Mathematical modeling of septic shock based on clinical data

Abstract

Background: 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.

Results: 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.

Conclusions: 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.

Keywords: Blood pressure; Immune system; Inflammation; Model-based therapy; Septic shock.

Fig. 1

Overview of the sepsis model

Fig. 2

Cardiovascular system model

Fig. 3

Overview of the immune system model [16]

Fig. 4

Experimental results of the effects of noradrenaline on mean arterial pressure. [27]

Fig. 5

Noradrenaline effect model (dose-response curve)

Fig. 6

Parameter classification

Fig. 7

(a) Patient with a mild infection (k_pg = 0.2). (b) Patient with a moderate infection (k_pg = 0.45). (c) Patient with a severe infection (k_pg = 1.50). Time courses of sepsis development

Fig. 8

(a) Mild infection with heart failure. (b) Moderate infection with heart failure. (c) Severe infection with heart failure. Time courses of sepsis development in patients with heart failure

Fig. 9

Clinical data for patient 1

Fig. 10

Clinical data for patient 2

Fig. 11

Clinical data for patient 3

Fig. 12

Comparison of the simulation results and clinical data for patient 1

Fig. 13

Simulation results after f_noise was introduced

Fig. 14

Comparison of the simulated and clinical data after f_noise was introduced

Fig. 15

Comparison of the simulation results and clinical data for patient 2

Fig. 16

Comparison of the simulation results and clinical data for patient 3