# A Multicellular Vascular Model of the Renal Myogenic Response

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

#### 2.1. Model Description

#### 2.1.1. Vascular Blood Flow

#### 2.1.2. Vascular Wall

#### 2.2. Numerical Methods

`ode15s`, and supply the Jacobian matrix to improve the solver’s efficiency. We note that this built-in MATLAB method is adaptive, so that we expect that the numerical error arising from time discretization is controlled. The absolute and relative tolerances we use for

`ode15s`are both ${10}^{-7}$.

#### 2.3. Parameter Values

#### 2.3.1. Vascular Geometry and Hemodynamics

#### 2.3.2. Electrophysiology

## 3. Model Results

#### 3.1. Responses to Steady Perturbations

#### 3.2. Responses to a Step Perturbation

#### 3.3. Responses to Sinusoidal Perturbation

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Model vasculature. Blood flows through the afferent arteriole from the cortical radial artery to the glomerulus and subsequently flows through the efferent arteriole from the glomerulus to the peritubular capillaries.

**Figure 2.**A segment of the model afferent arteriole. The vascular wall consists of a chain of smooth muscle cells (SMC). The local vascular radius ${R}_{i}$ is determined by the balance of ${T}_{P}^{i}$ and ${T}_{\mathrm{wall}}^{i}$ which depend on local blood pressure ${P}_{i}$ and the contractile state of the surrounding myocyte, respectively. Along the vascular wall, signals are conducted directly through the smooth muscles (${v}_{i-1}\leftrightarrow {v}_{i}\leftrightarrow {v}_{i+1}$) or indirectly through the endothelium (shaded region, ${u}_{i-1}\leftrightarrow {u}_{i}\leftrightarrow {u}_{i+1}$). For simplicity, we only show selected intercellular currents. See Section 2.1.2 for complete equations.

**Figure 4.**Predicted oscillations for the first cell in the afferent arteriole at an inflow pressure of 100 mmHg for the (

**A**) membrane potential; (

**B**) cytosolic concentration of Ca${}^{2+}$; and (

**C**) afferent arteriole local diameter.

**Figure 5.**Predicted myogenic response (

**blue**) compared to perfect autoregulation (

**red**) and no autoregulation (

**purple**) of blood flow through the vessel for a range of blood pressures. For illustrative purposes, we show a straight line for the case of no autoregulation that qualitatively captures the response under this assumption (actual slope may differ).

**Figure 6.**(

**A**) Predicted steady-state diameter (calculated by time-averaging for the first and last cells and calculated by space- and time-averaging for the mean) across the afferent arteriole cells for a range of time independent blood pressures; (

**B**) Predicted myogenic response of blood flow for five cells in the afferent arteriole for a range of blood pressures.

**Figure 7.**Predicted time profiles of (

**1**) inflow pressure; (

**2**) blood flow; (

**3**) membrane potential; (

**4**) cytosolic Ca${}^{2+}$ concentration; and (

**5**) local diameter for the first cell in the model afferent arteriole under (

**A**) pressure pulse decrease to 80 mmHg; and (

**B**) pressure pulse increase to 120 mmHg.

**Figure 8.**Predicted time profiles of (

**A**) membrane potential; (

**B**) cytosolic Ca${}^{2+}$ concentration; and (

**C**) local diameter for five cells in the afferent arteriole at an inflow pressure pulse increase to 140 mmHg. For clarity, we only show one in 5 cells along the AA; the behavior of neighboring cells is summarized in Section 3.2.

**Figure 9.**Predicted (

**A**) steady-state diameter (time- and space-averaged across all cells); and (

**B**) time-averaged outflow pressure (for the last afferent arteriole cell), for a range of luminal pressure pulses.

**Figure 10.**(

**1**) Inflow pressure; and predicted oscillations in (

**2**) local diameter for the first and last cells in the afferent arteriole; and (

**3**) outflow pressure when a sinusoidal perturbation is applied to the inflow pressure with amplitude $A=20$ mmHg and frequency (

**A**) 0.01 Hz; (

**B**) 0.1 Hz; and (

**C**) 1 Hz.

**Figure 11.**Normalized inflow pressure ${P}_{\mathrm{a}}/{P}_{\mathrm{a}}^{\mathrm{ref}}$ and normalized blood flow ${Q}_{\mathrm{AA}}/{Q}_{\mathrm{AA}}^{\mathrm{ref}}$ when a sinusoidal perturbation is applied to the inflow pressure with amplitude $A=20$ mmHg and frequency (

**A**) 0.01 Hz; (

**B**) 0.1 Hz; and (

**C**) 1 Hz.

Description | Parameter | Value | Units | Reference |
---|---|---|---|---|

Reference afferent arteriole inflow pressure | ${P}_{\mathrm{a}}^{\mathrm{ref}}$ | 100 | mmHg | [55,60] |

Reference afferent arteriole outflow pressure | ${P}_{\mathrm{g}}^{\mathrm{ref}}$ | 50 | mmHg | [55,60] |

Reference afferent arteriole radius | ${R}_{\mathrm{AA}}^{\mathrm{ref}}$ | 10 | $\mathsf{\mu}$m | [29,60] |

Reference afferent arteriole flow | ${Q}_{\mathrm{AA}}^{\mathrm{ref}}$ | 300 | nL/min | [55,60] |

Reference glomerular filtration rate | ${Q}_{\mathrm{g}}^{\mathrm{ref}}$ | 30 | nL/min | [7,60] |

Pressure in renal vein | ${P}_{\mathrm{v}}$ | 4 | mmHg | [7] |

Axial length of smooth muscle cell | h | 3 | $\mathsf{\mu}$m | [69] |

Number of smooth muscle cells | ${N}_{\mathrm{AA}}$ | 20 | - | present study |

Afferent arteriole length | ${L}_{\mathrm{AA}}$ | 60 | $\mathsf{\mu}$m | [69] |

Efferent arteriole length | ${L}_{\mathrm{EA}}$ | 60 | $\mathsf{\mu}$m | present study |

Efferent arteriole radius | ${R}_{\mathrm{EA}}$ | 11 | $\mathsf{\mu}$m | [71] |

Filtration fraction | ${f}_{\mathrm{g}}$ | 0.1 | - | [72] |

Blood viscosity | $\mu $ | 6.68 | mmHg ·$\mathsf{\mu}$s | present study |

Muscle contraction time constant | $\eta $ | 1.71 | s | [29,30] |

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**MDPI and ACS Style**

Ciocanel, M.-V.; Stepien, T.L.; Sgouralis, I.; Layton, A.T.
A Multicellular Vascular Model of the Renal Myogenic Response. *Processes* **2018**, *6*, 89.
https://doi.org/10.3390/pr6070089

**AMA Style**

Ciocanel M-V, Stepien TL, Sgouralis I, Layton AT.
A Multicellular Vascular Model of the Renal Myogenic Response. *Processes*. 2018; 6(7):89.
https://doi.org/10.3390/pr6070089

**Chicago/Turabian Style**

Ciocanel, Maria-Veronica, Tracy L. Stepien, Ioannis Sgouralis, and Anita T. Layton.
2018. "A Multicellular Vascular Model of the Renal Myogenic Response" *Processes* 6, no. 7: 89.
https://doi.org/10.3390/pr6070089