Optimizing irradiation conditions for low-intensity pulsed ultrasound to upregulate endothelial nitric oxide synthase

Abstract

Purpose

Here we aimed to develop a minimally invasive treatment for ischemic heart disease and demonstrate that low-intensity pulsed ultrasound (LIPUS) therapy improves myocardial ischemia by promoting myocardial angiogenesis in a porcine model of chronic myocardial ischemia. Studies to date determined the optimal treatment conditions within the range of settings available with existing ultrasound equipment and did not investigate a wider range of conditions.

Methods

We investigated a broad range of five parameters associated with ultrasound irradiation conditions that promote expression of endothelial nitric oxide synthase (eNOS), a key molecule that promotes angiogenesis in human coronary artery endothelial cells (HCAEC).

Results

Suboptimal irradiation conditions included 1-MHz ultrasound frequency, 500-kPa sound pressure, 20-min total irradiation time, 32–48-$\mathrm{\mu s}$ pulse duration, and 320-$\mathrm{\mu s}$ pulse repetition time. Furthermore, a proposed index, ${P}_{\mathrm{N}}$, calculated as the product of power and the total number of irradiation cycles applied to cells using LIPUS, uniformly revealed the experimental eNOS expression associated with the various values of five parameters under different irradiation conditions.

Conclusion

We determined the suboptimal ultrasound irradiation conditions for promoting eNOS expression in HCAEC.

Appendix: Derivation of approximated lines in Fig. 3b using the least squares method

By introducing new variables $x$ and $\mathrm{y}$ as

$$\begin{array}{c}x={\mathrm{log}}_{10}\overline{{E }_{\mathrm{q}}},\end{array}$$

(A1)

$$\begin{array}{c}y=EX,\end{array}$$

(A2)

let us define an approximated line graph (polyline) in Fig. 3b as

$$\begin{array}{c}y=\left\{\begin{array}{c}{a}_{1}\left(x-{x}_{0}\right)+{y}_{0}, \left(x<{x}_{0}\right)\\ {a}_{2}\left(x-{x}_{0}\right)+{y}_{0}, \left(x\ge {x}_{0}\right),\end{array}\right.\end{array}$$

(A3)

where ${a}_{1}$, ${a}_{2}$, ${x}_{0}$, and ${y}_{0}$ are the coefficients of the approximated polyline that should be estimated. For the measured values $\{{x}_{i},{y}_{i}\}$ (${x}_{i}<{x}_{0}$) and $\{{x}_{j},{y}_{j}\}$ (${x}_{j}\ge {x}_{0}$), the weighted mean squared error $\alpha ({a}_{1},{a}_{2},{y}_{0};\,\,{x}_{0})$ between the measured values and polyline in Eq. (A3) can be obtained as follows:

$$\begin{array}{c}\alpha \left({a}_{1},{a}_{2},{y}_{0};\,\,{x}_{0}\right)={\sum }_{i}{w}_{i}\cdot {\left|{y}_{i}-\left({a}_{1}\left({x}_{i}-{x}_{0}\right)+{y}_{0}\right)\right|}^{2}\\ +{\sum }_{j}{w}_{j}\cdot {\left|{y}_{j}-\left({a}_{2}\left({x}_{j}-{x}_{0}\right)+{y}_{0}\right)\right|}^{2}, \left({x}_{i}<{x}_{0},{x}_{j}\ge {x}_{0}\right)\end{array}$$

(A4)

where $w$ is the weight defined by the variance of $EX$, ${\sigma }^{2}$, as

$$\begin{array}{c}w=\frac{1}{{\sigma }^{2}},\end{array}$$

(A5)

where the standard deviation $\sigma$ of $EX$ is presented in Table 1 and Fig. 3b. By fixing the coefficient ${x}_{0}$ to an arbitrary value and defining ${x}{^\prime}$ as

$$\begin{array}{c}{x}^{\mathrm{^{\prime}}}\left({x}_{0}\right)=x-{x}_{0},\end{array}$$

(A6)

the condition minimizing $\alpha \left({a}_{1},{a}_{2},{y}_{0};\,\,{x}_{0}\right)$ is given by

$$\begin{array}{c}\frac{1}{2}\frac{\partial \alpha \left({a}_{1},\,\,{a}_{2},\,\,{y}_{0};\,\,{x}_{0}\,\right)}{\partial {a}_{1}}={a}_{1}{\sum }_{i}{w}_{i}{{x}_{i}^{\mathrm{^{\prime}}}\left({x}_{0}\right)}^{2}-{\sum }_{i}{w}_{i}{y}_{i}{x}_{i}^{\mathrm{^{\prime}}}\left({x}_{0}\right)+{y}_{0}{\sum }_{i}{w}_{i}{x}_{i}^{\mathrm{^{\prime}}}\left({x}_{0}\right)=0,\end{array}$$

(A7)

$$\begin{array}{c}\frac{1}{2}\frac{\partial \alpha \left({a}_{1},\,\,{a}_{2},\,\,{y}_{0};\,\,{x}_{0}\,\right)}{\partial {a}_{2}}={a}_{2}{\sum }_{j}{w}_{j}{x}_{j}^{\mathrm{^{\prime}}}{\left({x}_{0}\right)}^{2}-{\sum }_{j}{w}_{j}{y}_{j}{x}_{j}^{\mathrm{^{\prime}}}\left({x}_{0}\right)+{y}_{0}{\sum }_{j}{w}_{j}{x}_{j}^{\mathrm{^{\prime}}}\left({x}_{0}\right)=0,\end{array}$$

