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Mathematical Modelling Disproves Decades-old Hypothesis

A Potential New Way to Treat Heart Disease

By Alona Ben-Tal

Biological systems often display a built-in redundancy—referred to as “plasticity”—that enables more than one mechanism to support the same function. Assessing which mechanism is most important can be difficult, leading to misinterpretations. Here I describe one such misconception and explain mathematics’ crucial role in offering clarity and insight.

Our heart rate varies when we breathe, increasing during inspiration (inhalation) and decreasing during expiration (exhalation) — this is called respiratory sinus arrhythmia (RSA). RSA is typically present from birth; it gets stronger in early adulthood and decreases with age. It is also exaggerated during deep and slow breathing and as a result of physical training. RSA is a process that occurs naturally in healthy individuals, and its loss is linked with cardiac mortality. Although researchers have known about the phenomenon for over 150 years, the benefits of RSA have remained a puzzle. A 1996 study by Junichiro Hayano and colleagues [3] suggested that RSA improved oxygen uptake in the lungs of dogs by matching ventilation (the flow of air in and out of the lungs) with perfusion (the flow of blood through the lungs).

Hayano’s hypothesis, as the study’s conclusions came to be known, seemed very reasonable; increasing one’s heart rate as lungs fill up with fresh air could be expected to improve gas exchange in the lungs. The hypothesis became widely accepted by physiologists and clinicians. Consequently, I had no reason to doubt its accuracy when I was able to test it in a mathematical model (a study I conducted with Julian Paton and Sophie Shamailov). Our initial simulations were disappointing; introducing RSA into the model showed only slight, insignificant improvements in gas exchange. We first attributed the discrepancy to our choice of model parameters, and surmised that simplification of the original model would yield an optimal set of parameters. To achieve this, we reduced the model from six nonlinear equations to two linear ordinary differential equations (ODEs) using several assumptions [1]:

\[\frac{dh_1}{dt} = \alpha q_{in}(t) - D(h_1 - h_2) \\ \frac{dh_2}{dt} = D(h_1 - h_2) - \beta u(t).\]

As is often the case in mathematical studies, we realised that this simplified model can describe a completely different system (see Figure 1). \(h_1\) and \(h_2\) can represent either the heights of water in two interconnected containers or the partial pressures of oxygen (or alternatively, carbon dioxide) in lung air sacs and blood capillaries. \(q_{in}(t)\) can portray water flow into the left chamber or inspired air flow into the lungs, and \(u(t)\) can depict the action of a water pump or heart rate over time; \(D\), \(\alpha\), and \(\beta\) are appropriate parameters for each of these systems.

Figure 1. A hydrodynamic model analogy of gas transport in the lungs. 1a. Three given flow patterns; only one breathing period is shown for each pattern. 1b. The container on the left represents air sacs and the container on the right depicts blood capillaries in the lungs. 1c. The resulting optimal calculation of heart rate. The solid red line portrays slow, deep breathing and the black dotted line represents fast, shallow breathing. Image adapted from [1].

We could now test Hayano’s hypothesis in the hydrodynamic model: if we let water flow into the left container for the first half of the period and stop the flow during the second half, could we operate the pump in a way that would maximize the amount of water flowing out of the right container? Similarly, if the lungs are filled with oxygen during inspiration, could we operate the heart in a way that would transport more oxygen into the bloodstream? The answer is clearly no; if we wish to maintain steady state, the same amount of water that comes in over one period must also flow out during that same time period, regardless of the manner in which the pump functions. This understanding ruled out Hayano’s hypothesis in the simplified model and revived the initial question: what is the benefit of RSA?

We searched for another solvable optimization problem. First we estimated the amount of work carried out by the heart during one breathing period. Suppose the heart pumps a volume of blood \(V_c\) over each heartbeat against a resistance to flow \(R_b\). We can then show [1] that the work expended by the heart over one breathing period \(T\) is

\[W_T = \int_0^T V_c^2R_b HR^2dt.\]

If we assume that \(V_c\) and \(R_b\) are constant, minimizing \(W_T\) is the same as minimizing \(E = \int_0^T HR^2 dt\). This allowed us to state the following optimal control problem:

Find \(HR(t)\), such that \(E\) is minimized subject to the following constraints:

  1. The differential equations of the mathematical model are satisfied
  2. The system is in steady state
  3. The partial pressure of either oxygen or carbon dioxide in the blood has a given average value over one breathing period.

We transformed this optimal problem into a larger system of ODEs with boundary conditions, which we solved numerically [1]. To our delight, the optimal solution (shown in Figure 1c) had the form of RSA. Moreover, the calculations showed that the amplitude of RSA increases under slow and deep breathing (indicated by the solid red line in Figures 1a and 1c), consistent with experimental observations. We also found that maintaining blood carbon dioxide levels had a significantly stronger effect on RSA amplitude than maintaining oxygen levels.

These realizations led to a new hypothesis: RSA minimizes the work executed by the heart while maintaining physiological levels of blood partial pressures of carbon dioxide. We tested and validated the hypothesis on models with increasing degrees of complexity [1, 2]. Our calculations (over one breath) predicted RSA-related energy savings of around three percent. This prompted us to wonder whether the loss of RSA results in more work for the heart and hence contributes to heart damage; if so, would reinstating RSA help the heart recover? Our group is currently contributing to a new study that will test the effect of RSA on heart function and could potentially change the way cardiac pacemakers operate in people with heart disease. 

Why did Hayano’s hypothesis endure for so long? First, Hayano and his colleagues’ experiments in dogs were compelling, revealing a clear difference between mean oxygen consumption with and without RSA. Upon re-examination, it appears that the results might have been statistically insignificant; the margins of error were much larger than the difference in the means; calculation of the mean of the differences may have yielded more conclusive answers. Additionally, some of the experiments that confirmed Hayano’s hypothesis used deep and slow breathing to generate stronger RSA. However, our theoretical study illustrates that such breathing improves gas exchange efficiency regardless of RSA, which explains these previous experimental findings. Lastly, our intuition tells us that RSA should make gas exchange more efficient; it does, but the more complicated mathematical model reveals that the improvement is insignificant.

Ultimately, theoretical studies and mathematical modelling are essential for the quantification and comparison of the different effects that co-exist in biological systems.

Acknowledgments: This work was partly supported by NIH grant R01 NS069220. The research of my collaborator, Julian Paton, was supported by the University of Bristol and the Royal Society.

[1] Ben-Tal, A., Shamailov, S.S., & Paton, J.F.R. (2012). Evaluating the physiological significance of respiratory sinus arrhythmia: looking beyond ventilation-perfusion efficiency. J. Physiol., 590(8), 1989-2008.
[2] Ben-Tal, A., Shamailov, S.S., & Paton, J.F.R. (2014). Central regulation of heart rate and the appearance of respiratory sinus arrhythmia: New insights from mathematical modeling. Math. Biosci., 255, 71-82.
[3] Hayano, J., Yasuma, F., Okada, A., Mukai, S., & Fujinami, T. (1996). Respiratory sinus arrhythmia. A phenomenon improving pulmonary gas exchange and circulatory efficiency. Circ., 94(4), 842-847.

Alona Ben-Tal was initially trained as a mechanical engineer and worked for a few years in industry before pursuing a Ph.D. in mathematics, followed by a New Zealand Science and Technology Postdoctoral Fellowship at the Auckland Bioengineering Institute. She is now a senior lecturer in mathematics and Deputy Head of Institute at Massey University’s Institute of Natural and Mathematical Sciences in Auckland, New Zealand.

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