SIAM News Blog

Mathematical Models for Him (and for Her Too)

By Anita Layton

Picture someone having a heart attack. Do you imagine a man doubled over, clutching his chest? In Hollywood films, this “classic heart attack” is almost always portrayed by a male. Despite the fact that heart disease is the leading killer of women worldwide, a misconception that the ailment is a man’s illness persists. That incorrect assumption has unfortunate implications in medical care for women.

Gender biases and false impressions are by no means limited to heart attack symptoms. Such prejudices exist throughout the healthcare system, from medical research to disease diagnosis and treatment strategies. Historically, clinical research studies have primarily focused on men and utilized male animal models. The Physicians’ Health Study—a landmark Harvard Medical School analysis founded in 1982 to examine aspirin’s effect on heart disease—initially enrolled over 22,000 participants. None of these participants were female. In the 1970s, the U.S. Food and Drug Administration (FDA) banned women of childbearing age from participating in phase I clinical trials. The ban remained in effect for 20 years and was only lifted in 1993.

Why would researchers choose to exclude half of the population? Female menstrual cycles and fluctuating hormones, which scientists fear may limit the reliability and reproducibility of their findings, might be to blame. Cost is another likely deterrent, as replicating experiments in both sexes requires double the resources. For these reasons, researchers often conduct experiments in men and assume that the findings apply to women as well.

Does this mentality pose a problem? Consider a scenario in which the FDA approved a drug that was tested exclusively in cats. Given that cats share about 90 percent of our genes, would you be comfortable taking the drug without human data? Of course not; biological differences between people and animals could lead to unexpected, undesirable drug reactions.

Women face a similar dilemma when it comes to medical research. While the physiological differences between males and females are undoubtedly much smaller than those between humans and cats, important—albeit subtle—variations nonetheless exist. For example, men are generally larger than women. As a result, a recommended dosage calculated for an average-sized man may cause an overdose in small women. Major differences also exist in the kidneys, which can affect how the body excretes some drugs.

Because of these gender disparities, many diseases affect men and women in dissimilar ways and elicit different responses to treatment. One notable example is high blood pressure, also known as hypertension. According to a report from the American Heart Association, hypertension affects one in three U.S. adults and is the primary cause of premature death in the developed world. Scientists have long known that men generally experience higher blood pressure and are at greater risk for heart and kidney disease. Despite this fact, doctors often prescribe the same medication for both males and females suffering from high blood pressure. This one-size-fits-all approach is problematic; even though women with hypertension are more likely to seek treatment and follow their medication regimens, only 45 percent of treated women get their blood pressure under control (compared to 51 percent of treated men).

His and Her Blood Pressure Regulation Models

Blood pressure regulation involves multiple systems; as such, it can benefit from studies related to systems biology. In 1972, Arthur Guyton, Thomas Coleman, and Harris Granger pioneered computational modeling of the circulatory system for blood pressure regulation in their seminal work, often referred to as the Guyton model [2]. The model consists of a large set of coupled ordinary differential and algebraic equations that describe how different regulators operate synergistically in the circulatory system. Model components include cardiovascular function, circulatory dynamics, renal hemodynamics, kidney function, respiratory function, neurohormonal feedback, autonomic nervous system activity, and electrolyte balance. For instance, the kidneys effectively regulate blood pressure by decreasing reabsorption and thus increasing sodium and water excretion in response to elevated blood pressure. This causes extracellular fluid volume to go down, which in turn lowers blood pressure.

Even mathematical biologists often fail to account for sex differences. Indeed, the Guyton model and its many variants—published over the last four and a half decades—are all gender neutral. To investigate sexual dimorphism and its implications in antihypertensive therapy, our group recently published the first and only set of sex-specific computational models for blood pressure regulation [3]. These models represent sex differences in the renin-angiotensin system (RAS) [4], a signaling pathway that interacts with the kidneys and plays a crucial role in blood pressure regulation as well as the less excitable and more easily repressed female renal sympathetic nervous activity. The schematic in Figure 1 outlines sex-specific components in red.

Figure 1. Schematic model of blood pressure regulation. Pink nodes denote variables that describe cardiovascular function, green nodes denote renal hemodynamics, orange nodes denote renal sodium handling and urine production, and blue nodes denote the renin-angiotensin system. Each node typically consists of a system of ordinary differential and algebraic equations. Red outlines indicate sex-specific model components. Notations are defined as follows: anti-diuretic hormone (ADH); mean arterial pressure (MAP); atrial natriuretic peptide (ANP); renal sympathetic nerve activity (RSNA); plasma renin concentration (PRC); plasma renin activity (PRA); angiotensinogen (AGT); angiotensin I (Ang I); angiotensin II (Ang II); angiotensin II type 1 receptor bound angiotensin II (AT1R-bound Ang II); angiotensin II type 2 receptor bound angiotensin II (AT2R-bound Ang II); and aldosterone (ALD). Figure courtesy of [3].

We use the RAS to illustrate the structure of each model component. Figure 2 portrays the RAS reaction cascade, which starts with angiotensinogen (AGT). Renin, angiotensin-converting enzyme (ACE), and neutral endopeptidase activity facilitate AGT’s conversion into different forms of angiotensin. Renin catalyzes the first reaction in the RAS and is often considered to be the cascade’s driving force. The products of the RAS—angiotensin (1-7) (denoted  Ang (1-7)) and Ang II—bind to receptors and impact the brain, heart, kidneys, vasculature, and immune system.

