## Physical Principles of Cardiovascular Function

These web pages are being developed for students of cardiology and others who may be interested in physical principles that govern cardiovascular function.  Mathematical models are developed of varying complexity in conjunction with computerized image and animation production to allow users to visualize the results.  Software snippets referred to as Active Figures can be downloaded, allowing users to explore physical principles interactively.  While focusing on topics most germane to cardiovascular function, these principles may be of value in other branches of medicine or bioengineering.

Topics are presented for educational purposes only.  Not for medical or research applications.

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### Fluid Dynamics

The video shows a distorting fluid surface, determined using computational fluid dynamics, as it flows around a 90 degree bend in a conduit with a square cross section.  DOWNLOAD a more detailed (larger) version (Microsoft Video 1 Codec) If the video isn't visible to your browser, check it out on YouTube.

Below is the view from "behind", watching the fluid surface travel down the tube.  DOWNLOAD it.

The "videos" were created using Computation Fluid Dynamics software that's available to download at this website.  Make your own and learn about fluid dynamics in the process.

### SeeMOREabout the software and what it does.

Below – Fluid flow in a rectangular bifurcation.  While very unlike arterial bifurcations in the circulation, the model affords insights into the complexities of the flow.  The bifurcation below would be a very poor design for the circulation.  We'll find out why.

In this bifurcation, flow enters from the lower left and splits into the 2 branches with 75% of the flow going into the 90° side branch (Reynolds number 1000, k-epsilon turbulence model). The video starts by labeling a surface of fluid that has already entered the straight branch.  This shows how fluid that has entered into the straight branch actually ends up flowing down the side branch (towards the top of the figure) due to the flow complexity.

Here's a high-def example:

Because the fluid is close to the "suction zone" (a force due to a pressure gradient acts on the fluid), it flows upstream along the sides of the conduit and flows out the side branch.  Yep, that happens.

In the above videos, the model tracks a fluid surface as it deforms during the flow.  The surface remains intact throughout (and beyond), like a sheet of material that has been stretched, twisted, and folded.  The fluid that exits the side branch remains connected to the fluid that exited the straight branch.

### Left Ventricular Strain

Prolate ellipsoidal model of left ventricle showing fiber orientations (Active figure will allow you to highlight individual fiber shells). Watch how the deformation occurs!

The next 2 videos show animations suggestive of a deforming left ventricle. I hasten to clarify that these are not models per se; there is no linkage between physical events (pressures, forces) and the deformations depicted. This active figure will allow you to explore various aspects of motion and deformation - strain. Concepts to be explored include 4 aspects of motion (translation, rigid body rotation, extension, shear), continuity of solids (constant volume for incompressible materials), Eulerian versus Lagrangian frame of reference, principle strain, fiber strain, etc.

The videos suggest a deforming left ventricle. The active figure allows specification of fractional shortening, ejection fraction, and shape factors for the LV (e.g. simulating normals and various pathologic conditions). Part way through the video you'll see fiber orientations depicted in a sequence of concentric shells within the structure (as suggested by Streeter et al). Once the deformation and fiber orientation are known, fiber shortening (strain) can also be determined. Fibers are imbedded into and are part of the material. Once you see how they're arranged you can start to imagine why the LV deforms as it does. (Defomations in these videos are not tailored specifically to represent the observed and the software allows specifications wildly unlike reality. That's how I learn.)

The second video helps conceptualize an important aspect of how strain occurs in an incompressible material; every single piece of matter retains it's original volume.

Look closely and you'll observe that this requires material near the "endocardial" surfaces to deform more than those at the "epicardium". This fact becomes more apparent with larger deformations and with thicker walls (greater disparity in deformation between endocardium and epicardium). While the animation is not a model, this conservation principle is enforced to precision numerical accuracy in the depictions. The software allows you to run the sim as slow as you like or stop it so you can see differences between epi and endo. Also change the LV geometry, fractional shortening, twist, shear, etc.

### Fluid Dynamics Visualization

CFD 3D ViewB - An Active Figure for viewing results of Computational Fluid Dynamic solutions.  Now with PERSPECTIVE! See example figures below.  Here are some velocity vectors from outside the box.

Here are some velocity contours from INSIDE the box.

Velocity vectors (blue) and fluid displacement (grid) during pulsatile flow in a circular tube.

### Linear Hemodynamics Primer

Learn about pressure and flow wave transmission in the circulation, "pulse transformation" and why the pressure isn't the same everywhere in the arteries, wave reflection, why you can't tell how fast the pulse travels from one place to another (wave velocity), input impedance spectra and the factors that affect it, and a lot more.  An integrated "Active Figure" is nearly available to explore this complicated subject.

Above, pressure and flow waves traveling in a compliant tube (wave consists of 2 harmonics, with a second sinusoid at twice the fundamental frequency) with a strong reflection site at the right side of the figure (terminus) caused by clamping the end. The latter forces the flow to 0 at that location (node) but causes a doubling of the pressure (anti-node) and reflected pressure and flow waves resulting from a discontinuity in the impedance. Measured pressure and flow in the tube (and the circulation) are due to appropriate summation of antegrade and retrograde traveling waves.

Here's what the measured pressure and flow would look like at the input (left side of video figure above) plus the antegrade and retrograde pressure and flow waves that you can't measure.  Measured pressure, flow, and input impedance are entirely dependent on the location of measurement.  The Active Figure let's you choose multiple locations to compare waveforms and spectra.

### Impedance

And here's the input impedance spectrum (modulus and phase) determined at the far left end (input) of the simple tube.   Input impedance (blue) oscillates above and below the characteristic impedance (red) due to reflection of pressure and flow waves.  See how it happens!

### Why do Fluids Behave as They Do?

What does flow over an airfoil have to do with cardiology?  More than you might think....

### Computational Fluid Dynamics Solutions

Flow through a stenosis (side view above, 3D perspective below) showing velocity vectors; color also depicts velocity (warmer colors, higher velocity)

Velocity vectors near the entrance region of a rectangular conduit. Color depicts the velocity also showing velocity increasing near the center of the conduit as flow proceeds left to right.

Velocity vectors in a rectangular conduit with a sharp bend. Flow enters from the lower right.  Pressure is also represented by color and contour lines.

Three-dimensional flow in an asymmetrically lid-driven cube (a common test problem for computational fluid dynamics) showing velocity vectors and pressure as both color and contour lines. Difficult to comprehend? Yes it is!! Flow visualization software (Active Figure) will allow you to orient the image three dimensionally for a better understanding.