A theoretical approach to choosing the minimum number of multiple tumors required for assessing treatment response

Most advanced cancer patients have multiple tumors. Because the multiple tumors are from the same patient, the tumor sizes are expected to be correlated and the information contained in each additional tumor might not always have significant 'added value' toward the response assessment. Needing to measure only a subset of tumors would reduce workload for the study radiologist but is expected to increase the variability in response outcome. We compute this increment in variability and find a procedure for choosing the minimum number (m) of tumors among some fixed maximum number (M) of correlated tumors that must be considered to ensure precision of at least as high as a specified proportion of the precision obtained if one were to measure all M tumors.

The ratio V(m)(R)/V(M)(R) = M2[m + (m2 - m)rho(ICC)]/m2[M + (M2 - M)rho(ICC)] quantifies the percentage increment in variance of the response R, where rho(ICC) is the intra-class between tumors within patient correlation coefficient. The procedure for choosing the minimum number of tumors is demonstrated using data for 42 cancer patients with 10 or more tumors.

Using the criterion that >20% increase in variability due to selection of a subset out of M of 10 tumors is unacceptable, we find that m of 9, 6, 5, 3, and 2 tumors are needed when rho(ICC)=0.0 (no correlation), 0.2, 0.4, 0.6, and 0.8, respectively. If the criterion is made stricter to >10%, the number of tumors needed rise to 10, 8, 6, 4, and 3, respectively. For the example, 6 tumors out of 10 are found to provide sufficiently stable response categorization confirming the theoretical result.

If cancer research community can agree on a percentage of variability in response outcome that is unacceptable, it is mathematically possible to recommend a minimum number of tumors that should be used for response assessment.