Radiotherapy dose calculations have evolved from simple factor based methods performed with pen and paper, into computationally intensive simulations based on Monte Carlo theory and energy deposition kernel convolution. Similarly, in the field of positron emission tomography (PET), attenuation correction, which was originally omitted entirely, is now a crucial component of any PET reconstruction algorithm. Today, both of these applications – radiotherapy and PET – derive their needed in-tissue radiation attenuation coefficients from images acquired with X-ray computed tomography (CT). Since X-ray images are themselves acquired using ionizing radiation, the intensity at a point in an image will reflect the radiation interaction properties of the tissue located at that point. Magnetic resonance imaging (MRI), on the other hand, does not use ionizing radiation. Instead MRI make use of the net transverse magnetization resulting from the spin polarization of hydrogen nuclei. MR image contrast can be varied to a greater extent than CT and the soft tissue contrast is, for most MR sequences, superior to that of CT. Therefore, for many cases, MR images provide a considerable advantage over CT when identifying or delineating tumors or other diseased tissues. For this reason, there is an interest to replace CT with MRI for a great number of diagnostic and therapeutic workflows. Also, replacing CT with MRI would reduce the exposure to ionizing radiation experienced by patients and, by extension, reduce the associated risk to induce cancer. In part MRI has already replaced CT, but for radiotherapy dose calculations and PET attenuation correction, CT examinations are still necessary in clinical practice. One of the reasons is that the net transverse magnetization imaged in MRI cannot be converted into attenuation coefficients for ionizing radiation in a straightforward way. More specifically, regions with similar appearance in magnetic resonance (MR) images, such as bone and air pockets, are found at different ends of the spectrum of attenuation coefficients present in the human body. In a CT image, bone will appear bright white and air as black corresponding to high and no attenuation, respectively. In an MR image, bone and air both appear dark due to the lack of net transverse magnetization. The weak net transverse magnetization of bone is a result of low hydrogen density and rapid transverse relaxation. A particular category of MRI sequences with so-called ultrashort echo time (UTE) can sample the MRI signal from bone before it is lost due to transverse relaxation. Thus, UTE sequences permit bone to be imaged with MRI albeit with weak intensity and poor resolution. Imaging with UTE in combination with careful image analysis can permit ionizing-radiation attenuation-maps to be derived from MR images. This dissertation and appended articles present a procedure for this very purpose. However, as attenuation coefficients are radiation-quality dependent the output of the method is a Hounsfield unit map, i.e. a substitute for a CT image. It can be converted into an attenuation map using conventional clinical procedure. Obviating the use of CT would reduce the number of examinations that patients have to endure during preparation for radiotherapy. It would also permit PET attenuation correction to be performed on images from the new imaging modality that combines PET and MRI in one scanner – PET/MR.