Analysis of biomedical signals using differential geometry invariants

Filip StudničkaUniversity of Hradec Králové, Czech Republic

Biosignals• we measure human biosignals using

accelerometric and tensometric sensors placed in medical bed

• we obtain n-signals from those sensors• those signals can be described as 1D

manifolds (curves) embedded in Rn

• since human haemodynamics does not depend on rotation or translation of the body, those curves are invariant under those transformations – invariant under the action of special orthogonal group SO(3) x Rn

Arc length parametrization

Frenet-Serret theory

Cartan curvatures

Human cardiovascular system

•human cardiovascular system is a branching graph consisting of aorta, aorta branchings, arteries, etc., on which the pulse wave is propagating

Why curvature invariant?

Velocity of the pulse wave• the velocity of the pulse wave was measured using

aplanation tonometry and accelerometric sensors▫ the occurence of the propagating pulse wave

measured using a.t. was done on the carotid and on the femoral artery using pressure sensors

• the peaks in curvature correspond to the bend in the travel of pulse wave through aortic arch and to the repellation of the pulse wave on the biffurcation in the abdomen

• if we assume, that the distance between aortic arch and carotid equals the distance between biffurcation and femoral artery then we can compare the velocities measured using above mentioned methods

• the measurement was done on approx. 50 people

Result of the pulse wave measurement

Mass invariant• for monitoring of movement activity of the

patient the mass invariant was used• there were four weight sensors installed in

the medical bed• if we add all those signals we obtain the

mass invariant (total mass on the bed)• in theory if there is no external influence, the

integral of the mass invariant over some time is always zero

• using this property, we can distignuish between the movement of the patient himself and e.g. the turning of the patient by a nurse

Special affine curvature•since human body is continuously deformed by

the procesess inside it, we decided to explore other invariants than euclidean curvature

•we replaced the special orthogonal group by the special linear group, which is the group of volume preserving transformations: SL(3) x Rn

which is called affine special linear group•it shows, that using special affine curvature,

the small vibrations and noise are suppressed and the heamodynamics is highlighted

Thank you for the attention