulder function and no irritative symptoms of internal fixation and refused to undergo a second operation. The patient had a satisfactory prognosis after the treatment.
Lessons:
The treatment of bipolar clavicle fractures remains controversial. This study provides evidence of a feasible method to treat bipolar clavicle fractures: trans-sternoclavicular locking plate and lateral clavicular hook plate treatment.
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