There's a sign error in here somewhere. Either the slopes need opposite signs as one goes up and one goes down, or the denominator of VR/CO needs to be Rv-Rh (with Rh being negative as a reflection of the fact that the heart is an energy input rather than a dissipator). As it is, I think you switched sign conventions between the triangle image and the written formula.

Thank you for this observation. Initially, I had the very same concern and I had to convince myself! I think the math shakes out though as the opposing slopes are caked into the equations. If we think of the flow equations as linear: y = mx + b.

--Then the VR equation becomes VR = -(1/Rvr)Pra + VRmax, i.e., with Pra as x, the slope, m, is negative (i.e., -1/Rvr). To get Pmsf into this, we have to say that Pmsf = VRmax x 1/Rvr (to get the x-intercept) and VR becomes Pmsf/Rvr – Pra/Rvr, or (Pmsf – Pra)/Rvr.

--For the CO equation, mx = (1/Rh)Pra and the y-intercept, b, is (-Ppc/Rh). In this sense, if Ppc is positive, then the y-intercept is negative and if the Ppc is negative, the y-intercept is positive. So, the equation becomes CO = (1/Rh)Pra – Ppc(1/Rh) or, (Pra – Ppc)/Rh. Therefore, when we compare the slopes (i.e., m) between these equations, in the former it is negative (i.e., -1/Rvr) and in the latter, it is +ve (i.e., +1/Rh).

I hope that makes sense.

Models always oversimplify things and they are only models. Ontologically, Rcardiac (or Rh as you call it, i think tipping your hat to Professor Parkin) means that the heart can only "obstruct" flow. That is, the best it can do is get to an Rh of zero (i.e., completely get out of the way). When Rh is zero, then Ppc and Pra are equal and the equation reduces to (Pmsf - Pra)/Rvr. I appreciate Professor Brengelmann's criticism of this ontology. Caked into Pmsf is energy supplied by the heart, but i believe that the peripheral vasculature also imparts energy on the blood (by changing its capacitance, and there is good evidence that there are active changes in flow caused by the peripheral vasculature). In any case, Oscar Wilde said it best:

There's a sign error in here somewhere. Either the slopes need opposite signs as one goes up and one goes down, or the denominator of VR/CO needs to be Rv-Rh (with Rh being negative as a reflection of the fact that the heart is an energy input rather than a dissipator). As it is, I think you switched sign conventions between the triangle image and the written formula.

Hi Maarten:

Thank you for this observation. Initially, I had the very same concern and I had to convince myself! I think the math shakes out though as the opposing slopes are caked into the equations. If we think of the flow equations as linear: y = mx + b.

--Then the VR equation becomes VR = -(1/Rvr)Pra + VRmax, i.e., with Pra as x, the slope, m, is negative (i.e., -1/Rvr). To get Pmsf into this, we have to say that Pmsf = VRmax x 1/Rvr (to get the x-intercept) and VR becomes Pmsf/Rvr – Pra/Rvr, or (Pmsf – Pra)/Rvr.

--For the CO equation, mx = (1/Rh)Pra and the y-intercept, b, is (-Ppc/Rh). In this sense, if Ppc is positive, then the y-intercept is negative and if the Ppc is negative, the y-intercept is positive. So, the equation becomes CO = (1/Rh)Pra – Ppc(1/Rh) or, (Pra – Ppc)/Rh. Therefore, when we compare the slopes (i.e., m) between these equations, in the former it is negative (i.e., -1/Rvr) and in the latter, it is +ve (i.e., +1/Rh).

I hope that makes sense.

Models always oversimplify things and they are only models. Ontologically, Rcardiac (or Rh as you call it, i think tipping your hat to Professor Parkin) means that the heart can only "obstruct" flow. That is, the best it can do is get to an Rh of zero (i.e., completely get out of the way). When Rh is zero, then Ppc and Pra are equal and the equation reduces to (Pmsf - Pra)/Rvr. I appreciate Professor Brengelmann's criticism of this ontology. Caked into Pmsf is energy supplied by the heart, but i believe that the peripheral vasculature also imparts energy on the blood (by changing its capacitance, and there is good evidence that there are active changes in flow caused by the peripheral vasculature). In any case, Oscar Wilde said it best:

"The truth is rarely pure and never simple"

Literally just became a paid subscriber to get access to these old posts I recall from the prior version of this site. Great timing!

So glad you did- thanks for the support! -Matt