A unified model for the human response to lipopolysaccharide-induced inflammation Kristen A. Windoloski Elisabeth O. Bangsgaard Atanaska Dobreva Johnny T.

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A uniﬁed model for the human response to

lipopolysaccharide-induced inﬂammation

Kristen A. Windoloski, Elisabeth O. Bangsgaard, Atanaska Dobreva, Johnny T.

Ottesen, and Mette S. Olufsen

Abstract This study develops a uniﬁed model predicting the whole-body response

to endotoxin. We simulate dynamics using diﬀerential equations examining the re-

sponse to a lipopolysaccharide (LPS) injection. The model tracks pro- and anti-

inﬂammatory cytokines (TNF-𝛼, IL-6, IL-10), concentrations of corticotropin-

releasing hormone (CRH), adrenocorticotropic hormone (ACTH), and cortisol in

the hypothalamic-pituitary-adrenal (HPA) axis. Daily hormonal variations are inte-

grated into the model by including circadian oscillations when tracking CRH. Ad-

ditionally, the model tracks heart rate, blood pressure, body temperature, and pain

perception. Studied quantities function on timescales ranging from minutes to days.

To understand how endotoxin impacts the body over this vast span of timescales,

we examine the response to variations in LPS administration methods (single dose,

repeated dose, and continuous dose) as well as the timing of the administration and

the amount of endotoxin released into the system. We calibrate the model to literature

Kristen A. Windoloski

Department of Mathematics, North Carolina State University, 2311 Stinson Drive, Raleigh, NC

27607

e-mail: kawindol@ncsu.edu

Elisabeth O. Bangsgaard

Technical University of Denmark, Asmussens All´

e, 2800 Lyngby, Denmark

e-mail: eoba@dtu.dk

Atanaska Dobreva

Department of Mathematics, Augusta University - 1201 Goss Lane, Augusta, GA 30912

e-mail: adobreva@augusta.edu

Johnny T. Ottesen

Centre for Mathematical Modeling - Human Health and Disease, Roskilde University, Univer-

sitetsvej 1, 4000 Roskilde, Denmark

e-mail: johnny@ruc.dk

Mette S. Olufsen

Department of Mathematics, North Carolina State University, 2311 Stinson Drive, Raleigh, NC

27607

e-mail: msolufse@ncsu.edu

2 Authors Suppressed Due to Excessive Length

data for a 2 ng/kg LPS bolus injection. Results show that LPS administration during

early morning or late evening generates a more pronounced hormonal response.

Most of the LPS eﬀects are eliminated from the body 24 hours after administration,

the main impact of inﬂammation remains in the system for 48 hours, and repeated

dose simulations show that residual eﬀects remain more than 10 days after the initial

injection. We also show that if the LPS administration method or total dosage is

increased, the system response is ampliﬁed, posing a greater risk of hypotension and

pyrexia.

1 Introduction

The body has a wealth of regulatory mechanisms controlling vital functions that op-

erate on timescales that diﬀer by a factor of 107, ranging from milliseconds (action

potentials) to years (aging). Studying the eﬀects of diseases on these timescales can

be challenging, even for a well-deﬁned event such as the inﬂammatory response to

a low-dose endotoxin challenge (typically achieved by administering lipopolysac-

charides (LPS)). The immune system is complex, and its response to a pathogenic

threat entering the body through an external or internal wound varies signiﬁcantly

depending on the pathogen type, the degree of infection, the host’s age, sex, and eth-

nicity [33]. The body responds to the threat by activating local and systemic (innate)

signaling cascades to remove the pathogen.

Most studies examining inﬂammatory signaling cascades focus on the short-

term response (6-8 hours) using a combination of experimental and computational

approaches examining dynamics in both animals and humans. In both species, in-

ﬂammation can be stimulated by low-dose LPS administration. The eﬀects have

been studied both experimentally [20,42,67] and computationally [6,29,80,101]

as this stimulus provides an excellent controlled model of the inﬂammatory cas-

cade. But detailed experimental studies mapping inﬂammatory signaling pathways

have found signiﬁcant diﬀerences between animals and humans [30,85,108]. In

addition to the immune response, pathogens impact dynamic signaling within the

endocrine hypothalamic-pituitary-adrenal (HPA) axis, vascular systems, temperature

regulation, and pain perception threshold [33], which display hourly, daily, monthly,

and yearly variations. Long-term variations (monthly and yearly) are signiﬁcant for

chronic inﬂammation, but controlling the experimental environment is challenging.

