- Open Access
Mathematical modeling of septic shock based on clinical data
© The Author(s). 2019
- Received: 4 October 2018
- Accepted: 11 February 2019
- Published: 6 March 2019
Mathematical models of diseases may provide a unified approach for establishing effective treatment strategies based on fundamental pathophysiology. However, models that are useful for clinical practice must overcome the massive complexity of human physiology and the diversity of patients’ environmental conditions. With the aim of modeling a complex disease, we choose sepsis, which is highly complex, life-threatening systemic disease with high mortality. In particular, we focused on septic shock, a subset of sepsis in which underlying circulatory and cellular/metabolic abnormalities are profound enough to substantially increase mortality. Our model includes cardiovascular, immune, nervous system models and a pharmacological model as submodels and integrates them to create a sepsis model based on pathological facts.
Model validation was done in two steps. First, we established a model for a standard patient in order to confirm the validity of our approach in general aspects. For this, we checked the correspondence between the severity of infection defined in terms of pathogen growth rate and the ease of recovery defined in terms of the intensity of treatment required for recovery. The simulations for a standard patient showed good correspondence. We then applied the same simulations to a patient with heart failure as an underlying disease. The model showed that spontaneous recovery would not occur without treatment, even for a very mild infection. This is consistent with clinical experience.
We next validated the model using clinical data of three sepsis patients. The model parameters were tuned for these patients based on the model for the standard patient used in the first part of the validation. In these cases, the simulations agreed well with clinical data. In fact, only a handful parameters need to be tuned for the simulations to match with the data.
We have constructed a model of septic shock and have shown that it can reproduce well the time courses of treatment and disease progression. Tuning of model parameters for each patient could be easily done. This study demonstrates the feasibility of disease models, suggesting the possibility of clinical use in the prediction of disease progression, decisions on the timing of drug dosages, and the estimation of time of infection.
- Septic shock
- Model-based therapy
- Blood pressure
- Immune system
Sepsis is a highly complex, life-threatening systemic disease caused by infection and has a high mortality rate. The number of sepsis patients is estimated to be around 27 million per year globally, of whom approximately 8 million people die, and the number of sepsis patients is increasing . The disease is sometimes referred to as the most common but least recognized disease . In the most severe form of sepsis, called septic shock, underlying circulatory and cellular/metabolic abnormalities are profound enough to substantially increase mortality  and the effects of inflammation produced by the immune system spread systemically and induce an acute systemic disorder . Patients with septic shock must be treated urgently in an intensive care unit. Because of its complexity, the progression of septic shock varies from patient to patient, depending on age, sex, physical characteristics, physiological activity, underlying disease, and other factors. Therefore, treatment is largely based on doctors’ skill obtained through practical experience, as is usually the case in the treatment of other diseases. Although several standard guidelines are available [5, 6], more effective, versatile, and reliable therapeutic strategies for emergency medicine are currently being sought.
The art of medicine, which emphasizes the individuality of patients, must be supported by a solid scientific understanding of disease based on human physiology. The art and science of medicine should be integrated in clinical practice at a much higher level than at present.
In the physical sciences and engineering, most of the knowledge accumulated to date about devices, components, and systems has been represented by models, most of which are presented quantitatively (mathematically). These models are available in various forms, such as scientific papers, patents, and software packages, and are used extensively as a concise representation of accumulated scientific knowledge in the research and development of new devices, components, and systems.
Accurate models of disease based on physiology and pharmacology could contribute to improving the treatment of diseases. Doctors could use such models to estimate the physiological state of their patients, predict the disease progression, and decide on treatment strategies, including the administration of drugs. Models could therefore provide a unified scientific background to clinical practice. Rami et al. extensively discussed and presented persuasive reasoning along these lines based on a historical review of treatment for sepsis . They aimed to demonstrate the potential of disease models in therapy and open the door to model-based therapy.
One problem with models is to incorporate the individuality of patients. We anticipate that individual differences can be accommodated by choosing model parameters carefully based on the patients’ age, weight, sex, physiological status, underlying diseases, and other factors. Modern hospitals are well equipped with advanced diagnostic systems that would allow the easy customization of a disease model for each patient. In addition, mathematical models could help to promote a deeper understanding of diseases and establish a hypothesis of pathogenesis, improving our understanding of treatment methods.
