Doubling Time

Doubling time is an important concept in many life sciences and in epidemiology, in particular: It refers to the length of time required for some quantity to double in size. It is commonly used to refer to human population size, but it is also used to describe, for instance, growth of viral counts or antigens, tumors, and the number of cases of a particular disease. Suppose we have a quantity Q(t) growing in time t. The function Q(t) may stand for the population size or the amount of a substance at time t. Doubling time is the time it takes for Q(t) to double in size and is uniquely determined by the growth rate r(t). If Q(t) has an exponential growth, then the doubling time can be exactly calculated from the constant growth rate r(t) = r. However, without the assumption of exponential growth or knowledge of the growth rate, one still can closely estimate the doubling time using two data values of Q(t) at two time points. The estimation formulas are derived below:

The growth rate for Q(t), assumed to be a continuous function of t, is defined as the relative change of Q(t) with respect to t:

where ln stands for natural logarithm (loge). Equation 1 indicates that the growth rate r(t) has dimension ‘per unit time.’ Equation 1 is a simple linear firstorder differential equation, which can be easily integrated to yield

where Q(0) is the quantity at time t = 0 and

is the average growth rate over the time interval [0, t]. Solving Equation 2 for t we obtain

The doubling time is the value of t, say, τ, such that Q(t)= Q(0) = 2, which when substituted into Equation 3 yields the following relation between doubling time and average growth rate:

which is approximately equal to 70% divided by the average growth rate, the so-called rule of seventy. Equation 4 says that the doubling time τ is inversely proportional to the average growth rate. The larger the growth rate, the smaller the doubling time. Doubling the growth rate is equivalent to halving the doubling time. For example, a short tumor-doubling time implies that the tumor is growing rapidly in size (measured either in terms of tumor volume or diameter). In fact, tumor-doubling times are considerably shorter in cancer patients who developed metastases than in those who did not develop metastatic disease.

[Page 284]In real applications, the average growth rate rt in Equation 4 is unknown and has to be estimated from data in some time interval [0, T], where Q(0) and Q(T) are both known. From Equation 3, we have

The closer T is to τ, the better the approximation in Equation 5. When T = τ, the approximation becomes an equality. Substituting Equation 5 into Equation 4 yields an estimate of the doubling time as

If the growth rate r(t) = r > 0 is a positive constant, then rt = r and Q(t) has an exponential growth. In this case, strict equality instead of approximation holds in Equations 5 and 6, as stated in the first paragraph. We now illustrate an application of formula (Equation 6) to estimate a doubling time in infectious disease epidemiology. During the outbreak of severe acute respiratory syndrome (SARS) epidemic in the early months of 2003, the health authority in Hong Kong had reported 300 SARS cases as of March 30 and reported 1,425 cases up to April 28. So we take the time interval [0, T] as [3/30/2003, 4/28/2003] so that T = 29 days, Q(0) = 300, and Q(T) = 1,425, which when substituted into Equation 6 yields 12.9 days as the doubling time—the period of time required for the number of cases in the epidemic to double—during the early stage of the SARS epidemic. If Q(t) data are available for more than two time points, then the doubling time may be estimated using least-squares regression analysis.

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