(A8)

$$\begin{array}{c}\frac{1}{2}\frac{\partial \alpha \left({a}_{1},\,{a}_{2},\,{y}_{0};\,{x}_{0}\,\right)}{\partial {y}_{0}}={y}_{0}\left({\sum }_{i}{w}_{i}+{\sum }_{j}{w}_{j}\right)-\left({\sum }_{i}{w}_{i}{y}_{i}+{\sum }_{j}{w}_{j}{y}_{j}\right)\\ +{a}_{1}{\sum }_{i}{w}_{i}{x}_{i}^{\mathrm{^{\prime}}}\left({x}_{0}\right)+{a}_{2}{\sum }_{j}{w}_{j}{x}_{j}^{\mathrm{^{\prime}}}\left({x}_{0}\right)=0.\end{array}$$

(A9)

By substituting ${a}_{1}$ into Eq. (A7), and ${a}_{2}$ in Eq. (A8) into Eq. (A9), $\widehat{{y}_{0}}({x}_{0})$, which minimizes $\alpha \left({a}_{1},{a}_{2},{y}_{0};\,\,{x}_{0}\right)$ with fixed ${x}_{0}$, is estimated by

$$\begin{array}{c}\widehat{{y}_{0}}\left({x}_{0}\right)=\frac{{B}_{I}{E}_{I}{D}_{J}+{B}_{J}{E}_{J}{D}_{I}-\left({\sum }_{i}{w}_{i}{y}_{i}+{\sum }_{j}{w}_{j}{y}_{j}\right){D}_{I}{D}_{J}}{{{B}_{I}}^{2}{D}_{J}+{B}_{J}^{2}{D}_{I}-\left({\sum }_{i}{w}_{i}+{\sum }_{j}{w}_{j}\right){D}_{I}{D}_{J}},\end{array}$$

(A10)

where

$$\begin{array}{c}{B}_{I}={\sum }_{i}{w}_{i}{x}_{i}^{\mathrm{^{\prime}}}\left({x}_{0}\right), { B}_{J}={\sum }_{j}{w}_{j}{x}_{j}^{\mathrm{^{\prime}}}\left({x}_{0}\right),\\ {D}_{I}={\sum }_{i}{w}_{i}{{x}_{i}^{\mathrm{^{\prime}}}\left({x}_{0}\right)}^{2}, { D}_{J}={\sum }_{j}{w}_{j}{{x}_{j}^{\mathrm{^{\prime}}}\left({x}_{0}\right)}^{2},\\ {E}_{I}={\sum }_{i}{w}_{i}{y}_{i}{x}_{i}^{\mathrm{^{\prime}}}\left({x}_{0}\right),{E}_{J}={\sum }_{j}{w}_{j}{y}_{j}{x}_{j}^{\mathrm{^{\prime}}}\left({x}_{0}\right).\end{array}$$

(A11)

Subsequently, $\widehat{{a}_{1}}\left({x}_{0}\right)$ and $\widehat{{a}_{2}}\left({x}_{0}\right)$ minimizing $\alpha \left({a}_{1}, {a}_{2}, {y}_{0};\, {x}_{0}\right)$ with a fixed ${x}_{0}$ are estimated by substituting Eq. (A10) into Eq. (A7) and (A8), respectively.

$$\begin{array}{c}\widehat{{a}_{1}}\left({x}_{0}\right)=\frac{{E}_{I}-{B}_{I}\widehat{{y}_{0}}\left({x}_{0}\right)}{{D}_{I}},\end{array}$$

(A12)

$$\begin{array}{c}\widehat{{a}_{2}}\left({x}_{0}\right)=\frac{{E}_{J}-{B}_{J}\widehat{{y}_{0}}\left({x}_{0}\right)}{{D}_{J}}.\end{array}$$

(A13)

Using $\widehat{{a}_{1}}\left({x}_{0}\right)$, $\widehat{{a}_{2}}\left({x}_{0}\right)$, and $\widehat{{y}_{0}}\left({x}_{0}\right)$, $\alpha \left(\widehat{{a}_{1}}\left({x}_{0}\right),\widehat{{a}_{2}}\left({x}_{0}\right),\widehat{{y}_{0}}\left({x}_{0}\right);\,{x}_{0}\right)$ is calculated for each of the various values $\left\{{x}_{0}\right\}$. Thus, the coefficient $\widehat{{x}_{0}}$ that minimizes $\alpha \left({a}_{1},{a}_{2},{y}_{0};\,\,{x}_{0}\right)$ in Eq. (A4) can be obtained as follows:

$$\begin{array}{c}\widehat{{x}_{0}}=\mathrm{arg}\underset{{x}_{0}}{\mathrm{min}}\alpha \left(\widehat{{a}_{1}}\left({x}_{0}\right),\widehat{{a}_{2}}\left({x}_{0}\right),\widehat{{y}_{0}}\left({x}_{0}\right);\,\,{x}_{0}\right).\end{array}$$

(A14)

Finally, the optimum values of ${a}_{1}$, ${a}_{2}$, and ${y}_{0}$ for the minimum condition can be determined by $\widehat{{a}_{1}}\left(\widehat{{x}_{0}}\right)$, $\widehat{{a}_{2}}\left(\widehat{{x}_{0}}\right)$, and $\widehat{{y}_{0}}\left(\widehat{{x}_{0}}\right)$, respectively.