To address this challenge, we focused on developing a uniﬁed mathematical model

examining the hourly and daily whole-body response to LPS, accounting for ultradian

and circadian variation.

The immune, hormonal, and cardiovascular systems have historically been stud-

ied individually and often at diﬀerent timescales. Mathematical modeling of the

inﬂammatory cascade has been investigated on the timescale of hours using either

models that lump inﬂammation components into broad categories (such as general

pro-inﬂammatory and anti-inﬂammatory states) [26,31,53,83] or more detailed

models including speciﬁc immune response cells or cytokines [10,18,76].

A uniﬁed endotoxin model 3

Cardiovascular dynamics are typically studied over seconds or minutes to predict

ﬂow to a speciﬁc organ [21] or examine the control of blood ﬂow in response to a

challenge, such as the Valsalva maneuver [82]. While these models provide excellent

predictions of hemodynamics, they do not address how these predictions vary daily,

weekly, or monthly. Moreover, cardiovascular dynamics studies typically exclude

inﬂuences from other systems, even though it is well known that the immune and

hormonal systems impact dynamics. For example, the formation of atherosclerotic

lesions involves an immune response [37,16], and inﬂammation developing into

sepsis depends on vagal responses [15,104]. Additionally, elevated cortisol levels

during stress result in increases in heart rate and anti-inﬂammatory reactions [60],

and the transition to an advanced disease state is often accompanied by noticeable

physiologic immune responses. Furthermore, morbidity is transformed into comor-

bidities often due to couplings by compromised immune or endocrine systems [34]

Previous studies have investigated the coupling of stress to inﬂammation [6,66],

and inﬂammation to cardiovascular dynamics, temperature, and pain perception

[29,94]. However, mathematical model coupling interactions between inﬂamma-

tion, stress, cardiovascular, pain, and thermal dynamics have yet to be investigated.

Therefore, our study is the ﬁrst to develop a mathematical model, henceforth denoted

as the uniﬁed model (depicted in Figure 2) mapping the LPS response to the im-

mune cascade, the HPA axis, and the cardiovascular system as well as temperature

and pain dynamics on a timescale of hours to days. The uniﬁed model has several

components: (1) an inﬂammation model that tracks concentrations of tumor necro-

sis factor-alpha (TNF-𝛼), interleukin 6 (IL-6), and interleukin 10 (IL-10) as well

as resting and activated monocytes released in response to LPS; (2) an endocrine

HPA axis model tracking concentrations of corticotropin-releasing hormone (CRH),

adrenocorticotropic hormone (ACTH), and cortisol; (3) a cardiovascular model pre-

dicting heart rate, nitric oxide concentrations, vascular resistance, blood ﬂow, and

blood pressure using a circulation model integrated with a simple autonomous nerve

system model; (4) a temperature model; and (5) a pain perception model. These

systems operate on multiple timescales but are modeled on the timescale of hours.

Figures 2and 2show the coupling between the models including the stress

hormone, cortisol, having a stimulating eﬀect on heart rate, autonomous nerve system

signaling aﬀecting CRH and cytokine production, and inﬂammation aﬀecting heart

rate, temperature, and the HPA axis hormones. To test the validity of our uniﬁed

model, we ﬁt dynamics to data from Janum et al. (2016) and Clodi et al. (2008).

Physiology is the science of functions and mechanisms of the living. It dates

back to Hippocrates in the late 5th century BC. The word comes from the An-

cient Greek word φύσιη(ph´usis), meaning ”nature” and λογία (log´

ıa), meaning

”study of”. The modern term, coined by pioneers including Jean Ferne (1497-

1558), William Harvey (1578-1657), Claude Bernard (1813–1878), and Au-

gust Krogh (1874-1949) refers to a model-based point of view.

The human body consists of a wealth of interconnecting mechanisms and

subsystems, but these may be considered isolated or only weakly connected

for many purposes. Thus, researchers have studied the circulation of blood,

4 Authors Suppressed Due to Excessive Length

endocrine systems, the immune system, and other mechanisms independently.

However, the development of pathologies and comorbidities often needs such

subsystems to be bridged. In the current paper, we illustrate and discuss this

bridging in the case of infection with an engineered coli bacterium denoted

LPS. While the subsystems are well-known for speciﬁc purposes, the coupling

of these quantitatively is only vaguely known due to the lack of direct in vivo

measurement methodologies.

