This paper presents the design of a new Positron Emission Tomography (PET) with improved time resolution, Time-Of-Flight PET (TOF-PET). Using the time information improves image quality and reduces the radiation dose inoculated to patients. Extended Field-Of-View (FOV) total-body scanner built from low-radioactivity liquid argon (LAr) scintillator (LAr-TOF-PET) and silicon photomultipliers (SiPMs) are readout by fast cryogenic electronics. This new concept is based on the emerging technology coming from the field of direct dark matter searches for weakly interacting massive particles (WIMPs). The DarkSide Collaboration has been particularly successful in demonstrating the true power of the advancing LAr detector technology. SiPM technology has many broader applications in science and industry, most notably in medical physics, where exquisite timing resolution of SiPMs could revolutionize PET, one of the key tools in a diagnosis of cancer, allowing one to substantially improve precision and at the same time minimize adverse side-effects to the patient. We simulated this design using Geant4 toolkit while following the National Electrical Manufacturers Association's evaluation tests for performance assessment. We will present results that highlight a 200-fold increase in sensitivity, spatial resolutions comparable to commercial PET scanners and produce PET images from 15-30 second scans, faster than traditional 30-35-minute scans. Further studies will involve optimizing the layer thickness of LAr+Xe. Moreover, scintillation induced ionization electrons can produce Cherenkov radiation along with the LAr+Xe scintillation light. With the LAr+Xe scintillator and SiPMs of 3Dπ, we can use the precise TOF info of gamma rays to improve the localization of individual positron annihilations, and as one example benefit, provide low-dose PET scans for patients who may be at high risk for exposure to radiation.