Unbiased volume estimation with coaxial sections: an application to the human bladder

A problem of current medical interest is how to estimate the volume of an object from coaxial sections. Such images can be produced in ultrasound scanning, when the planar scan sections all emanate from a common axis. Two unbiased volume estimators for this sampling regime, previously published without derivation, are derived and presented here in some detail. The estimators are based on an ancient theorem of Pappus of Alexandria (c. AD 320). One of the estimators was used to estimate the volume of urine in the bladder of six human volunteers. The case example was attractive because the corresponding volumes could be voided and directly measured after the scanning session, thus confirming the unbiasedness of the estimator used. Sample size guidelines are also given.