While deep learning based speech enhancement systems have made rapid progress in improving the quality of speech signals, they can still produce outputs that contain artifacts and can sound unnatural. We propose a novel approach to speech enhancement aimed at improving perceptual quality and naturalness of enhanced signals by optimizing for key characteristics of speech. We first identify key acoustic parameters that have been found to correlate well with voice quality (e.g. jitter, shimmer, and spectral flux) and then propose objective functions which are aimed at reducing the difference between clean speech and enhanced speech with respect to these features. The full set of acoustic features is the extended Geneva Acoustic Parameter Set (eGeMAPS), which includes 25 different attributes associated with perception of speech. Given the non-differentiable nature of these feature computation, we first build differentiable estimators of the eGeMAPS and then use them to fine-tune existing speech enhancement systems. Our approach is generic and can be applied to any existing deep learning based enhancement systems to further improve the enhanced speech signals. Experimental results conducted on the Deep Noise Suppression (DNS) Challenge dataset shows that our approach can improve the state-of-the-art deep learning based enhancement systems.