Receptor-bound Ang II regulates blood pressure via its effects on the kidneys. ACE and chymase convert Ang I to Ang II at rates \(c_\textrm{ACE}\) and \(c_\textrm{Chym}\) respectively. ACE2 then converts Ang II into Ang (1-7) at rate \(c_\textrm{ACE2}\). Ang II also binds to the receptors AT1R and AT2R at corresponding rates of \(c_\textrm{AT1R}\) and \(c_\textrm{AT2R}\). Assuming that Ang II has a half-life of \(h_\textrm{AngII}\), the rate of change of Ang II concentration is given by

\[\frac{d[\textrm{Ang II}]}{dt}=(c_{\textrm{ACE}}+c_{\textrm{Chym}})[\textrm{Ang I}] - (c_{\textrm{ACE2}}+c_{\textrm{Ang II=Ang IV}}+c_{\textrm{AT1R}}+c_{\textrm{AT2R}})[\textrm{Ang II}]- \frac{\textrm{ln}(2)}{h_\textrm{Ang II}}[\textrm{Ang II}].\]

We specify analogous rate equations for AGT, Ang I, Ang (1-7), Ang IV, AT1R-bound Ang II, and AT2R-bound Ang II.

To formulate sex-specific RAS models, we consider the system at steady state and seek to determine the reaction rate parameters (i.e., the \(c\)s), which will presumably differ between the sexes. We first identified male and female hormone levels (i.e., Ang I, Ang II, etc.) from experimental literature. Half-lives (the \(h\)s) for the hormones are known, and solving a linear system separately for males and females yields the rate parameters (or \(c\)s) [4].

Figure 2. The renin-angiotensin reaction cascade. Notations are analogous to Figure 1. Figure courtesy of [4].

We can then incorporate these sex-specific RAS models into the blood pressure regulation models, which are useful for understanding contrasting male and female responses to various hypertensive stimuli [3]. Simulation results suggest that the severity of hypertension induced for a given pathophysiological perturbation may vary significantly between men and women. They also indicate that stronger renal sympathetic nervous activity-mediated regulation of afferent arteriole tone in women is primarily responsible for their resistance to hypertension. Renal sympathetic nervous activity is elevated in response to high blood pressure. This results in a higher degree of afferent arteriole dilation in females, leading to a larger increase in renal blood flow and consequential urine output.

Sex as a Biological and Mathematical Variable

Sex and gender differences exist in many other ailments besides high blood pressure. Heart disease is a classic example, as men and women have disparate prevalences, symptoms, comorbidities, and treatment responses. For instance, women are more likely to report pain associated with heart attack somewhere other than the chest. Multiple sclerosis is another example: females are more susceptible to the condition, but it progresses more severely in males. Pain can also affect men and women differently. Failure to properly account for these gender discrepancies often leads to misdiagnosis and inappropriate treatment in women. Computational modeling thus plays an important role in identifying the most suitable treatment for each sex or gender.

Policymakers have attempted to close the gender gap in medical research. Recently-implemented rules from the National Institutes of Health (NIH) mandate the incorporation of sex as a biological variable in NIH-sponsored research, and have increased the number of females in experimental and clinical studies. While modeling has seen similar progress, the overall number of sex-specific computational models remains low. A search for “sex-specific computational model kidney” on PubMed yields only two publications, both of which belong to our group [1, 5].

A comprehensive understanding of sex and gender’s impact on health and disease is key to the ultimate development of effective sex-based therapies, and mathematical modeling can be a major contributor. Model analysis that highlights sex and gender differences will facilitate the larger effort of precision medicine.

[1] Chen, Y., Sullivan, J.C., Edwards, A., & Layton, A.T. (2017). Sex-specific computational models of the spontaneously hypertensive rat kidneys: Factors affecting nitric oxide bioavailability. Am. J. Physiol. Renal Physiol., 313, F174-F183.
[2] Guyton, A., Coleman, T., & Granger, H. (1972). Circulation: Overall regulation. Ann. Rev. Physiol., 34, 13-46.
[3] Leete, J., & Layton, A.T. (2019) Sex-specific long-term blood pressure regulation: Modeling and analysis. Comput. Biol. Med., 104, 139-148.
[4] Leete, J., Gurley, S., & Layton, A.T. (2018). Modeling sex differences in the renin angiotensin system and the efficacy of antihypertensive therapies. Comput. Chem. Engin., 112, 253-264.
[5] Li, Q., McDonough, A.A., Layton, H.E., & Layton, A.T. (2018). Functional implications of sexual dimorphism of transporter patterns along the rat proximal tubule: Modeling and analysis. Am. J. Physiol. Renal. Physiol., 315, F692-F700.

Anita Layton is the Canada 150 Research Chair in Mathematical Biology and Medicine and a professor of applied mathematics, pharmacy, and biology at the University of Waterloo. She is also the associate dean (Research and International) for the Faculty of Mathematics at Waterloo. Layton’s main research interest is the application of mathematics to biological systems. Her lab collaborates with physiologists and clinicians to formulate detailed computational models to better understand the mechanisms of action and effectiveness of novel therapeutic treatments for diabetes and hypertension.

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