# Cardiovascular regulation in response to multiple hemorrhages: analysis and parameter estimation

## Abstract

Mathematical models can provide useful insights explaining behavior observed in experimental data; however, rigorous analysis is needed to select a subset of model parameters that can be informed by available data. Here we present a method to estimate an identifiable set of parameters based on baseline left ventricular pressure and volume time series data. From this identifiable subset, we then select, based on current understanding of cardiovascular control, parameters that vary in time in response to blood withdrawal, and estimate these parameters over a series of blood withdrawals. These time-varying parameters are first estimated using piecewise linear splines minimizing the mean squared error between measured and computed left ventricular pressure and volume data over four consecutive blood withdrawals. As a final step, the trends in these splines are fit with empirical functional expressions selected to describe cardiovascular regulation during blood withdrawal. Our analysis at baseline found parameters representing timing of cardiac contraction, systemic vascular resistance, and cardiac contractility to be identifiable. Of these parameters, vascular resistance and cardiac contractility were varied in time. Data used for this study were measured in a control Sprague-Dawley rat. To our knowledge, this is the first study to analyze the response to multiple blood withdrawals both experimentally and theoretically, as most previous studies focus on analyzing the response to one large blood withdrawal. Results show that during each blood withdrawal both systemic vascular resistance and contractility decrease acutely and partially recover, and they decrease chronically across the series of blood withdrawals.

## Keywords

Blood withdrawal Cardiovascular regulation Mathematical modeling Parameter estimation Left ventricular volume Left ventricular pressure## Notes

### Funding

This material is based upon work supported by the Mathematics Research Communities of the American Mathematical Society, under National Science Foundation Grant Number DMS 1321794. The project was initiated during the Mathematics Research Community on Mathematics in Physiology and Medicine workshop (2016). Follow-up visits and workshops were supported by the John N. Mordeson Endowed Chair in Mathematics at Creighton University and by the Mathematical Biosciences Institute. In addition, MVC was supported in part by the Mathematical Biosciences Institute and the National Science Foundation under Grant Number DMS 1440386, and by NSF Grant Number DMS 1408742. SSD was supported by a Joint University of California Davis and Lawrence Livermore National Laboratory Graduate Mentorship Award. CD, BEC and MSO were been supported in part by NIH NIGMS Grant Number P50-GM094503-02, and CD was supported by was supported by the U.S. Department of Veterans Affairs BLR&D program by a Merit Review Award # BX001863. CD and BEC were also supported by NIH NHLBI Grant Number U01 HL109505-01.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflicts of interest.

### Ethical approval

All applicable international, national, and/or institutional guidelines for the care and use of animals were followed. All procedures performed in studies involving animals were in accordance with the ethical standards of the institution at which the studies were conducted.

### Data

The datasets generated during and/or analyzed during the current study are available on request from the corresponding author.

## Supplementary material

## References

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