By Kristen A. Windoloski and Mette S. Olufsen
The cardiovascular system is a structurally complex network in the human body that interacts with other bodily systems, including the immune system. When an individual becomes critically ill, cardiovascular measurements such as heart rate (HR) and blood pressure (BP)—along with temperature (Temp) and markers of pain like the pain perception threshold (PPT)—are among the first quantities that medical professionals document. They can continually measure cardiovascular quantities via readily available, non-invasive techniques; the resulting rich signals have the potential to determine a patient’s state without the use of complex, invasive blood samples.
Figure 1. A diagram of the mathematical model that describes inflammatory-cardiovascular-pain-thermal dynamics in response to a bolus dose of lipopolysaccharide (LPS). Figure courtesy of the authors.
Sepsis, a severe inflammatory illness that is characterized by its high mortality and strain on hospitals [9], is still not fully understood within the medical community. Advancing overall knowledge of inflammatory dynamics and inflammation’s effects on cardiovascular dynamics is critical for improving early detection of sepsis and lowering patient mortality. Mathematical modeling can help clinicians better understand cardiovascular responses to inflammatory events and guide the development of new treatments that achieve optimal patient outcomes.
Here we discuss recently published results [2, 5] that analyze a patient-specific mathematical model of inflammatory response to better comprehend an inflammatory event’s impact on cardiovascular, pain, and thermal dynamics. The coupled mathematical model comprises 20 equations with 88 parameters (see Figure 1). It consists of three major components: (i) an inflammatory submodel that predicts the concentrations of pro- and anti-inflammatory cytokines in response to a bolus dose of lipopolysaccharide (LPS); (ii) a submodel that relates inflammatory, pain, thermal, and nitric oxide (NO) dynamics; and (iii) a cardiovascular model that predicts the effects of other model states on the cardiovascular system.
The inflammatory submodel contains seven ordinary differential equations that govern LPS, resting and activated monocytes, and pro- (TNF-α, IL-6, IL-8) and anti- (IL-10) inflammatory cytokine levels. This submodel is coupled with a model that predicts the temperatures that will be induced by an increased concentration of pro-inflammatory markers and increased pain perception, which is directly stimulated by the LPS. Increased cytokine signaling also induces NO synthase, though the effect is not apparent until two to four hours after the onset of inflammation; we capture this phenomenon by introducing a delay in the differential equation. These factors all impact the cardiovascular system and its control, which we also model as a system of differential equations.
Figure 2. Individual data for a bolus injection (thin black shaded lines) [6] and a continuous infusion (thin red shaded lines) of lipopolysaccharide (LPS) [1] in healthy young men. The bold lines represent the mean bolus and continuous infusion data. The boxplots display sepsis clinical data (green dots) from 16 septic patients who were admitted to the intensive care unit [1]. Figure courtesy of the authors.
We calibrated the mathematical model with experimental human data from a study wherein 20 healthy male volunteers received an intravenous 2 ng/kg bolus dose of LPS [6]. The study recorded the participants’ physiological functions (HR, BP, Temp, PPT) and cytokines (TNF-α, IL-6, IL-8, IL-10) for two hours before LPS administration and periodically for six hours afterwards (see Figure 2). The large variation in cytokine levels between subjects—which is considered normal—presents a challenge when attempting to interpret inflammatory response.
We employed local sensitivity analysis to determine sensitive and insensitive parameters, then applied subset selection via the structured correlation method [8] to find 18 identifiable, sensitive parameters that we can estimate given the model and data. We identified six population-uniform and 12 patient-specific parameters using nested optimization via MATLAB’s built-in optimization routine and calculated the goodness of fit \((R^2)\) for each parameter combination. We repeated this process until \(R^2\) went down and the coefficient of variation stopped decreasing. The model retains its key cytokine and physiological function behavior when estimating six population-uniform parameters and 12 patient-specific parameters versus 18 patient-specific parameters (see Figure 3).
Figure 3. Comparison of the model calibration for patient 6 when estimating 18 patient-specific parameters (gray lines) versus 12 patient-specific and six population-uniform parameters (black lines). The blue dots represent the individual data for patient 6. Figure courtesy of the authors.
Our model identified and estimated hemodynamic and regulatory parameters that have the potential to assist in the classification of early biomarkers for inflammatory illnesses like sepsis. These parameters include the rate of change of a subject’s HR, the peripheral vascular resistance decay rate, the rate of change in peripheral vascular resistance in response to a pain threshold, and the pain threshold’s rate of change in response to LPS (or other inflammatory stimuli).
