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Artificial intelligence-based methods have generated substantial interest in nuclear medicine. An area of significant interest has been using deep-learning (DL)-based approaches for denoising images acquired with lower doses, shorter acquisition times, or both. Objective evaluation of these approaches is essential for clinical application. DL-based approaches for denoising nuclear-medicine images have typically been evaluated using fidelity-based figures of merit (FoMs) such as RMSE and SSIM. However, these images are acquired for clinical tasks and thus should be evaluated based on their performance in these tasks. Our objectives were to (1) investigate whether evaluation with these FoMs is consistent with objective clinical-task-based evaluation; (2) provide a theoretical analysis for determining the impact of denoising on signal-detection tasks; (3) demonstrate the utility of virtual clinical trials (VCTs) to evaluate DL-based methods. A VCT to evaluate a DL-based method for denoising myocardial perfusion SPECT (MPS) images was conducted. The impact of DL-based denoising was evaluated using fidelity-based FoMs and AUC, which quantified performance on detecting perfusion defects in MPS images as obtained using a model observer with anthropomorphic channels. Based on fidelity-based FoMs, denoising using the considered DL-based method led to significantly superior performance. However, based on ROC analysis, denoising did not improve, and in fact, often degraded detection-task performance. The results motivate the need for objective task-based evaluation of DL-based denoising approaches. Further, this study shows how VCTs provide a mechanism to conduct such evaluations using VCTs. Finally, our theoretical treatment reveals insights into the reasons for the limited performance of the denoising approach.