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Exploring Biomedical Signals with Differential Geometry Invariants

Investigating human biosignals using differential geometry invariants for cardiovascular analysis. Measuring pulse wave velocity and movement activity in patients, highlighting the importance of curvature invariance.

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Exploring Biomedical Signals with Differential Geometry Invariants

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  1. Analysisofbiomedicalsignalsusingdifferential geometry invariants Filip Studnička University of Hradec Králové, Czech Republic

  2. Biosignals • wemeasurehumanbiosignalsusingaccelerometricand tensometricsensorsplaced in medicalbed • weobtain n-signalsfromthosesensors • thosesignalscanbedescribedas 1D manifolds(curves) embedded in Rn • sincehumanhaemodynamicsdoes not depend on rotationortranslationofthe body, thosecurves are invariant underthosetransformations – invariant undertheactionofspecialorthogonalgroup SO(3) x Rn

  3. Arc length parametrization

  4. Frenet-Serret theory

  5. Cartan curvatures

  6. Human cardiovascular system • humancardiovascularsystemis a branchinggraphconsistingof aorta, aorta branchings, arteries, etc., on whichthe pulse waveispropagating

  7. Why curvature invariant?

  8. Velocity of the pulse wave • thevelocityofthe pulse wavewasmeasuredusingaplanation tonometry and accelerometricsensors • theoccurenceofthepropagating pulse wavemeasuredusing a.t. was done on thecarotid and on thefemoralarteryusingpressuresensors • thepeaks in curvaturecorrespond to thebend in thetravelof pulse wavethroughaortic arch and to therepellationofthe pulse wave on thebiffurcation in the abdomen • ifweassume, thatthe distance betweenaortic arch and carotidequalsthe distance betweenbiffurcation and femoralarterythenwecancomparethevelocitiesmeasuredusingabovementionedmethods • themeasurementwas done on approx. 50 people

  9. Resultofthe pulse wavemeasurement

  10. Mass invariant • for monitoring ofmovementactivityofthepatientthemass invariant wasused • therewerefourweightsensorsinstalled in themedicalbed • ifweaddallthosesignalsweobtainthemass invariant (totalmass on thebed) • in theoryifthereis no external influence, theintegralofthemass invariant oversometime isalwayszero • usingthisproperty, wecandistignuishbetweenthemovementofthepatienthimself and e.g. theturningofthepatient by a nurse

  11. Special affine curvature • sincehuman body iscontinuouslydeformed by theprocesessinsideit, wedecided to exploreotherinvariantsthaneuclideancurvature • wereplacedthespecialorthogonalgroup by thespeciallineargroup, whichisthegroupofvolumepreservingtransformations: SL(3) x Rn whichiscalledaffinespeciallineargroup • itshows, thatusingspecialaffinecurvature, thesmallvibrations and noise are suppressed and theheamodynamicsishighlighted

  12. Thank you for the attention

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