We validated our model with both the mean data from the study that we used for model calibration [6] along with data from an independent study that also administered a 2 ng/kg bolus dose of LPS to volunteers [4]. After validation, we simulated various treatment methods for subjects with sustained inflammation and found that a multimodal treatment—a combination of LPS absorption, antipyretics, and vasopressors—resulted in the quickest return to normal cytokine and physiological behavior levels (see Figure 4).
The patient-specific mathematical model that we present here can reproduce key features of inflammatory and physiological function data during a bolus dose endotoxin challenge. It can also simulate therapeutic treatments that produce desirable outcomes for subjects with sustained inflammation. Analysis of the model’s parameters indicates that clinicians can potentially utilize several cardiovascular and regulatory patient-specific parameters that are associated with HR, vascular resistance, and pain thresholds as biomarkers for earlier diagnosis of inflammatory illnesses like sepsis.
Figure 4. Model simulation of the best course of treatment—a multimodal treatment of lipopolysaccharide (LPS) absorption, antipyretics, and vasopressors (blue lines)—introduced at hour four after sustained endotoxemia. Transient endotoxemia (gray lines) refers to the nominal inflammation case where LPS is introduced and all cytokine and physiological levels return to baseline readings within 12 hours. Sustained endotoxemia (black lines) refers to endotoxin that remains at a constant elevated level (2 ng/kg) for 12 hours. Figure courtesy of the authors.
While many researchers have investigated bolus inflammation models that portray short-term inflammation, sepsis and other critical inflammatory illnesses generally exhibit excessive inflammation levels over a longer time frame (on a scale of hours or days) [7]. A continuous infusion human endotoxemia model—where volunteers are infused with inflammatory stimuli (LPS) over several hours instead of minutes—better represent these sustained inflammation levels. Therefore, our current and future work focuses on extending our bolus inflammation model to a continuous infusion inflammation model that is calibrated to experimental data (see Figure 2). With this new model, we plan to investigate the effect of different dosing strategies on inflammatory response and its subsequent impact on cardiovascular dynamics. We aim to extend our investigation to a mathematical model of sepsis that is calibrated to clinical data from sepsis patients. To fully capture sepsis and the severe damage that it can inflict on the body’s vital organs, we plan to include a tissue damage component in our model [3]. Preliminary analysis of sepsis data shows that sepsis patients exhibit elevated IL-6 levels up to 48 hours after initial diagnosis; this observation agrees with the murine data and model simulations of endotoxemia [3].
Investigating different dosing strategies’ effects on inflammation via a continuous infusion inflammation model will help advance our clinical understanding of the human body’s response during a prolonged inflammatory event. Extending the proposed continuous infusion model to a mathematical model for sepsis that is calibrated to inflammatory and cardiovascular data will allow clinicians to better comprehend inflammation’s impact on cardiovascular dynamics during sepsis. This knowledge can in turn help practitioners identify cardiovascular markers that will enable earlier diagnosis and improve patient outcomes.
Mette Olufsen presented this research during a minisymposium at the 2021 SIAM Conference on Computational Science and Engineering, which took place virtually earlier this year.
References
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[2] Brady, R., Frank-Ito, D.O., Tran, H.T., Janum, S., Møller, K., Brix, S., …, Olufsen, M.S. (2018). Personalized mathematical model of endotoxin-induced inflammatory responses in young men and associated changes in heart rate variability. Math. Model. Nat. Phenom., 13(5), 42.
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[5] Dobreva, A., Brady‐Nicholls, R., Larripa, K., Puelz, C., Mehlsen, J., & Olufsen, M.S. (2021). A physiological model of the inflammatory‐thermal‐pain‐cardiovascular interactions during an endotoxin challenge. J. Physiol., 599(5), 1459-1485.
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[8] Olufsen, M.S., & Ottesen, J.T. (2013). A practical approach to parameter estimation applied to model predicting heart rate regulation. J. Math. Biol., 67, 39-68.
[9] Paoli, C.J., Reynolds, M.A., Sinha, M., Gitlin, M., & Crouser, E. (2018). Epidemiology and costs of sepsis in the United States — an analysis based on timing of diagnosis and severity level. Crit. Care Med., 46(12), 1889.
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Kristen A. Windoloski is a Ph.D. candidate in applied mathematics at North Carolina State University, where she is advised by Mette S. Olufsen. Her research focuses on the mathematical modeling of inflammation and its effects on cardiovascular dynamics. |
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Mette S. Olufsen is a professor of mathematics at North Carolina State University. Her current interests include patient-specific modeling of cardiovascular and inflammatory dynamics, sensitivity analysis, parameter inference, and methods for the construction of spatial networks from imaging data. She is director of the Directed Research for Undergraduates in Mathematics and Statistics (DRUMS) Research Experiences for Undergraduates (REU) program, an adjunct professor in biomedical engineering, and a contributor to the biomathematics graduate program. |