Question

I am an academic researcher who studies fluid mechanics of the left ventricle (the primary chamber of the heart that actually pumps blood to the rest of your body). The majority of my work focuses on analysis of intra-ventricular velocity fields as measured by phase-contrast MRI and subject-specific computational fluid dynamics models. However, I am also interested in idealised assessments of the ventricle that utilise measurements that are commonly made in a clinical setting.

Most of these more basic analysis utilise the temporal profile of ventricular volume V(t)
and its derivative V˙(t)
, where V(t)
is measured by image segmentation of MRI scans.

V˙(t)
is interesting to examine because the ventricle has two valves, and only one of them is open at a time (unless something is very wrong). Consequently, V˙(t)
is equal to the volume flowrate across either valve.

Comparing features of V˙(t)
between a few subject groups I have data for has yielded some interesting results. So, I'd like to see if there's anything I can do by comparing the second derivative of volume V¨(t)
.

However, I'm hoping to get some help building my physical intuition of the second derivative beyond it being the flowrate acceleration. I suspect that the second time derivative of volume should be related to the inertia of the ventricle walls, at least in some idealised or averaged sense because the wall motion is highly non-uniform, and that I should be able to approximate some aggregate forces acting normal to the internal surface. However, I haven't been able to convince myself of anything concrete in this regard.

So! If anyone has some suggestions for how I might use the second time derivative of volume to get at something interesting, particularly related to the net (normal) forces acting on the ventricular walls, I would be very grateful.

As a final note, the ventricle is often geometrically idealised as a half-prolate spheroid/spheroidal shell. So, provided that any geometric assumptions need to be made, I think this is a reasonable shape to use. However, to get things started, any shape that has a clear axial component, such as a circular cylinder, is likely a reasonable first approximation.

Thanks in advance,

John

Answer 1

The second derivative of volume (V¨(t)) can provide insights into the net forces acting on the ventricular walls in fluid mechanics of the left ventricle.

Step-by-step explanation:

The second derivative of volume (V¨(t)) in the context of fluid mechanics of the left ventricle can provide insights into the net (normal) forces acting on the ventricular walls.

The second derivative represents the rate of change of the volume flowrate across the valve, which can be related to the inertia of the ventricle walls.

In an idealized sense, the second time derivative of volume can be used to approximate the aggregate forces acting normal to the internal surface of the ventricle.

Answer 2

Utilizing the Second Derivative of Volume in Left Ventricle Analysis:

The second derivative of volume (V¨(t)) in left ventricle assessment offers an opportunity to explore the dynamics beyond flowrate acceleration. It likely reflects the inertia of the ventricular walls, hinting at aggregate forces acting on the internal surface. This may be vital in understanding the normal forces acting on the walls, especially in geometric idealizations like a half-prolate spheroid/spheroidal shell.

Step-by-step explanation:

1.Understanding V¨(t) Beyond Flowrate Acceleration:

The second derivative of volume, V¨(t), moves beyond examining flowrate acceleration. Instead, it seems to encapsulate the inertia of the ventricular walls. This perspective offers insights into the forces at play beyond the flow dynamics, potentially shedding light on wall motion and related forces.

2.Aggregate Forces and Wall Inertia:

The suspicion that V¨(t) might relate to the inertia of the ventricle walls suggests a connection to aggregate forces. This hypothesis hints at the possibility of approximating the net (normal) forces acting on the internal surface of the ventricle. These forces might play a crucial role in understanding the wall's behavior and interactions within the ventricle.

3.Geometric Idealization and Shape Considerations:

Considering the ventricle as a half-prolate spheroid/spheroidal shell offers a relevant geometric idealization. This shape allows for assessing the axial components effectively. While a circular cylinder might serve as a preliminary approximation due to its clear axial component, the half-prolate spheroid/spheroidal shell aligns better with the ventricle's actual geometry