Introduction Coronary arterial inflow varies in time during the cardiac cycle. Systolic inflow is smaller than diastolic inflow, demonstrating that the pulsatility of coronary flow is not caused by variation of the arterial venous pressure difference. Instead, the phasic pattern of coronary inflow has been attributed to changes in cross-sectional area of the myocardial vessels. For example, a transient decrease in cross-sectional area affects coronary flow in two ways. First, coronary inflow decreases and outflow increases because blood is squeezed out of the coronary bed. Second, both coronary inflow and outflow decrease because of the increase of coronary resistance. 11 16 15 17 15 20 18 18 9 1 2 6 22 12 et al 5 29 30 et al 26 27 15 17 et al With the increasing complexity of the models, more experimental observations have been replicated. Simultaneously, the number of model parameters has increased, making it difficult to perform a detailed parameter sensitivity analysis and identify the critical model parameters. Therefore, our aim was to study the primary relations between left ventricular pressure and volume, wall stress in fiber and transverse direction, intramyocardial pressure and the coronary blood flow, with a mathematical model with a limited number of parameters, and to assess the sensitivity of the model results to the model parameter settings. Material and methods Figure 1.  λ σ m,f σ m,r 19  Ventricular Wall Mechanics 3 1 \documentclass[12pt]{minimal}
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\begin{document}$$ \frac{\partial\sigma_{\rm r}}{\partial{ r}}+\frac{2\sigma_{\rm r}}{r} -\frac{1}{r}(\sigma_{\rm l}+\sigma_{\rm c}) = 0 $$\end{document}  σ r  σ c  σ l r \documentclass[12pt]{minimal}
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\begin{document}$$ \varvec{\bf \sigma} = -p_{\rm im}\varvec{I}+\varvec{\bf \sigma}_{\rm m}+\sigma_{\rm a}\vec{e}_{\rm f}\vec{e}_{\rm f} $$\end{document} p im \documentclass[12pt]{minimal}
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\begin{document}$$ \begin{aligned} \sigma_{\rm l}&=- p_{\rm im}+(\sigma_{\rm m,f}+\sigma_{\rm a})\sin^2 \alpha = - p_{\rm im}+\sigma_{\rm f}\ \sin^2\alpha\\ \sigma_{\rm c}&=- p_{\rm im}+(\sigma_{\rm m,f}+\sigma_{\rm a})\cos^2 \alpha = - p_{\rm im}+\sigma_{\rm f}\ \cos^2\alpha\\ \sigma_{\rm r}&=- p_{\rm im}+\sigma_{\rm m,r} \end{aligned} $$\end{document}  σ m,r  σ f 4 \documentclass[12pt]{minimal}
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\begin{document}$$ \frac{\partial\sigma_{\rm r}}{\partial { r}}+\frac{2 \sigma_{\rm m,r}}{r}-\frac{\rm 1}{r}\sigma_{\rm f}= 0 $$\end{document}  σ f  σ m,r  σ r r r o r r i 5 \documentclass[12pt]{minimal}
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\begin{document}$$ \sigma_r= -p_{\rm lv}\quad\hbox{for}\quad r=r_{\rm i};\quad\sigma_{\rm r}= 0\quad\hbox{for}\quad r=r_{\rm o} $$\end{document} 3  σ f \documentclass[12pt]{minimal}
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\begin{document}$$ \int_{r_{\rm i}}^{r_{\rm o}}{d\sigma_{\rm r}}= \int_{r_{\rm i}}^{r_{\rm o}}{\frac{1}{r}(\sigma_{\rm f}- 2 \sigma_{\rm m,r})dr} = (\sigma_{\rm f} - 2 \sigma_{\rm m,r}(\bar{r})) \int_{r_{\rm i}}^{r_{\rm o}}{\frac{1}{r}dr} $$\end{document} 7 \documentclass[12pt]{minimal}
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\begin{document}$$ p_{\rm lv}=(\sigma_{\rm f}-2\sigma_{\rm m,r}(\bar{r})) \ln \left({\frac{r_{\rm o}}{r_{\rm i}}}\right)=\frac{1}{3}(\sigma_{\rm f}-2 \bar{\sigma}_{\rm m,r})\ln \left({1+\frac{V_{\rm w}}{V_{\rm lv}}}\right) $$\end{document} r i r o V lv V w \documentclass[12pt]{minimal}
