Micro-structural parameters of the thoracic or lumbar spine generally carry insufficient accuracy and precision for clinical in vivo studies when assessed on quantitative computed tomography (QCT). We propose a 3D convolutional neural network with specific loss functions for QCT noise reduction to compute micro-structural parameters such as tissue mineral density (TMD) and bone volume ratio (BV/TV) with significantly higher accuracy than using no or standard noise reduction filters. The vertebra-phantom study contained high resolution peripheral and clinical CT scans with simulated in vivo CT noise and nine repetitions of three different tube currents (100, 250 and 360 mAs). Five-fold cross validation was performed on 20466 purely spongy pairs of noisy and ground-truth patches. Comparison of training and test errors revealed high robustness against over-fitting. While not showing effects for the assessment of BMD and voxel-wise densities, the filter improved thoroughly the computation of TMD and BV/TV with respect to the unfiltered data. Root-mean-square and accuracy errors of low resolution TMD and BV/TV decreased to less than 17% of the initial values. Furthermore filtered low resolution scans revealed still more TMD-and BV/TV-relevant information than high resolution CT scans, either unfiltered or filtered with two state-of-the-art standard denoising methods. The proposed architecture is threshold and rotational invariant, applicable on a wide range of image resolutions at once, and likely serves for an accurate computation of further micro-structural parameters. Furthermore, it is less prone for over-fitting than neural networks that compute structural parameters directly. In conclusion, the method is potentially important for the diagnosis of osteoporosis and other bone diseases since it allows to assess relevant 3D micro-structural information from standard low exposure CT protocols such as 100 mAs and 120 kVp.