Due to the complexity of disease physiology, it is difficult to model human diseases, and most mathematical models of disease physiology have so far focused on experimental animals, except models of diabetes and Parkinson’s disease. Treatment strategies have been developed for diabetes based on mathematical models . The model developed by Kovatchev et al.  was approved by the US Federal Drug Administration as an alternative to animal research for the approval of diabetes medications. Recently, we have constructed a diabetes model that includes brain-centered glucose metabolism and suggested an alternative therapeutic strategy for diabetes . A mathematical model based on brain metabolism has been constructed for Parkinson’s disease and is recognized as a useful tool for investigating its pathogenesis [11, 12]. The importance of mathematical models in understanding the basic physiology in the progression of sepsis has been highlighted in previous work . In addition, Kendrick et al. published a clear description of the immune response to sepsis , and Shi et al. discussed a bifurcation analysis of sepsis based on an immune system model .
In this study, we aimed to construct a new mathematical model of septic shock based on clinical data. Among the diverse symptoms of sepsis patients, we focused on the damage caused to the cardiovascular system because septic shock most frequently damages the cardiovascular system. Our model combines the cardiovascular system, immune system, and pharmacological models, and we used existing models of these systems as our guiding tools [16, 17]. We focused on how inflammation resulting from immune activity affected the cardiovascular system and caused septic shock. Among the many possible effects of inflammation on the cardiovascular system, we selected increased vessel permeability, vasodilation, and reduced stroke volume [18, 19]. We incorporated these three factors into the combined model of the cardiovascular and immune systems, making the resulting model highly nonlinear. Through simulations, we showed that these three factors are sufficient to reproduce septic shock.
To complete the sepsis model, the nervous system and pharmacological responses to drug administration must be incorporated because they are crucial to the disease model. We could not measure the activity of the nervous system, but we incorporated qualitative physiological and empirical data to achieve a quantitative description in our model to reflect realistic physiological effects. The activity of the nervous system is weaker in patients with sepsis than in healthy people; thus, we introduced fatigue as a parameter of the sympathetic nervous system [20, 21]. In addition, the effects of drugs are reduced in sepsis patients compared with healthy people. Therefore, we used experimental data showing the reduced effects of an antihypertensive medicine in sepsis patients.
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.  because it is simple but captures the essential features of the immune system that are relevant to our sepsis model. We incorporated the effect of antibiotics into this model, following the proposal of Kitamura .
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.
Next, we briefly describe each model.
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 .
If ar increases above a0r, then Kr, cr and resistance R decrease.
The normal arterial blood pressure, Pra0, and systemic blood pressure, Psa0, depend on individual patients.
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.  because their model is simple but captures some essential features of the immune system that are relevant to sepsis.
The interactions among these variables are described as follows.
Here, X1 denotes antibiotic dosage, X2 its blood concentration, ka the absorption coefficient and ke the degradation coefficient.
The first term of equation (24) is a simplified representation of the initiation of inflammation caused by P, D, and N∗ represented by their linear combination in equation (25), and snr and μnr are Michaelis–Menten parameters for the inflammatory reactions. Function g introduced in equation (20), which represents the saturating factor due to the presence of anti-inflammatory mediator CA, is also used to represent the initiation of inflammation. The second term represents the degradation.
Here, Sc denotes a source of CA and the second term represents the production of CA from damage D and inflammation N∗ by a Michaelis–Menten term with inhibition mediated by CA itself. The third term represents the degradation. More detailed descriptions are found in Reynolds et al. .
Lowered blood pressure in septic shock is treated by infusion and drugs. Infusion is represented by the term Qinf in equation (4). An infusion may contain many blood components and varies according to the condition of the patient. However, we omitted a detailed description of the components and assume the infusion to be 0.9% saline.
There are several drugs used to treat severe hypotension in sepsis patients, of which noradrenaline and dopamine are the most commonly used in clinical practice. Antibiotics are also used to dispose the pathogen and are represented by the fourth term of equation (19).
The parameters in group 1 are used throughout the whole simulation, even before infection. They represent the physical characteristics of the patient, such as weight, sex, and underlying diseases. All parameters in the cardiovascular system are taken from the paper by Ursino and Innocenti . Their numerical values are shown in Table A1 of the Appendix.
The parameters related to the nervous system used in equations (9)–(18) are shown in Table A2 in the Appendix. Some of them are taken from Ursino and Innocenti’s paper , and others are estimated mainly based on the literature. Some parameters are fitted based on MATLAB tools to minimize the gap between data and simulation. .
The initial value of the total blood volume V depends on sex and weight. We assume that the total blood volume is 8% of body weight for men and 7% for women . We also consider the possibility of heart failure as an underlying disease. The quantitative description of heart failure is presented in the Results section.
The parameters in group 2 are used after the initiation of infection and include the immune system parameters. We take these parameters from the paper by Reynolds et al.  (Table A3). This group also contains parameters that represent the effects of infection on the cardiovascular system. The most important parameters in this group are those that represent the increase in blood permeability La denoted by equation (5). There are several papers that report attempts to measure blood permeability. It was reported that the maximum value of La during infection is almost 6 times greater than normal La [30–32], and we used this observation in our model. The other parameters in equation (29) are chosen by tuning and are listed in Table A4 in the Appendix.
