Nonlinear mechanics of the inner ear and its relation to otoacoustic emissions: two steps on the way to a mathematical model of DPOAE generation.

摘要

Among clinical users of the registration of distortion product otoacoustic emissions (DPOAE), the understanding of the basic causality and interpretation of the phenomenon is not yet widely spread, nor is the expected influence of the middle ear and ear canal clear. On the other side, the effort in mathematical modeling of middle and inner ear structures is driven very far by now. We are convinced, though, that the essentials of an effect as DPOAE generation must be understandable from quite simple models. In a first step de Boer's one-dimensional model was adopted and expanded by a weak frictional and a weak elastic nonlinearity, respectively. By means of perturbation theory the weakly nonlinear problem is converted in an approximation series of linear problems. So it is solvable by the common methods of linear differential equations (DEs), above all the superposition principle can be used. At the same time a structure of causality is introduced: Sources for outgoing waves are in first order approximation formed by incoming waves, and so they can be localized. The calculations show clearly that of all six cubic distortions only the 2f(1) - f(2) term does have a source in its 'allowed' region and so can travel outward. We can use the calculated DPOAE to study the influence of middle ear, external ear canal and probe plug. Some problems remain: the weakly nonlinear model in first order does not give account for proper L(dp) = f(L(1), L(2)) and L(dp) = f(f(2)/f(1)) dependency, nor does it deliver additional sources or the effect of additional suppressor tones. In a second step, therefore, we replace de Boer's simple model basilar membrane (BM) by a doubly resonant, coupled tectorial/basilar membrane (TM/BM) system. By feedback now we introduce a strong nonlinearity, which we can mathematically care for by an iterative feedback loop. The algorithm shapes the incoming waves according to strong compressive nonlinearity. More relastic incoming waves yield better source terms, and after optimization of the mistuning function between TM and BM the model now is able to deliver qualitatively correct L(dp) (L(1),L(2)) and L(dp)(f(2)/f(1)) dependencies.