Quantitative PET attenuation correction (AC) for cardiac PET/CT and 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 reconstruct μ -maps based on physics principles alone. The low 30 cm3 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 ≤ 5 % bias in activity concentrations for PET images generated with sTX-MLAA and a TX source strength ≥ 12.7 MBq, relative to CT-AC. PET background variability (from noise and sparse sampling) was substantially reduced with sTX-MLAA compared to using counts segmented from the transmission source alone for AC. Results suggest that sTX-MLAA will enable quantitative PET during cardiac PET/CT and PET/MR of human patients.