A simple model of maximum tumor metabolic burden

The proliferation and survivorship dynamics of normal cells of a multicellular organism are governed by tissue control mechanisms and cell intrinsic properties aimed at serving the whole organism. Not so for cancer. Cancer can be defined as uncontrolled proliferation by transformed cells subject to natural selection. Therefore, cancer exhibits ecological and evolutionary dynamics and conforms to the laws and principles of ecology and evolution. First, under ideal conditions cancer cells have the capacity to grow exponentially. For most cancers, this results in a doubling time of every 24-48 hours. Fortunately, in patients the net growth rates of cancer cells are much less: ranging between 0.1 – 5% per day. Because of limits to growth no population can grow exponentially forever. Models of limits to growth can be phenomenological (e.g., logistic and Gompertz) or mechanistic (e.g., consumer-resource). Here, I develop a simple model for three scales of limits to growth within patients. First, within a tumor there are limits to how densely packed cells can become. This “carrying capacity” is likely reached quickly (days or weeks) and manifests as lumps in normal tissue. Second, as the tumor volume grows (months to years) it acquires new blood flow at a rate lower than the increase in “mouths to feed” placing a limit on the maximize size of a given tumor. Third, the human body turns over about 0.75% of its cells per day with a maximum capacity of 1.5% setting an upper limit on total body tumor burden (years). Even in the absence of organ failure, simply supporting the cancer cells can indirectly lead to patient death via cachexia and hyper-metabolic demands on the patient. This invites mathematical models of whole patient tumor dynamics and therapy that include these three important scales.