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A mathematical model of pressure and flow waveforms in the aortic root

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Abstract

The differences in the pressure and flow waveforms in the aortic root have not been explained so far in a satisfactory mathematical way. It is a generally accepted idea that the existence of the reflected wave causes the differences in shapes of pressure and flow. In this paper, a mathematical model is proposed that explains the blood pressure and flow waveforms based on changes in left ventricular volume during blood ejection into the aorta. According to the model, a change in volume of the left ventricle during contraction can be mathematically presented with solutions of differential equations that describe the behavior of a second-order system. The proposed mathematical equations of pressure and flow waveforms are derived from left ventricular volume change and basic equations of fluid dynamics. The position of the reflected wave depends on the age and elasticity of arteries, and has an effect on the flow and pressure waveforms. The model is in acceptable agreement with the experimental data available.

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Correspondence to Dejan Žikić.

Appendix

Appendix

From the equation for energy conservation:

$$\frac{1}{2}m{\text{d}}v^{2} = \frac{\Delta p \cdot \Delta V}{2} + \frac{{\sigma^{2} }}{2E} \cdot V$$
(1)

(The change in kinetic energy: \(E_{\text{k}} = \frac{1}{2}m{\text{d}}v^{2}\); Strain energy of the fluid (stored by a volume and pressure change):\(\frac{\Delta p \cdot \Delta V}{2}\); Strain energy of the elastic arterial wall: \(\frac{{\sigma^{2} }}{2E} \cdot V\); Stress for thin arterial wall (d 0 diameter, t thickness) \(\sigma = \frac{{\Delta pd_{0} }}{2t}\))

Volume of the arterial wall:

$$V = \pi \cdot d_{0} \cdot L \cdot t$$

Pulse wave velocity:\(c^{2} = \frac{A}{\rho }\frac{{{\text{d}}A}}{{{\text{d}}P}}\)

After substitution in Eq. (1)

$$\frac{1}{2}\rho {\text{ALd}}v^{2} = \frac{{\Delta p\Delta {\text{AL}}}}{2} + \left( {\frac{{\Delta pd_{0} }}{2t}} \right)^{2} \cdot \frac{{\pi d_{0} tL}}{2E}$$
(2)
$$\frac{1}{2}\rho {\text{ALd}}v^{2} = \frac{{\Delta p\Delta {\text{AL}}}}{2} \cdot \frac{\Delta pA}{\Delta pA} + \frac{{\left( {\Delta p} \right)^{2} d_{0}^{2} }}{{4t^{2} }} \cdot \frac{{\pi d_{0} tL}}{2E}$$
(3)

Equation (3) maybe re-written as follows (with substitutions \(\Delta p = {\text{d}}p\), \(\Delta A = {\text{d}}A\) and \(A = \frac{{\pi d_{0}^{2} }}{4}\))

$$\rho {\text{d}}v^{2} = \frac{{{\text{d}}p^{2} {\text{d}}A}}{{{\text{d}}pA}} + \frac{{{\text{d}}p^{2} }}{t} \cdot \frac{{d_{0} }}{E}$$
(4)

From Hooke’s law σ =  and after substitutions

$$\frac{{pr_{0} }}{t} = E\frac{\Delta r}{{r_{0} }} = > \frac{\Delta r}{{r_{0} }} = \frac{{pr_{0} }}{Et}$$

The change of cross-sectional area

$$A = A_{0} + {\text{d}}A = \pi \left( {r_{0} + \Delta r} \right)^{2} = \pi r_{0}^{2} \left( {1 + \frac{\Delta r}{{r_{0} }}} \right)^{2} = \pi r_{0}^{2} \left( {1 + \frac{{pr_{0} }}{Et}} \right)^{2}$$
(5)

After differentiating Eq. (5) with respect to p =>

$$\frac{{{\text{d}}A}}{{{\text{d}}p}}\sim \frac{{2\pi r_{0}^{3} }}{Et}\sim \frac{{Ad_{0} }}{Et}$$
(6)

Substituting Eq. (6) in Eq. (4) and after some simplification

$$\rho {\text{d}}v^{2} = \frac{{{\text{d}}p^{2} {\text{d}}A}}{{{\text{d}}pA}} + \frac{{{\text{d}}p^{2} }}{t} \cdot \frac{{d_{0} }}{E} = {\text{d}}p^{2} \left( {\frac{{{\text{d}}A}}{{{\text{d}}pA}} + \frac{{d_{0} }}{tE}} \right) = {\text{d}}p^{2} \left( {\frac{{{\text{d}}A}}{{{\text{d}}pA}} + \frac{{{\text{d}}A}}{{{\text{d}}pA}}} \right) = {\text{d}}p^{2} \frac{{2{\text{d}}A}}{{{\text{d}}pA}}$$
(7)

After rearranging Eq. (7)

$${\text{d}}p^{2} = \frac{1}{2}\frac{A}{\rho }\frac{{{\text{d}}p}}{{{\text{d}}A}}\rho^{2} {\text{d}}v^{2}$$
(8)

and finding the square root

$${\text{d}}p = \pm \sqrt {\frac{1}{2}} \cdot c \cdot \rho \cdot {\text{d}}v = \pm \sqrt {\frac{1}{2}} \cdot c \cdot \rho \cdot \frac{{{\text{d}}Q}}{A}$$
(9)

For arteries

$$E^{\prime} = \frac{E}{{1 - \nu^{2} }},\nu = \frac{1}{2}.$$
(10)

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Žikić, D. A mathematical model of pressure and flow waveforms in the aortic root. Eur Biophys J 46, 41–48 (2017). https://doi.org/10.1007/s00249-016-1133-2

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  • DOI: https://doi.org/10.1007/s00249-016-1133-2

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