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The feasibility of K-edge imaging using energy-resolved, photon-counting transmission measurements in X-ray computed tomography (CT) has been demonstrated by simulations and experiments. The method is based on probing the discontinuities of the attenuation coefficient of heavy elements above and below the K-edge energy by using energy-sensitive, photon counting X-ray detectors. In this paper, we investigate the dependence of the sensitivity of K-edge imaging on the atomic number Z of the contrast material, on the object diameter D , on the spectral response of the X-ray detector and on the X-ray tube voltage. We assume a photon-counting detector equipped with six adjustable energy thresholds. Physical effects leading to a degradation of the energy resolution of the detector are taken into account using the concept of a spectral response function R(E,U) for which we assume four different models. As a validation of our analytical considerations and in order to investigate the influence of elliptically shaped phantoms, we provide CT simulations of an anthropomorphic Forbild-Abdomen phantom containing a gold-contrast agent. The dependence on the values of the energy thresholds is taken into account by optimizing the achievable signal-to-noise ratios (SNR) with respect to the threshold values. We find that for a given X-ray spectrum and object size the SNR in the heavy element's basis material image peaks for a certain atomic number Z. The dependence of the SNR in the high-Z basis-material image on the object diameter is the natural, exponential decrease with particularly deteriorating effects in the case where the attenuation from the object itself causes a total signal loss below the K-edge. The influence of the energy-response of the detector is very important. We observed that the optimal SNR values obtained with an ideal detector and with a CdTe pixel detector whose response, showing significant tailing, has been de- ermined at a synchrotron differ by factors of about two to three. The potentially very important impact of scattered X-ray radiation and pulse pile-up occurring at high photon rates on the sensitivity of the technique is qualitatively discussed.