A mathematical treatise on the fit of crown castings

A mathematical analysis was given to calculate the marginal cement thickness, Dm, of a cemented crown casting as a function of the minimum cement film thickness, F; the width, W, of the base of the tapered region; the width, W', at the bevel (if present) of the tooth preparation; the taper angle, theta; the marginal angle, beta; and the compensatory expansion, X, of the metal investment system: Dm = [F/sin theta - 0.5(X - 1) (W cot theta - W' cot beta)] sin beta. The elevation of the cemented casting, delta H, above the tooth preparation may also be obtained from this equation by substituting beta = 90 degrees, in which case Dm = delta H. The effect of the preparation of a chamfer margin or bevel on the marginal film thickness may be calculated from this equation. An additional analysis showed that the minimum marginal cement thickness, Dm = F, can be obtained for a shoulder preparation and X values of common metal-investment systems by applying a spacer of appropriate thickness on the entire occlusal and axial walls of the die. This renders the preparation of bevels superfluous. The thickness of the spacer should allow for the cement film thickness, roughness of the tooth and casting surfaces, dimensional inaccuracies of the die, and distortions of the wax pattern. If the spacer is thicker than required, a minimum value of Dm is still obtained for preparations without bevels. If the horizontal part of the die shoulder is covered as well with the spacer, a greater spacer thickness is required to obtain a minimum value of Dm than if the spacer extends to the end of the axial walls. In this case the spacer thickness should be greater on the occlusal die wall than on other regions of the die for values of X smaller than 1. The spacer should also be thicker in the occlusal region if restriction of the investment setting expansion by the wax pattern occurs mainly in this region.