Blog Post

Why I like π.

…or why I dislike τ.

…or I should have called it πEk.

This is a site for a physiology lab, and there is physiology later in this post! But, my undergraduate degree was a dual major in physics and mathematics.  So, as an established nerd, I have a long history of using π.  …or should I say 2π cause its #^@&’n everywhere in physics. 

Now, there’s a trend in physics and math(s) that is trying to use τ as a short hand in lieu of 2π.  Here’s my short and sweet reason why I don’t plan to use τ instead of 2π.   

π looks like two τs together. 

Seriously… π ~ ττ = 2τ

So π should be 2τ, not the other way around.

I could maybe be okay with calling 2π pound or hash, but # is more like two πs on top of each other, i.e. π / π =1. So maybe not.

Or what about kappa? ϰ. In some font renderings, I could imagine the 2 being jammed in on the π.

Does anyone have any better ideas for a good symbol here? 

Okay, let’s get back to physiology.  ♥

Those who have studied cardiac physiology know that τ is used as a relaxation time constant (Weiss 1976).  Of course, its easy to see why, just look at a pressure trace.  The isovolumic relaxation looks roughly like an exponential decay! 

(Fig 1, Chung 2015)

This plot is in the time domain (e.g. as a function of time), if you take the derivative of it, the same part still looks like an exponential decay: 

(From charles’s disseration) Top: Pressure versus time; middle: dP/dt versus time; bottom: Pressure Phase Plane (PPP) that plots dP/dt against pressure. For the top and middle ones, the shape of the curve after dP/dtmin is roughly exponential.

In the last panel, this is plotted in the phase plane (dP/dt vs P(t)). From dP/dtmin to minimum pressure, you can see that it looks roughly linear. This can be explained by the math of a phase plane.  As Raff and Glanz (1981 )and others put it mathematically: 

Now you see how τ, well, 1/τ, plots a straight line, i.e. the slope (the m of y=mx+b) is -1/τ.

Well, low and behold, others found that relaxation wasn’t always linear!  Originally characterized in in canine hearts, a new equation was suggested, one called the Logistic time (τL) constant (Matsubara 1995). 

While τ can only give you a straight line in the PPP, the τL can only give you a curve.  But τ does a poor job fitting when the PPP is curved and τL does a poor job fitting when the PPP is linear. Not only that, but there might be something in between the straight and logistic curvatures!

(Fig 1, Chung 2008). Here the solid straight line (Panels A,C) is a fit of τ, the solid curved line is a fit for τL (Panels B,C). Note that neither fit well on the right

I found this very frustrating (not the least because there is no mechanism). So is there something better?

Well, maybe a wandering mind will help! 

During my graduate work, I first started studying diastolic filling using a damped harmonic oscillator model.  You can read about why it works a bit more in a review (Chung 2015), but essentially, the model characterized transmitral flow velocity as a spring that was damped. The model has a stiffness (k) and a damping term (c).

(Fig 3 Chung 2015)

But we had a lot of high fidelity (Millar) in vivo pressure catheterization data from humans and I was wondering why we didn’t use the model during isovolumic relaxation, instead of just filling! 

So I tried it. I set up an model equation that looks like:

Then fit it to multiple data points.

(Fig 2, Chung 2008) Here the solid lines are the model fit.

Well, low and behold it worked (Chung 2008)!  And interestingly, it did a bit better than both τ and a τL for a few reasons: 

Second, the model can characterize data prior to peak negative dP/dt, sometimes as early as AVC. 

It does better than either τ or τL because the model has parameters that address both τ and the τL. 

(A slide from charles’s thesis defense)
(Yes, this is a screen grab of a PowerPoint slide from 2007.)
Note that Equations 1 (monoexponential τ) and 2 (logistic τL) have been re-written into a differential form, without offsets.

Anyways, as the equation for the pressure relaxation model is again a damped harmonic oscillator, doing some approximation (physicist!), when d2P/dt2 is small, τ ~ 1/μEk. This is why it fits the linear one well. (A similar but more complicated argument can be made about the logistic equation.)

