Hemodynamic Energy Dissipation in the Cardiovascular System: Generalized Theoretical Analysis on Disease States

Abstract

Background We present a fundamental theoretical framework for analysis of energy dissipation in any component of the circulatory system and formulate the full energy budget for both venous and arterial circulations. New indices allowing disease-specific subject-to-subject comparisons and disease-to-disease hemodynamic evaluation (quantifying the hemodynamic severity of one vascular disease type to the other) are presented based on this formalism. Methods and Results Dimensional analysis of energy dissipation rate with respect to the human circulation shows that the rate of energy dissipation is inversely proportional to the square of the patient body surface area and directly proportional to the cube of cardiac output. This result verified the established formulae for energy loss in aortic stenosis that was solely derived through empirical clinical experience. Three new indices are introduced to evaluate more complex disease states: (1) circulation energy dissipation index (CEDI), (2) aortic valve energy dissipation index (AV-EDI), and (3) total cavopulmonary connection energy dissipation index (TCPC-EDI). CEDI is based on the full energy budget of the circulation and is the proper measure of the work performed by the ventricle relative to the net energy spent in overcoming frictional forces. It is shown to be 4.01 ± 0.16 for healthy individuals and above 7.0 for patients with severe aortic stenosis. Application of CEDI index on single-ventricle venous physiology reveals that the surgically created Fontan circulation, which is indeed palliative, progressively degrades in hemodynamic efficiency with growth (p < 0.001), with the net dissipation in a typical Fontan patient (Body surface area = 1.0 m²) being equivalent to that of an average case of severe aortic stenosis. AV-EDI is shown to be the proper index to gauge the hemodynamic severity of stenosed aortic valves as it accurately reflects energy loss. It is about 0.28 ± 0.12 for healthy human valves. Moderate aortic stenosis has an AV-EDI one order of magnitude higher while clinically severe aortic stenosis cases always had magnitudes above 3.0. TCPC-EDI represents the efficiency of the TCPC connection and is shown to be negatively correlated to the size of a typical “bottle-neck” region (pulmonary artery) in the surgical TCPC pathway (p < 0.05). Conclusions Energy dissipation in the human circulation has been analyzed theoretically to derive the proper scaling (indexing) factor. CEDI, AV-EDI, and TCPC-EDI are proper measures of the dissipative characteristics of the circulatory system, aortic valve, and the Fontan connection, respectively.

Appendix

Dimensional Analysis

Re-writing Eq. (1) in the general form:

$$ \psi \left( {\varepsilon ,Q,\rho ,\nu ,BSA,S} \right) = 0 $$

(1)

With each of the variable with the following dimensions:

$$ \begin{gathered} \left[ \varepsilon \right] = ML^{2} T^{ - 3} \hfill \\ \left[ Q \right] = M^{0} L^{3} T^{ - 1} \hfill \\ \left[ \rho \right] = ML^{ - 3} T^{0} \hfill \\ \left[ \nu \right] = M^{0} L^{2} T^{ - 1} \hfill \\ \left[ {BSA} \right] = M^{0} L^{2} T^{0} \hfill \\ \left[ S \right] = M^{0} L^{0} T^{0} \hfill \\ \end{gathered} $$

(A1)

According to the Buckingham π theorem,3 any relationship between n variables spanning d dimensions may be reduced to an equivalent relationship between k = n − d dimensionless groups π₁, π₂,…, π _k . Equation (1) has six variables spanning three dimensions (i.e. mass, [M], length, [L], and time, [T], dimensions). Therefore, it can be expressed as a relationship between 6 − 3 = 3 dimensionless variables given as:

$$ \varphi \left( {\Uppi_{1} ,\Uppi_{2} ,\Uppi_{3} } \right) = 0 $$

(A2)

Choosing Q, ρ, and BSA as our fundamental variables that span M, L, and T, we can obtain a specific form for Eq. (A2) by solving the following equations:

$$ \left[ {Q^{{a_{1} }} \rho^{{b_{1} }} BSA^{{c_{1} }} \nu } \right] = \left[ {M^{0} L^{0} T^{0} } \right] $$

(A3)

$$ \left[ {Q^{{a_{2} }} \rho^{{b_{2} }} BSA^{{c_{2} }} \varepsilon } \right] = \left[ {M^{0} L^{0} T^{0} } \right] $$

(A4)

$$ \left[ {Q^{{a_{3} }} \rho^{{b_{3} }} BSA^{{c_{3} }} S} \right] = \left[ {M^{0} L^{0} T^{0} } \right] $$

(A5)

Solving for Eq. (A3) for a ₁, b ₁, c ₁:

$$ 3a_{1} - 3b_{1} + 2c_{1} + 2 = 0 $$

(A6)

$$ - a_{1} - 1 = 0 $$

(A7)

$$ b_{1} = 0 $$

(A8)

Gives: $a_{1} = - 1,b_{1} = 0,c_{1} = \frac{1}{2}$

Solving for Eq. (A4) for a ₂, b ₂, c ₂

$$ 3a_{2} - 3b_{2} + 2c_{2} + 2 = 0 $$

(A9)

$$ - a_{2} - 3 = 0 $$

(A10)

$$ b_{2} + 1 = 0 $$

(A11)

Gives: a ₂ = −3, b ₂ = −1, c ₂ = 2

And finally solving Eq. (A5) for a ₃, b ₃, c ₃ gives a ₃ = 0, b ₃ = 0, c ₃ = 0, as S by definition is dimensionless.

Therefore the specific forms of the three dimensionless groups are::

$$ \Uppi_{1} = \frac{\nu }{{QBSA^{ - 1/2} }} $$

(A12)

$$ \Uppi_{2} = \frac{\varepsilon }{{\rho \frac{{Q^{3} }}{{BSA^{2} }}}} $$

(A13)

$$ \Uppi_{3} = S $$

(A14)

Examination of Π₁ indicates that it is a form of a special Reynolds number, $Re = \frac{{Q \times BSA^{ - 1/2} }}{\upsilon },$ where the characteristic velocity scale is $Q/BSA,$ and the characteristic length scale is $\sqrt {BSA}.$ Reynolds number is always an important dimensionless group for any fluid flow problem dictating the dependence on the fluid flow regime (i.e. laminar, to turbulence).

Examination of Π₂ indicates that the energy dissipation rate has been non-dimensionalized by a “body-level” kinetic energy scale given by $\rho \frac{{Q^{3} }}{{BSA^{2} }}$.

The shape variable, S, is by definition dimensionless and thus is directly third dimensionless group without need for non-dimensionalization.

Solving for Π₂ in Eq. (A2) and using results (A12)–(A14) finally gives:

$$ \frac{\varepsilon }{{\rho \frac{{Q^{3} }}{{BSA^{2} }}}} = f(Re,S) $$

(A3)