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On a possible theory for the design of tissue expanders. (English) Zbl 0656.73046

Silastic implants have been used in plastic surgery since about 1976 to expand tissue which is used to reconstruct neighbouring defects. In order to reduce the amount of bleeding and ultimate scarring it seems worthwhile to design carefully the tissue expander to be used. The analysis presented in this paper is intended to go someway in this direction.
The plan of the paper is as follows. In section 2 we formulate the problem based on the assumption that the expanded skin will try to minimize its surface area (in the discussion section we discuss some possible generalizations of this); the problem is then posed as a problem in the calculus of variations subject to a constraint. The corresponding Euler-Lagrange equations lead to a nonlinear differential equation with a parameter \(\lambda\) (the Lagrange multiplier of the variational principle) which can be related to the amount of saline in the implant. In section 2.1 we look for an approximate solution via the direct method of the calculus of variations for an assumed family of shapes for the base of the implant. In this way we can investigate the usefulness of a concave plan form such as is shown for skin advancement to cover a neighbouring defect. A simple criterion is also given for determining a stage at which ‘full expansion’ is achieved. In practice too much expansion may lead to complications such as breaking of the skin.
In section 3 dual variational principles are considered and used in the calculations to give us some confidence in our numerical results. In section 4 the differential-equation version of the problem is considered and a maximum principle is obtained by analogy with results from nonlinear torsion. In section 5 we present some numerical results together with some plots of the (theoretical) shape of the expanded skin surface and a description of how the advancement would take place. Note that in the authors and O. M. Fenton’s submitted article (ibid.) an account is given of an operation in which a similar plan-form construction to that considered here has been successfully performed, although more work needs to be done to develop this further and to understand the characteristics of the skin.

MSC:

74L15 Biomechanical solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74P99 Optimization problems in solid mechanics
49M15 Newton-type methods
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