The growth rate of the pathogen, given by parameter kpg, is used to represent the severity of the infection. Other parameters are taken from the model reported by Reynolds et al. .
The parameters in group 3 are pharmacological parameters that represent drug efficacy . The dose-response curve of noradrenaline is represented by sigmoid function in equation (32) and the numerical values of the associated parameters have been experimentally obtained for healthy subjects . The dose-response curve for sepsis patients may differ from that for healthy people. The numerical values of the parameters in equation (30) are listed in Table A5.
We validated the model in two steps. In the first step, we established a standard patient model capturing some essential features of sepsis progression and treatment effect, at least qualitatively. For this purpose, the relationship between the severity of the infection and the difficulty of recovery was important in the disease model. We represented the severity of infection through the value of parameter kpg in equation (19), which describes the growth rate of the pathogen.
We took heart failure as a representative example of an underlying disease in patients due to the strong link between sepsis and cardiac insufficiency [33, 34]. A typical consequence of heart failure is a reduction in stroke volume. According to the European Society of Cardiology guidelines published in 2016 , heart failure is defined as a circulatory condition in which the ejection fraction (EF) is below 40%, where EF is defined as the ratio of the left heart cardiac stroke volume to the left heart blood volume. Normal EF is between 50 and 60%. We noticed that if the left cardiac stroke volume Sl in equation (3) was reduced by 22%, we obtained a 40% drop in EF, which is consistent with the definition of heart failure with reduced EF. Therefore, we used this reduction in Sl to represent heart failure as the underlying disease.
We classified the severity of infection as mild, moderate, and severe, based on the range of parameter kpg. Sepsis progression was represented in the time courses of mean arterial pressure (MAP) and heart rate. The recovery can be judged when the time course of MAP and heart rate returned to the normal or original level.
For moderate infection (kpg = 0.45), the internal immune system alone cannot control the effect of inflammation and the blood pressure continues to decrease (Fig. 7(b)). However, infusion can prevent the decrease in blood pressure. Blood pressure does not decrease even after the infusion rate is reduced to the minimum level after 1 h of intensive infusion, which is consistent with clinical data.
For severe infection with a high kpg (kpg = 1.50), infusion alone is not enough to raise the blood pressure and noradrenaline is needed (Fig. 7(c)).
The two in silico experiments show that our model reproduced the progression of septic shock and the outcome of the treatment, at least qualitatively.
Basic Information about Patients
A sudden drop in blood pressure occurred 17 h after treatment began, which the model did not reproduce.
This patient was stable with low blood pressure and high heart rate when given infusion therapy. The administration of dopamine contributed to the elevation of the heart rate, which was reproduced in the simulation.
The gradual recovery of the blood pressure and the associated normalization of the heart rate due to infusion and noradrenaline administration are closely reproduced by the simulation.
Patients parameter information
Normal value of heart rate f(f0)
Normal value of arteriovascular radius r(r0)
Initial total blood Volume(V0)
Rate of vessel radiusr dilation(kEX)
Compliance of sa(Csa)
Resistance of the systemic circulation upstream of arteriolar capillaries(Rs2)
Resistance of the systemic circulation downstream of arteriolar capillaries(Rs3)
Resistance of sv(Rsv)
Resistance of pa(Rpa)
Resistance of pv(Rpv)
Rate at which activated phagocytes(N*) consume pathogen(kpn)
An important parameter that is not listed in Table 2 is the time interval between the time of infection and the start of treatment. At the start of the simulation, the initial value of pathogen P is set to be positive. This implies that the starting time of the simulation is the time of infection. We must decide when treatment is started (when the initial data was obtained) based on the goodness of fit between data and simulations. This tuning is one-dimensional and was not difficult. It conveys, however, valuable information about when the patient was infected before hospitalization.
Severity and intensity of treatment
Infusion and medicine
The results show that the severity of the infection matched the intensity of treatment required for recovery.
Many sepsis patients have underlying diseases; thus, we used heart failure as an example of an underlying disease that can be modeled by changing some parameters. We showed that the patient with heart failure does not recover from even a mild infection without treatment, which is consistent with clinical experience. These simulations show that our model captures the essential features of sepsis and that the interactions quantified in our model among the immune, cardiovascular, and nervous system submodels are justified, at least qualitatively.