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\begin{document}$$ \lambda_{\rm f}=\left(\frac{V_{\rm lv}+\frac{1}{3}{V_{\rm w}}}{V_{\rm lv0}+ \frac{1}{3}{V_{\rm w}}}\right)^\frac{1}{3} $$\end{document} λ r 9 \documentclass[12pt]{minimal}
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\begin{document}$$ \lambda_{\rm r} = \lambda_{\rm f}^{-2} $$\end{document} V lv p lv λ f  λ r  σ f \documentclass[12pt]{minimal}
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\begin{document}$$\bar{\sigma}_{\rm m,r}.$$\end{document} Myocardial Constitutive Properties  σ a c l s t a v s 10 \documentclass[12pt]{minimal}
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\begin{document}$$ \sigma_{\rm a}(c,l_{\rm s},t_{\rm a},v_{\rm s})=c\sigma_{\rm ar}f(l_{\rm s})g(t_{\rm a}) h(v_{\rm s}) $$\end{document} c c  σ ar 11 \documentclass[12pt]{minimal}
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\begin{document}$$ f(l_{\rm s})=\left\{\begin{array}{ll} 0 & l_{\rm s}\leq l_{\rm s,a0}\\ \frac{l_{\rm s}-l_{\rm s,a0}}{l_{\rm s,ar}-l_{\rm s,a0}}& l_{\rm s} > l_{\rm a0}\\ \end{array}\right. $$\end{document} 12 \documentclass[12pt]{minimal}
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\begin{document}$$ g(t_{\rm a})=\left\{\begin{array}{ll} 0 & t_{\rm a}< 0 \\ \sin^2 \left(\pi\frac{t_{\rm a}}{t_{\rm max}}\right)&0\leq t_{\rm a}\leq t_{\rm max}\\ 0&t_{\rm a} > t_{\rm max} \\ \end{array} \right. $$\end{document} 13 \documentclass[12pt]{minimal}
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\begin{document}$$ h(v_{\rm s})=\frac{1-(v_{\rm s}/v_0)}{1+c_{\rm v}(v_{\rm s}/v_0)} $$\end{document} l s,a0 l s,ar  σ ar t a t max v 0 c v 1 8 13 Table 1. Reference settings for parameters in the model of the left ventricle. Parameter Value Unit Parameter Value Unit V w 200 −6 3 σ ar 55 3 V lv,0 60 −6 3 c 1 – l s,0 1.9 −6 l s,a0 1.5 −6 σ f0 0.9 3 l s,ar 2.0 −6 σ c0 0.2 3 t max 400 −3 c f 12 – v 0 10 −6 −1 c r 9 – c v 0 – 14 \documentclass[12pt]{minimal}
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\begin{document}$$ \sigma_{\rm m,f}(\lambda_{\rm f})=\left\{\begin{array}{ll} \sigma_{\rm f0} (\hbox{exp}[c_{\rm f}(\lambda_{\rm f} -1)]-1) & \lambda_{\rm f} \geq 1\\ 0 & \lambda_{\rm f} < 1 \end{array} \right. $$\end{document} 15 \documentclass[12pt]{minimal}
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\begin{document}$$ \sigma_{\rm m,r}(\lambda_{\rm r}) = \left\{ \begin{array}{ll} \sigma_{\rm r0}(\exp[c_{\rm r}(\lambda_{\rm r}-1)]-1)&\lambda_{\rm r}\geq 1\\ 0&\lambda_{\rm r}< 1 \end{array} \right. $$\end{document}  σ f0 c f  σ r0 c r 19 1 1 1 Figure 2. R art R per R ven C art C ven L ven L art R art,c R myo,1 R myo,2 R ven,c C art,c C myo,c C ven,c p art,c p myo,c p ven,c p ao p la q ao q m q art,c q ven,c \documentclass[12pt]{minimal}
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\begin{document}$$ p_{\rm im}(\bar{r})=\bar{\sigma}_{\rm m,r}+\frac{r_{\rm o}-\bar{r}}{r_{\rm o} - r_{\rm i}} p_{\rm lv}=\bar{p}_{\rm im} $$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$ r_{\rm i} = \left(\frac{3 V_{\rm lv}}{4 \pi}\right)^{\frac{1}{3}};\quad r_{\rm o} = \left(\frac{ 3 (V_{\rm lv}+V_{\rm w})}{4 \pi}\right)^{\frac{1}{3}};\quad \bar{r} = \left(\frac{3 (V_{\rm lv}+V_{\rm w}/3)}{4 \pi}\right)^{\frac{1}{3}} $$\end{document}  σ m,r λ r Systemic and Coronary Circulation 2 R L C p 19 \documentclass[12pt]{minimal}
