Cell detection in histopathology images is of great interest to clinical practice and research, and convolutional neural networks (CNNs) have achieved remarkable cell detection results. Typically, to train CNN-based cell detection models, every positive instance in the training images needs to be annotated, and instances that are not labeled as positive are considered negative samples. However, manual cell annotation is complicated due to the large number and diversity of cells, and it can be difficult to ensure the annotation of every positive instance. In many cases, only incomplete annotations are available, where some of the positive instances are annotated and the others are not, and the classification loss term for negative samples in typical network training becomes incorrect. In this work, to address this problem of incomplete annotations, we propose to reformulate the training of the detection network as a positive-unlabeled learning problem. Since the instances in unannotated regions can be either positive or negative, they have unknown labels. Using the samples with unknown labels and the positively labeled samples, we first derive an approximation of the classification loss term corresponding to negative samples for binary cell detection, and based on this approximation we further extend the proposed framework to multi-class cell detection. For evaluation, experiments were performed on four publicly available datasets. The experimental results show that our method improves the performance of cell detection in histopathology images given incomplete annotations for network training.