In the second part of the simulation, we validated our model to fit clinical data of three sepsis patients (Figs. 9–11). As stated in the Background section, the progression of sepsis differs among patients, and the symptoms are also different. Although it is necessary to customize the models for each patient by choosing appropriate model parameters, there are a large number of parameters making customization appear difficult. However, as was shown in Table 3, the number of parameters tuned for fitting was not very many and most parameters were unchanged from the general model of a standard patient used for the first part of simulations (Figs. 7 and 8). The most obvious differences were body weight and sex, which affect the total blood volume. Age differences were taken into account by choosing the vessel flexibility and radius, and the strength of the sympathetic nerve activity was tuned slightly to accommodate the data. They are natural, easy, and reasonable customizations for individuality.
Heart failure is included as an underlying disease in patient 1. The sudden drop in blood pressure is accommodated by a sudden drop in heart rate, which is usually a symptom of heart failure. This suggests the possibility that the model can explain sudden events occuring during the course of treatment. For patient 3, we ignored diabetes as an underlying disease, but still obtained a good fit between the model and data.
The most finely balanced and important parameter for fitting the model to real data is the time of infection or the starting time of the simulation. Typically, a patient has already been infected when admitted to the hospital and does not know when they were infected. As discussed in the Method sections, we could estimate the time of infection through a one-dimensional search. Estimating the time of infection with the model by finding the most appropriate initial time of simulation gives valuable information for determining the treatment strategy for a patient. This is an important benefit of disease modeling.
Because sepsis is a serious disease that affects almost all parts of the body, it may lead to other subsequent diseases, which we have not incorporated in the model. However, even when an unexpected physical event occurred, the model could identify the cause. Patient 1 had a sudden drop in blood pressure during treatment (Fig. 12), which was explained by a heart attack (Fig. 14). Although this was an estimate, it could provide valuable information for medical staff.
We have constructed a simple mathematical model of septic shock to represent and predict disease progression and the effects of treatments. The model combined the cardiovascular and immune systems through the effects of inflammation, which are represented by increases in vessel permeability, vasodilation, and stroke volume reduction. We assumed the following three effects of sympathetic nerve activity responding to severe hypotension caused by infection: elevated heart rate, increase in vessel resistance, and decrease in blood volume unloading. We also introduced the fatigue effect of the sympathetic nervous system. The weaker effects of drugs in sepsis patients were also considered.
We demonstrate that our model is a reasonable model of septic shock and represents the therapeutic effects of treatment through in silico experiments. We also show the reduced therapeutic effects in patients with sepsis who have underlying heart failure.
We validated our model based on the clinical data of three sepsis patients and showed that the model reproduced the treatment course. Moreover, the model reproduced sudden physiological events in patients.
Although our model represents specific aspects of septic shock, which is complex and involves almost every organ in diverse ways, we show that we can construct a model that captures the essential features of this disease. We discuss the potential of the model to help with clinical decision making and promoting a deeper understanding of sepsis.
Because the model contained a number of parameters that must be set for simulations, the difficulty in determining appropriate numerical values has been identified as one of the main barriers to using mathematical models in medicine. The customization of models for individual patients is an additional difficulty. However, we found that a standard patient model can be constructed based on the existing physiological, medical, and pharmacological knowledge, as described in the first part of the Results section, although some parameters had to be taken from experimental data on animals. We were able to customize the standard patient model for three patients based on their age, sex, weight, and underlying disease, by tuning only several parameters of the standard patient model. The simulations well reproduced the data.
Our results suggest that disease modeling could help medical staff predict the patient’s condition and establish a clinical strategy for recovery. The possibility of estimating the infection time before treatments start is another benefit of the disease model.
The disease model extracts knowledge about human physiology relevant to the target disease, and the model can be customized by selecting relatively small number of parameters. We consider that the general and individual data are accommodated well in the model and that their integration can bring great benefits to clinical practice. We hope that our model will play a role in guiding practitioners toward model-based therapies. To achieve this goal, our model must be more reliable and versatile, and must be validated using a larger amount of clinical data, which we intend to tackle in the next step of our research.
This research is a part of a joint project between Toyota Motor Co., Waseda University, and Tokyo Women’s Medical University, titled Disease Modeling for Clinical Use.
This work was supported by the Toyota Corporation.
Availability of data and materials
Clinical data can be supplied on demand. All other data are available as indicated in the references.
YY: Simulations and modeling. KU: Regular and continuous inspection and discussion. MA: Modeling, simulation and literature research. YW: Simulations and modeling. AY: Medical data acquisition and overall advice. SS: Medical data acquisition. SS: Modeling. HY: Modeling. MY: Modeling. HK: Initiation of the project and overall supervision. All authors read and approved the final manuscript.
Ethics approval and consent to participate
The study protocol in this paper was approved by the Waseda University Ethics Review Committee on research with human subjects (No. 2016–086) and the Tokyo Women’s Medical University Institutional Review Board (No. 4276), and the need for patient consent was waived.
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