The only way the couplings of the subsystems can be described quantita-

tively is through mathematical modeling coupled with data for the subsystems,

a strategy denoted the mathematical microscope [72]. If these couplings are of

signiﬁcant strength, non-linearities and multiple times scale become crucial.

These complications may introduce unexpected phenomena. The current paper

presents the state of such rising research.

2 Methods

To understand how daily (ultradian and circadian) rhythms and stress impact in-

ﬂammation and how inﬂammation impacts cardiovascular dynamics, we develop a

uniﬁed model (shown in Figures 2and 2) integrating and adapting Dobreva et al.’s

inﬂammatory-cardiovascular-temperature-pain model [29] and Bangsgaard et al.’s

inﬂammatory-HPA axis model [6]. The uniﬁed model is calibrated to data from

human studies administrating a low dose of LPS. Below, we describe the data used

for model calibration and each model component. Model parameter values, units,

and initial conditions for all state variables are listed in the Appendix, Table 1.

Inﬂammation The immune response of the human body is often divided

into two sub-systems: the innate immune response and the adaptive immune

response. Innate immunity is the ﬁrst line of defense that meets a foreign

substance. It has no immunological memory and is a mechanism that naturally

occurs in the body. The adaptive immune response is the next line of defense

if innate immunity fails, providing a complete immune response. When the

ﬁrst exposure to a particular virus or pathogen occurs, the adaptive immune

response creates an immunological memory to ﬁght future threats. The body’s

innate immune response is the focus of our inﬂammation submodel presented

in this paper.

The innate immune system’s ﬁrst defense is generally a physical barrier to

block an intruder from entering the body. This could be the skin, epithelial

barriers, or chemical mechanisms such as blood clotting (coagulation). The

innate immune response enacts a cascade of events when the foreign substance

enters the body’s tissues. Phagocytic cells, whose central role is to engulf and

A uniﬁed endotoxin model 5

eliminate foreign substances, destroy most of the intruders. A speciﬁc type of

phagocytic cell is the macrophage, which is derived from blood monocytes.

During the innate immune response, macrophages release cytokines (signaling

proteins), among other chemical mediators, to not only help recruit additional

immune cells to the site of infection to eliminate the threat but to also help the

body repair any damage caused by the foreign substance. [68]

The immune system aﬀects every system in the body. The response to an

infection is followed by an increase in temperature (fever) and pain perception.

However, other mechanisms may also be aﬀected, including heart rate and

blood pressure increases. Another essential system aﬀected is the endocrine

system, which regulates stress. Many studies have examined these systems,

but little work is done to understand how they work together. Studying how

the diﬀerent subsystems interact can be done experimentally or, as we do here,

using a complex mathematical model.

Fig. 1 Diagram showing interactions between the immune system (yellow), the HPA axis (pink),

and the cardiovascular system (red and dark blue) during an endotoxin challenge. A bolus or

continuous LPS dose is administered, prompting the activation of immune cells and the secretion of

pro- and anti-inﬂammatory cytokines. LPS administration instigates the release of pro-inﬂammatory

cytokines, stimulating the HPA axis to produce CRH, ACTH, and cortisol. Cortisol exhibits negative

feedback on CRH and ACTH and positive feedback on anti-inﬂammatory cytokine IL-10 and heart

rate. The cytokine production also causes an increase in body temperature (light blue), which

upregulates heart rate. The heart rate exhibits positive feedback on the pro-inﬂammatory cytokine

TNF-𝛼. Additionally, LPS administration inhibits the pain perception threshold (green), which

upregulates peripheral vascular resistance. The latter aﬀects nitric oxide production, which is

upregulated by TNF-𝛼and downregulated by IL-10. The stimulation between elements is denoted

by solid black lines and inhibition by dotted lines.

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Aunifiedmodelforthehumanresponsetolipopolysaccharide-inducedinflammationKristenA.Windoloski,ElisabethO.Bangsgaard,AtanaskaDobreva,JohnnyT.Ottesen,andMetteS.OlufsenAbstractThisstudydevelopsaunifiedmodelpredictingthewhole-bodyresponsetoendotoxin.Wesimulatedynamicsusingdifferentialequationsexaminingthe...

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A unified model for the human response to lipopolysaccharide-induced inflammation Kristen A. Windoloski Elisabeth O. Bangsgaard Atanaska Dobreva Johnny T.

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