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Brachial Artery Differential Characteristic Impedance: Contributions from Changes in Young’s Modulus and Diameter

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Cardiovascular Engineering

Abstract

This examination of brachial artery (BA) differential characteristic impedance, ΔZ c, illustrates that changes in Z c can occur from changes in either BA wall stiffness (Young’s modulus, E) and/or its diameter, D. Furthermore, we assessed how changes in both E and D combine in either an isolated, synergistic, or antagonistic manner to yield the net change in BA Z c. The basis of this analysis is a partial differential equation which approximates ΔZ c as a total differential. The effects on BA ΔZ c of acetylcholine, atenolol, fenoldapine, nitroglycerin, hydrochlorothiazide and other medications are examined using data from previously published studies. Clinical situations which alter BA Z c, such as congestive heart failure, hypertension, and hyperemia, are also analyzed. Results illustrate the usefulness of the present approach in differentiating how medications, hyperemia, and pathological conditions affect BA ΔZ c by causing independent changes to E and/or D.

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Correspondence to Glen Atlas.

Appendix: Derivation of the Approximate Total Differential for Characteristic Impedance

Appendix: Derivation of the Approximate Total Differential for Characteristic Impedance

The definition of characteristic impedance, based upon pulse wave velocity, from Eq. 4 is:

$$ Z_{\text{c}} = \frac{{\rho V_{\text{pw}} }}{A}. $$
(28)

Substituting diameter (D), for area (A), where \( A = \pi D^{2} /4 \) yields:

$$ Z_{\text{c}} = \frac{{4\rho V_{\text{pw}} }}{{\pi D^{2} }}. $$
(29)

Substituting the Moens–Korteweg equation (8), for pulse wave velocity, V pw:

$$ Z_{\text{c}} = \sqrt {\frac{16\rho Eh}{{\pi^{2} D^{5} }}} . $$
(30)

Squaring both sides of (30) yields:

$$ Z_{\text{c}}^{2} = \frac{16\rho Eh}{{\pi^{2} D^{5} }}. $$
(31)

Using implicit differentiation:

$$ 2Z_{\text{c}} {\text{d}}Z_{\text{c}} = \frac{16\rho h}{{\pi^{2} D^{5} }}{\text{d}}E - \frac{80\rho Eh}{{\pi^{2} D^{6} }}{\text{d}}D . $$
(32)

The total differential is then approximated as:

$$ \widetilde{\Updelta Z}_{\text{c}} = \frac{{\partial Z_{\text{c}} }}{\partial E}\Updelta E - \frac{{\partial Z_{\text{c}} }}{\partial D}\Updelta D. $$
(33)

Where:

$$ \frac{{\partial Z_{\text{c}} }}{\partial E} \approx \frac{8\rho h}{{\bar{Z}_{\text{c}} \pi^{2} \bar{D}^{5} }} $$
(34)

and

$$ \frac{{\partial Z_{\text{c}} }}{\partial D} \approx \frac{{40\rho \bar{E}h}}{{\bar{Z}_{\text{c}} \pi^{2} \bar{D}^{6} }}. $$
(35)

Note that the following terms are also used:

$$ \bar{E} = (E_{1} + E_{2} )/2\,{\text{and}}\,\bar{D} = (D_{1} +D_{2} )/2\,{\text{and}}\,\bar{Z}_{\text{c}} = (Z_{{{\text{c}}_1}}+ Z_{{{\text{c}}_2}} )/2. $$
(36)

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Atlas, G., Li, J.KJ. Brachial Artery Differential Characteristic Impedance: Contributions from Changes in Young’s Modulus and Diameter. Cardiovasc Eng 9, 11–17 (2009). https://doi.org/10.1007/s10558-009-9071-6

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  • DOI: https://doi.org/10.1007/s10558-009-9071-6

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