= Pulmonary-vein backflow component
The \b[pulmonary-vein backflow component] comprises represented <blood> that leaves the <left atrium> through a <pulmonary vein>, toward the lungs. In <atrial flow component analysis>, the exit boundary distinguishes it from <blood> delivered through the <mitral valve>. The Bäck study includes both newly entering <blood> that returns to a <pulmonary vein> and previously resident atrial <blood> taking that exit.<Comprehensive left atrial flow component analysis reveals abnormal flow patterns in paroxysmal atrial fibrillation by Sophia Bäck and others>[Methods, Flow Component Analysis; Figure 1]{ref}
An original signed-flow example explains why forward and backward transport should be recorded separately. Orient a <pulmonary vein>'s boundary normal toward the <left atrium>, so positive <volumetric flow rate> $Q(t)$ denotes inflow. Define gross entering and leaving volumes over an observation interval $I$ by
$$
V_+=\int_I\max(Q(t),0)\,dt,\qquad V_-=\int_I\max(-Q(t),0)\,dt.
$$
The net transferred volume is $V_+-V_-$. Constructed values $V_+=60$ mL and $V_-=5$ mL give 55 mL net inflow. Reporting only that net value hides the 5 mL reversal. Summing over all <pulmonary veins> gives a chamber-level boundary balance.
A boundary integral and a parcel history nevertheless answer different questions. If the same represented parcel crosses outward twice after re-entering, a gross boundary integral counts both crossings. A <pathline>-based component needs an explicit rule for repeated crossings and its observation window before its volume can be equated with that integral. Similarly, this route is not <mitral regurgitation>: the latter concerns reverse transport across the <mitral valve>, a different boundary. An outlet direction alone does not establish a diagnosis or the age of the <blood> taking that route.
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