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Quantitative PET attenuation correction (AC) for combined cardiac PET/MR is a challenging problem. We propose and evaluate an AC approach that uses coincidences from a relatively weak and physically fixed sparse external source, in combination with that from the patient, to correct for PET attenuation based on physics principles alone. The low 30 ml volume of the source makes it easy to fill and place, and the method does not use prior image data or attenuation map assumptions. Our supplemental transmission aided maximum likelihood reconstruction of attenuation and activity (sTX-MLAA) algorithm contains an attenuation map update that maximizes the likelihood of terms representing coincidences originating from tracer in the patient and a weighted expression of counts segmented from the external source alone. Both external source and patient scatter and randoms are fully corrected. We evaluated performance of sTX-MLAA compared to reference standard CT-based AC with FDG PET/CT phantom studies; including modeling a patient with myocardial inflammation. Through an ROI analysis we measured less than 5% bias in activity concentrations for PET images generated with sTX-MLAA relative to CT-AC. PET background variability (from noise and sparse sampling) was substantially reduced with sTX-MLAA compared to using coincidences segmented from the transmission source alone for AC. The study suggests that sTX-MLAA will produce PET images on PET/MR with quantification comparable to PET/CT results during human cardiac exams.