Dose escalation studies lay the foundation for successful clinical development. With patient centric drug development, the focus of dose finding switches from targeting the maximum tolerated dose to more comprehensive evaluation of the drug effects. I.e., in addition to the dose limiting toxicity as main outcome of drug toxicity effect, one could report different toxicity grades as a total toxicity burden. A patient experiencing six grade 2 toxicity events may have more overall toxicity burden than a patient with one grade 3 event. In addition, accounting for time-to-event outcomes may allow faster enrollment speed, leading to shorter trials. We propose a Bayesian design to incorporate time-to-event total toxicity burden in dose finding. The design allows rolling patient enrollment based on the assessment of total toxicity burden. We show that the proposed design can speed up the trial process and better characterize the toxicity profile of each dose. Through simulation study, we report the operating characteristics of the design in terms of duration of the trial and the success in finding the most promising dose with acceptable toxicity profile for later drug development.