We interpreted this the same way we did for filling, in other words, we stated that the unit 1/μ that was viscous and Ek was elastic.  (We’ll come back to that…) 

So that’s it. I don’t think τ is a great parameter (even if I use it still) because I hypothesize that it is a simplification of two physiologic parameters (elasticity and viscosity).

….but today is π day, which made me thin that I should have used π instead of μ! 

But what’s the physiologic relavence?

A spoiler about some (hopefully) future research:  The interpretation in this study was that 1/μ would characterize viscous relaxation (crossbridges and more), and Ek would characterize elastic relaxation (titin, ECM, etc).   

….but I don’t believe this anymore, I think it is opposite!   

Many of you might have seen me talking about Mechanical Control of Relaxation (more on this site or check ref’s like Chung 2017).  If a muscle is stretched just prior to or during relaxation the relaxation rates are accelerated.  This is the core of what Brutseart and Sys (1984) claimed was afterload control of relaxation (nee: Relaxation Loading), but the stretch is necessary and sufficient (independent of aferload) to modify relaxation. 

My current hypothesis is that during relaxation, myosin detachment is the typical rate limiting step.  In ex vivo trabeculae, the relaxation of an isometric muscle is actually curved, but dominated by the decay (viscous resistance) of myosin.  If a muscle is stretched, Mechanical Control of Relaxation can commence and the fully elastic component can drive the relaxation- causing a more linear relaxation rate! 

So now I think the μ term has to do with other recoil components and the Ek has to do with crossbridge deactivation.  Yes, this interpretation is different that the model during filling (where k is elastic and c is viscous).  I think that the model of the phases differ because of the difference in muscle length changes (strain). 

If you made it this far and especially if you’re a trainee, here’s a final message: I could have figured this out years ago! 

Before creating the μEk model, I co-authored a paper showing how a Huxley model of crossbridge kinetics can replicate an isovolumic pressure decay (Zhang 2006).  Well, it showed that crossbridge models do really well with curved PPP traces and we didn’t know why! 

It didn’t sink in to me that the curved μEk model should match this until I started working on Mechanical Control of Relaxation about 10 years after I published the relaxation work.  Sometimes its good to walk away and come back to a project to really understand what it means! 


Weiss JL, Frederiksen JW, Weisfeldt ML, (1976) Hemodynamic determinants of the time-course of fall in canine left ventricular pressure. J Clin Invest 58:751-60. 

Raff GL, Glantz SA, (1981) Volume loading slows left ventricular isovolumic relaxation rate. Evidence of load-dependent relaxation in the intact dog heart. Circ Res 48:813-24. 

Matsubara H, Takaki M, Yasuhara S, Araki J, Suga H, (1995) Logistic time constant of isovolumic relaxation pressure-time curve in the canine left ventricle. Better alternative to exponential time constant. Circulation 92:2318-26. 

Chung CS, Shmuylovich L, Kovacs SJ, (2015) What global diastolic function is, what it is not, and how to measure it. American journal of physiology. Heart and circulatory physiology 309:H1392-406. 

Chung CS, Kovacs SJ, (2008) Physical determinants of left ventricular isovolumic pressure decline: model prediction with in vivo validation. American journal of physiology. Heart and circulatory physiology 294:H1589-96. 

Chung CS, Hoopes CW, Campbell KS, (2017) Myocardial relaxation is accelerated by fast stretch, not reduced afterload. J Mol Cell Cardiol 103:65-73. 

Brutsaert DL, Rademakers FE, Sys SU, (1984) Triple control of relaxation: implications in cardiac disease. Circulation 69:190-6. 

Zhang W, Chung CS, Kovacs SJ, (2006) Derivation and left ventricular pressure phase plane based validation of a time dependent isometric crossbridge attachment model. Cardiovasc Eng 6:132-44. 

Related Posts