  The study of moving particles (e.g. molecules, virus, vesicles, organelles,
or whole cells) is crucial to decipher a plethora of cellular mechanisms within
physiological and pathological conditions. Powerful live-imaging approaches
enable life scientists to capture particle movements at different scale from
cells to single molecules, that are collected in a series of frames. However,
although these events can be captured, an accurate quantitative analysis of
live-imaging experiments still remains a challenge. Two main approaches are
currently used to study particle kinematics: kymographs, which are graphical
representation of spatial motion over time, and single particle tracking (SPT)
followed by linear linking. Both kymograph and SPT apply a space-time
approximation in quantifying particle kinematics, considering the velocity
constant either over several frames or between consecutive frames,
respectively. Thus, both approaches intrinsically limit the analysis of complex
motions with rapid changes in velocity. Therefore, we design, implement and
validate a novel reconstruction algorithm aiming at supporting tracking
particle trafficking analysis with mathematical foundations. Our method is
based on polynomial reconstruction of 4D (3D+time) particle trajectories,
enabling to assess particle instantaneous velocity and acceleration, at any
time, over the entire trajectory. Here, the new algorithm is compared to
state-of-the-art SPT followed by linear linking, demonstrating an increased
accuracy in quantifying particle kinematics. Our approach is directly derived
from the governing equations of motion, thus it arises from physical principles
and, as such, it is a versatile and reliable numerical method for accurate
particle kinematics analysis which can be applied to any live-imaging
experiment where the space-time coordinates can be retrieved.