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\begin{document}$$ \Delta p_L = L \frac{dq}{dt} $$\end{document} V q V 0 2 Table 2. Reference settings for parameters in the circulation model; coronary resistance values in parentheses represent maximum vasodilation. Systemic circulation Coronary circulation Parameter Value Unit Parameter Value Unit R art 5 6 −3 R art,c 700 (200) 6 −3 R per 120 6 −3 R myo,1 900 (100) 6 −3 R ven 5 6 −3 R myo,2 900 (100) 6 −3 C art 20 −9 3 −1 R ven,c 200 6 −3 C ven 800 −9 3 −1 C art,c 0.03 −9 3 −1 V art,0 500 −6 3 C myo,c 1.4 −9 3 −1 V ven,0 3000 −6 3 C ven,c 0.7 −9 3 −1 L art 60 3 −3 V art,c0 6 −6 3 L ven 60 3 −3 V myo,c0 7 −6 3 V blood 5000 −6 3 V ven,c0 10 −6 3 C myo,c et al 23 −1 −1 −1 23 et al 7 −1 R art C art R per Simulations Performed et al 16 17 et al 16 17 et al 20 \documentclass[12pt]{minimal}
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\begin{document}$$ \hbox{NFA} = \frac{\hat{q}_{\rm art,c,max}-\hat{q}_{\rm art,c,min}}{\hat{q}_{\rm art,c,max}} $$\end{document} Figure 5. 4 \documentclass[12pt]{minimal}
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\begin{document}$$\hat{q}_{\rm art,c,min}$$\end{document} c  Results The Normal Beating Heart 3 −1 −1 Constant Perfusion Pressure and Maximal Vasodilation 3 −1 −1 −1 −1 The Isovolumic Beating Heart et al 15 – 17 4 5 5 4 5 The Isobaric Beating Heart c 4 5 Sensitivity Analysis 6 c r0 6 c r0 c r0 c r0 c r0 t max 6 c v l s,a0 6 Figure 6. c r0 R art,c R myo,1 R myo,2 R ven,c C myo,c  R art,c R myo,1 R myo,2 R ven,c 6 C myo,c C myo,c 6 Discussion Model Setup The aim of this study was to design a model with a limited number of parameters for investigation of the primary relations between left ventricular pressure and volume, wall stress in fiber and transverse direction, intramyocardial pressure and the coronary blood flow. Central to the model are Eq. (7), which shows how fiber stress, that ultimately drives the cardiac cycle, is converted into both LV pressure and radial wall stress, and Eq. (17), that describes how LV pressure and radial wall stress contribute to intramyocardial pressure. Finally, it is the variation of intramyocardial pressure that causes a variation in arterial coronary inflow through a change in coronary volume, represented by intramyocardial compliance. Since the use of a simple model is not without danger, we will address the impact of the main model simplifications. 21 24 \documentclass[12pt]{minimal}
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\begin{document}$$\bar{\sigma}_{\rm m,r},$$\end{document}  σ m,r 4 10 8 13 1 28 26 9 1 2 6 22 et al 12 et al 5 22 30 18 6 The simplifications in the model may be seen as limitations, if one has the goal to explain all experimental observations available. In this study, we consider it a strength of the model, in view of our aim to investigate primary interactions and dependence on parameter settings. Results 15 5 4 et al 16 5 5 et al 14 et al 20 20 15 5 4 4 17 c 5 −1 −1 8 −3 18 4 Sensitivity Analysis c r0 6 c r0 6 15 6 15 6 6 3 6 Relation with Other Models 9 1 2 6 22 et al 5 29 30 et al 26 27 et al Conclusion In conclusion, a mathematical model of the interaction between coronary flow and cardiac mechanics is presented, with a limited number of model parameters. The model replicates the experimental observations, that the phasic character of coronary inflow is virtually independent of maximum left ventricular pressure, that the amplitude of the coronary flow signal depends linearly on cardiac contractility, and that intramyocardial pressure in the left ventricular wall may exceed left ventricular pressure. The normalized amplitude of coronary inflow is mainly determined by contractility, reflected in dependence of active fiber stress on sarcomere length, and maintained at low ventricular volumes by radial wall stress. The sensitivity of the NFA to myocardial coronary compliance and resistance, and to the relation between active fiber stress, time, and sarcomere shortening velocity is low.