RTD and FLOWs at steady and pulsatile flow in systemic circulation. Motivation Dispersion of matter in systemic circulation. Stimulus response method for system identification. Flow distribution and flowrate evaluated from transit time distributions.
Dispersion of matter in systemic circulation.
Stimulus response method for system identification.
Flow distribution and flowrate evaluated from transit time distributions.
What is the effect of flow pulsation upon flowrate distribution in a branched system
Task prepared within the project FRVS 90/2010
Fast particle (short residence time)
Residence time = time spend by a particle inside a continuous system between inlet and outlet.
RTD Residence time distribution E()= histogram of residence times of all particles passing through outlet.
Applications of E(): Prediction of chemical reactions yield, transport of farmaceuticals, evaluation of active and stagnant volumes, diagnostics,… There are generally different RTDs for plasma and RTDs of blood particles. Different RTD exists at steady and pulsatile flow.
All particles passing through inlet at time interval (tI,tI+dt)are marked “red”
Stimulus response experiment
Experiments: “Red particles” – tracers (dyes, radionuclides, salts,…) are quickly injected to inlet. Detectors monitor concentration of tracer at outlet (impulse response). In our case the tracer is KCl solution and reflective particles injected from syringe, detectors are electric conductivity probes.
Convolution enables to calculate response to an arbitrary stimulus function. It is possible to calculate impulse response E(t) by deconvolution from measured stimulus and response.
cin(t), cout(t) concentration of tracer at inlet and outlet of a continuous system
Convolution of stimulus and RTD function
Fourier transform of convolution
Mean time of response = mean time of stimulus + mean residence time
Variance of response = variance of stimulus + variance of E
Laminar flow and short residence times: diffusion can be neglected and residence times can be calculated from velocity field. The diffusion free regime is characterized by
where Q-flow rate, L-pipe length, Dm-diffusion coefficient.
Example: C~1000 at aorta, arteries, vena cava.
Residence time distribution E() is the response to infinitely short impulse (delta function)
Analytical solution exists also for response to a pulse of a finite width (violet line). Time of the first appearance is not affected, only mean response time is increased.
First appearance time =4s
Simulated part - pressure or conductivity transducers alternatively
Data file defines the coordinates x,y,z of all 29 nodes
Conductivity probes. Q-flowrate, C-response (defined in form of table)
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Data files define connectivity of
PIPES (2-node elements)
DIVISION (3-node elements)
x2,y2,z2 Q2 ,c2(t)
x1,y1,z1 ,Q1, c1(t)
Convolution realized by FFT, and by inverse transformation.
It is alternatively possible to characterize the whole wye as a perfectly mixed vessel. In this case the both responses c3 and c2 will be identical
1. Derive impulse response of perfectly mixed vessel Emixed(t)
2. How to modify the responses in the case that detectors record not the tracer concentration but the flowrate of tracer (e.g. radiotracer – counter records rate of decays in the whole outlet).
MATLAB M-files available at http:
MATLAB M-files available at http:
Input data files:
xyzq.txt x y z q (nodal coordinates and flowrates)
seq.txt e1 e2 … (sequence of evaluated elements, +pipes, -wyes)
cp.txt i j d (pipe -indices of nodes and diameter)
cv.txt i j k l d1 d2 d3(wye - indices of nodes and diameters)
1. Modify input files according to parameters of your experiment
2. Compare calculated and measured concentration responses
3. Evaluate flowrates in branches from the shortest/mean transit times
Ultrasound Doppler effect for measurement velocity profiles