Doubling time: toward understanding an abstruse concept.

• Author: Stark, James J.
• Position: Tumor growth; litigation concerning misdiagnosis

A MAN is diagnosed with lung cancer. X-ray reveals a 6-cm tumor that is inoperable because it invades the trachea, the main airway from the mouth to the lungs. He tells his attorney that he was complaining of a cough two years earlier. He tried to convince his doctor to take an x-ray, but the doctor, under economic constraints from the local health maintenance organization, refused.

What would an x-ray have shown?

Would the cancer have been visible?

Would finding the cancer then have made a difference to his outcome?

What is the role of "doubling time" in building a defense to a medical malpractice action?

What is the doubling time of a tumor?

What is the impact of doubling time on proximate cause in a case involving the alleged missed diagnosis of cancer?

If doubling time is known, how can it help defense counsel construct a defense that shows that any alleged delay in diagnosis had no relation to a poor outcome?

To answer these questions, we must know what doubling time is and its strengths and limitations in the defense of an allegation of a delay in the diagnosis of cancer.

What is doubling time?

Doubling time (DT) is the time it takes a tumor to double in volume--not diameter. The volume of a tumor is [Pi]/6 (ABC) where A, B and C are the measurements of the tumor in three dimensions. The doubling time of a tumor is the net result of the effects of cell division and cell death. As an abstract mathematical concept, however, it does not depend on those individual rates but rather on the over-all growth rate. It is best viewed simply as a number that quantifies that growth seen on serial x-rays or physical examinations.

Although most experts agree that cancers do not grow constantly but rather in short spurts, the spurts average out over time to produce smooth growth. Hence the concept of steady growth is reasonable over the time frame of the average malpractice case, which is months to years, whereas it would not necessarily be for much shorter intervals, such as hours to days, when a tumor might grow a little for a few hours and then rest.

The DT can be calculated by knowing the volume of the tumor at two different moments and the time between measurements. The formula used to do the calculation is:

final volume = (initial volume) [e.sup.kt]; k = ln2/doubling time and t is time

e is the universal growth constant and roughly equals 2.718. You can, by inserting initial and final volume, solve the equations for DT. Or if you know DT and either initial or final volume, you can insert one and solve for the other. Such equations are easily programmed into a graphing calculator or personal computer. I have placed such software on my personal computer.

By way of simple example (without needing to resort to these equations): if a tumor at time A has a volume of 3 [cm.sup.3] and at time B has a volume of 12 [cm.sup.3], in the interval the tumor has doubled in volume twice--once to get to 6 [cm.sup.3] and again to get to 12). If that interval is 700 days, the DT is 350 days.

More definitions

"Exponential growth" describes simple biological growth, e.g., when one cell divides and becomes two; two divide and become four; four become eight. Exponential growth occurs when the interval between each cell division is constant--i.e., it takes as much time to go from one to two cells as from 1 to 12 [cm.sup.3]. The equation above assumes simple exponential growth. Doubling times quoted in the oncology literature for a variety of tumors typically rely on the assumption of exponential growth. For brief periods of observation (typically less than a year), this assumption is fairly accurate. The equations above describe exponential growth.

"Gompertzian growth" describes a special type of exponential growth in which the doubling time gradually increases (i.e., the tumor slows its growth) as the tumor enlarges and ages. It is described mathematically by an equation more complex than that for simple exponential growth, in which there are two growth constants: one to describe growth and the other the slowdown in growth.

Gompertzian growth is described by the so-called sigmoid curve of tumor size versus time over the entire life of the tumor. (See...