A mathematical analysis on the biological zero problem in laser Doppler flowmetry

The biological zero (BZ) problem is a critical issue inherent in laser Doppler flowmetry (LDF). It causes confusion when measuring low tissue blood flows. Many experimental studies have been done on the question of whether the BZ flux should be subtracted from the normally measured flux in various situations. However this problem can only be solved after a proper mathematical analysis. Only then can one clearly define and formulate what flux is truly meaningful in blood perfusion measurement and what movement generates the BZ flux and how can one correctly remove it. Following this motivation, the movement of moving blood cells (MBCs) is decomposed into a net translation and a random wandering based on in vivo observations. This important step leads to a clear definition of the BZ and net perfusion flux and reveals that subtraction of BZ flux from the normal flux will certainly cause an underestimation of the net flux. Using this decomposition, the relationship between the net, BZ and normal flux is established which leads to the correct formula to recover the net flux from the BZ and normal fluxes. This recovered net flux is shown to be bounded by the normal flux and the normal flux minus the BZ flux. Numerical studies, preliminary phantom model and clinical evaluations manifest that the new approach is more accurate and reasonable at measuring low net fluxes. In contrast, subtracting BZ flux causes a systematic underestimation of perfusion and is apparently inappropriate even from a methodological point of view. In addition to the novel BZ solution, a general density function of the speed of MBC's is given which is more faithful than the Maxwell density used in R. Bonner and R. Nossal's paper (Appl. Opt., vol. 20, 1981). This general density function offers new possibilities for further theoretical developments in LDF.