Approximations for the optimization problem for medical microneedle systems

Microneedle systems are used for transdermal (hypodermic) medicine injections at the treatment of different diseases. The efficiency of using such systems depends significantly on the size and parameters of microneedles. The problem of determining such dependencies and optimal parameters is considered as the problem of optimizing the interaction of microneedle systems with an elastic surface. Minimization problems for integral functional, whose solutions are approximations for solutions to the interaction problem, are obtained by the homogenization theory methods. Such problems are formulated in the form of classical problems with obstacles .


A model for elastic interaction of a microneedle system with a surface
In order to represent a model of a real microneedle system, consider, for example, a square is the simplest model of the microneedle system from (1). Minimum of functional () l Fu  defines the optimal configuration of the surface (corresponding to a skin area if injections are considered) above the domain  under the action of the simplest system of microneedles () l x   from (1).
It is known [12] that the minimum ()

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The last case of this theorem is quite illustrated in Fig. 1. This situation is natural, since the minimum of problem (2) is strongly oscillating, characterizing very large energy of elastic resistance of the surface above the square  under the assumption of the absence of transversal displacements. The first case of this theorem is quite right represented in Fig. 2. This situation is also natural, since the minimum of problem (2) is zero, which characterizes a very small energy of elastic surface resistance over the square  under the assumption that for very thin microneedles there are no transverse displacements. This surface configuration should ideally be realized when using microneedle systems for drug delivery in the treatment of diseases in modern medicine. The fulfillment of these cases is not in doubt, since the latter is used in real medicine, and the first case can be visualized theoretically, for example, when we select as the sets ij radius. This visualization is shown in Fig. 1. Thus, there must be a case averaging the considered cases. The surface configuration for this case must be characterized by the averaged surface between the surfaces shown in Figures 1 and 2. This situation corresponds to case 2 of Theorem 1. The averaged surface can be calculated explicitly, for example, in the case when homogeneous Dirichlet boundary conditions in (1) are replaced by periodic boundary conditions in one of the coordinate directions. These calculations are given in [11]. In general, numerical methods can be used to calculate this surface. Integral functional () l Fu from Theorem 1 is usually called homogenized, since the minimum of this functional approximates the minimum of the considered functional of problem (2) for sufficiently small  or, which is equivalent by definition, for a sufficiently large number of microneedles in the system under consideration. Moreover, it follows from results of book [12] and the theorem that statements of Theorem 1 are fulfilled simultaneously for the sets holding the inclusions hold. Assuming the fulfillment of one of the conditions of the theorem and passing to the limit for 0  , we conclude that these inequalities transform into equalities, which proves the above formulated independence of the theorem statements of the shape microneedle base. The ascertained independence of the given statements of the needle base shapes is caused, first of all, by microthickness of needles, located periodically and forming the considered systems. The given statements clarify also that systems with circular cylindrical microneedles have the most optimal parameters since they have optimal contact surface and the best transmission capacity.

Independence of homogenized configurations from microneedle bases
Conclusion. Thus a new variational method of modeling and computation of the parameters of transdermal (hypodermic) injection of medicines on the use of a microneedle system is presented in the report. By the homogenization theory methods, we studied the problems of minimization for integral functionals with oscillating obstacles, which model the considered systems of microneedles. Such problems describe elastic resistance of surfaces on interaction with microneedle systems.
Invariance of the adduced statements from the shape of microneedle system base is proved. Such invariance is caused first of all by microthikness of needles located central-symmetrically and forming the considered microneedle systems. The given statements also show that systems with circular cylindrical microneedles are the most optimal since such needles have the optimal square of contact and the best transmission capacity for medicine injections at the treatment of various diseases.