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Therefore, we used experimental data showing the reduced effects of an antihypertensive medicine in sepsis patients. We constructed a mathematical model that represents the physiological dynamics of septic shock after infection and comprises cardiovascular system, immune system, nervous system, and pharmacological submodels. An overview of our model is shown in Fig. There are various cardiovascular, nervous, and immune system models for different uses in the literature.
Most of these models are closed in the single-target domain, although they must be connected to represent the disease. In this study, we focused on integrating these models, based on choosing appropriate existing models for the sepsis model. We used the cardiovascular system model proposed by Ursino and Innocenti [ 17 ], which is comprehensive and includes the solute kinetics of each constituent in blood, as well as the sympathetic nervous system. Because the increase in vascular permeability is an important effect of inflammation on the cardiovascular system, the solute kinetics of the systemic capillaries in the model are essential in our sepsis model.
The immune system is complex, and quantitative models are still incomplete [ 22 , 23 ]. We based our sepsis model on the model reported by Reynolds et al. We incorporated the effect of antibiotics into this model, following the proposal of Kitamura [ 24 ]. The core of our sepsis model is in the link between the cardiovascular and immune systems. In other words, we model how inflammatory responses damage the cardiovascular system.
As stated in the Background section, we considered the three effects of inflammation on the cardiovascular system—increased vessel permeability, vasodilation, and reduced stroke volume—all of which contribute to reducing blood pressure. To quantify these effects, we represented the three parameters as functions of inflammation. Because inflammation manifests in diverse ways, it is hard to represent as a simple physical quantity; it is more an abstract and collective quantity.
In contrast, permeability, vasodilation, and stroke volume are tangible physical parameters with clear units of measurement. The model connected these physical parameters with an abstract representation of the severity of inflammation. This was an unavoidable difficulty and an intriguing aspect of sepsis modeling. The cardiovascular system model is composed of five compartments, namely, the pulmonary atrium pa , right atrium ra , left atrium la , systemic arteries sa , and the systemic veins sv Fig. Each compartment is described by its volume V , pressure P , incoming flow rate q in , outgoing flow rate q out , and compliance C representing the compartment capacity, subject to the conservation of mass.
The right cardiac output q r and left cardiac output q l are represented by. The solute kinetics of the capillary system that transports the blood components to the tissues are important in our model. We focus on the material exchange between vessels and the interstitial fluid. The total blood volume V is subject to the following transport law:. Outflow F a from the vessel to the interstitial space and inflow R v in the opposite direction in eq. Coefficient L a in equation 5 denotes vessel permeability, which is important in our model, whereas coefficient L v in equation 6 denotes another permeability characterizing the opposite blood flow, which is considered to be irrelevant to damage.
In reality, there are more inputs and outputs that affect the total blood volume, such as the blood carried to the kidneys. However, we neglected these other factors because their contributions are relatively small.
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The more detailed solute dynamics associated with equations 5 and 6 are described by Ursino and Innocenti [ 17 ]. The unloaded volume is the part of blood reservoir in the heart that does not circulate. The baroreflex is governed by the sympathetic nervous system, which elevates the blood pressure when the baroreceptors detect a decrease in blood pressure.
The increase in blood pressure is achieved via elevation of the heart rate, increased vascular resistance, and increased venous blood volume unloading [ 25 ]. Let a be the action of the sympathetic nervous system. The elevation in heart rate mediated by sympathetic action is described as. We assume that sympathetic nerve activity decreases the vessel radius as.
If a r increases above a 0 r , then K r , cr and resistance R decrease. Finally, the unloaded blood volume V u in equation 7 is assumed to be reduced by the sympathetic nervous system in the same way as in equation 10 ,. The reduction of the unloaded volume implies an increase in circulating volume V f due to equation 5 , assuming that total volume V is fixed. Now we quantify the baroreflex and its fatigue.
Let X be the average output of the baroreceptors that detect the right arterial blood pressure, P ra , and the systemic arterial blood pressure, P sa , which is assumed to be.
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The sympathetic nervous system responds to the decreasing pressure signal represented by. The normal arterial blood pressure, P ra 0 , and systemic blood pressure, P sa 0 , depend on individual patients. We assume that a changes between its minimum, a min , and maximum, a max , due to a change in X.
Hence, equation 15 implies.
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If sympathetic nerve activity is sustained above its normal level for a long time, then the action gradually decreases due to fatigue e. Equations 17 and 16 are nonlinear differential equations. Sepsis is caused by excessive inflammation triggered by the immune system after infection. The dynamics of the immune system play an important role in evaluating the progression of sepsis.
However, because the immune system is complex, mathematical models of immune system dynamics are not well developed, although there have been several attempts to quantify the dynamics [ 23 , 25 ]. We base our sepsis model on the model proposed by Reynolds et al. Overview of the immune system model [ 16 ].
The second term represents the non-specific local immune response toward P characterized by the Michaelis—Menten equation [ 16 ]. The forth term represents the effect of antibiotic dosage proposed by Kitamura [ 24 ]. Here, C f denotes the free concentration of antibiotic, which is subject to the following dynamics. Function g introduced in equation 20 , which represents the saturating factor due to the presence of anti-inflammatory mediator C A , is also used to represent the initiation of inflammation.
The second term represents the degradation. The third term represents the degradation. More detailed descriptions are found in Reynolds et al. Damage D is an abstract quantity in the paper by Reynolds et al. There are several ways to identify cardiovascular damage, and we take reduced stroke volume S l , introduced in equation 3 , because it affects the whole system substantially. We describe the damage as. Next, we quantify how inflammation lowers blood pressure.
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The most important factor is the increase in the permeability of the capillaries due to inflammation [ 16 , 18 ]. In our model, capillary permeability is represented by coefficient L a in equation 5. Equation 29 is in the same form as equation Finally, we assume that inflammation damages the function of the heart substantially [ 13 , 23 ]. We assume that inflammation decreases the left stroke volume S l defined by equation 3 as.
Lowered blood pressure in septic shock is treated by infusion and drugs. Infusion is represented by the term Q inf in equation 4. An infusion may contain many blood components and varies according to the condition of the patient.