This study explores the simplification of a complex and complicated systems pharmacology model and its application to modeling PKPD data. A comprehensive model of the blood coagulation network was simplified and used to model data from a large study of snake envenomations. The simplified version of the model was shown to describe the time course of changes in fibrinogen concentrations following brown snake envenoming, which was comparable to that with the original version of the model.
Use of proper lumping as a model order reduction technique in this study showed that a 62-state blood coagulation network model could be significantly reduced to a 5-state model that described the fibrinogen consumption and recovery after envenomation from a brown snake comparable to the original model. Brown snake venom–fibrinogen serves as an example of an input–output relationship to show the application of proper lumping to reduce the number of states in a systems pharmacology model. The technique, however, can be applied to any other input–output relationship.
The ODEs of the reduced system had to be written explicitly, and the resulting extracted model only describes the brown snake venom–fibrinogen relationship. Lumped models are specific to the input–output being studied and other lumped models will be required from the same systems model for different input–output relationships. Use of an Information Theoretic Approach to assess the simplified model resulted in the list of parameters that were identifiable and thus could be estimated precisely. The structural identifiability method used in this work was based on that proposed by Shivva et al
This method is a local identifiability method based on an information theoretical framework. This could be termed as a “pragmatic” identifiability solution.
In this study, the half-life of fibrinogen was estimated to be 1.5 days. This compares to a value of 1 day from the original 62-state model (as per Wajima et al
Also, this value is similar to another study of taipan bites in which the half-life was found to be 1 day.24
Others have found longer values of half-life between 4 and 5 days,25
and it may need to be explored whether there are other components of snake venom that may eliminate fibrinogen via other mechanisms or there are other processes that get stimulated following snake envenoming. The half-life of brown snake venom was estimated as equal to 55
min. There are no previous studies of brown snake venom in humans that have investigated the half-life of the venom. However, in other snake species like red-bellied black snake, the half-life has been reported to be ~8–12
h (G.K. Isbister, personal communication). This was only able to be determined because the antivenom was not given in many cases of red-bellied black snakebites. In addition, the half-life refers to the presence of venom as a whole. The half-life of brown snake venom estimated in this study refers in particular to the activity of the prothrombin activator in the venom that is Xa:Va like. However, it is possible that the half-life of the functional activity of the procoagulant toxin is different to the half-life measured as the presence of whole venom in serum.
The use of lumping principles has been illustrated previously for systems models in the area of PKPD. These works have concentrated on physiologically based pharmacokinetic models, and only limited attention has been placed specifically on systems models describing pharmacological effects. The studies on physiologically based pharmacokinetic models characterized the pharmacokinetics of a homologous series of barbiturates in rat17
and for compounds like 1,3-butadiene15
that are largely used in the production of plastics and synthetic rubber. Proper lumping principles have also been applied to a systems biology model describing the signaling pathways of NF-κB.19
These studies, however, only discuss the illustration of the lumping process. The work carried out in our study appears to be the first time that a systems pharmacology model has been simplified to model PKPD data.
It might be theoretically possible to perform an identifiability analysis on the original model and then fix unidentifiable parameters. However, this model (i) would be cumbersome to use and computationally intensive to model and (ii) would not be able to be described simply in any report or future work. The theoretical benefit of the methods presented here is the observation that a model can be “extracted” from the original model to accommodate potentially any input–output relationship that is of interest. In addition, the use of mechanistic elements in models provides for greater predictive performance to settings outside of which the model was created. Using the simplification techniques discussed in this study, any multiscale mechanistic model that includes the necessary input–output relationship of interest can be used as a starting point for model extraction and then building. The resulting simplified model preserves the mechanism as seen in the original systems model. This allows for rapid input–output model building that effectively eliminates lengthy structural model selection processes. However, it should be noted that model order reduction techniques are locally dependent on the parameter space. Care should be taken to ensure that the performance of the final reduced model meets the intended needs.
To conclude, a simplified version of a multiscale model was created for exploring fibrinogen recovery after snake envenomation. The simplified model mechanistically aligned with the coagulation systems pharmacology model and was extracted for estimation purposes. The simplification process is multistaged and could benefit from some level of automation. A population PKPD model for fibrinogen concentration–time data based on the mechanisms apparent in the simplified model was developed and was able to describe the recovery of fibrinogen following brown snake envenomation without the need for further structural model development.