**Simplification of a multi-scale systems coagulation model with an application to modelling PKPD data**

Abhishek Gulati (1), Geoffrey K Isbister (2, 3), Stephen B Duffull (1)

(1) School of Pharmacy, University of Otago, Dunedin, New Zealand; (2) Department of Clinical Toxicology and Pharmacology, Calvary Mater Newcastle, NSW, Australia; (3) School of Medicine and Public Health, University of Newcastle, NSW, Australia

**Background**

A comprehensive systems pharmacology model of the coagulation network was recently shown to describe the time course of changes in coagulation factors in response to Australian elapid envenoming [1]. The model consists of 62 ordinary differential equations (ODEs) and 178 parameters with multiple inputs and outputs. Based on any given set of available data relating to a specific input-output process, it is possible that some compartments are either less important or have no influence at all. Fixing the parameters that are not informed by the data would solve the issue of identifiability but not resolve model complexity. In this work, we describe the simplification of a multi‑scale systems coagulation model and its application to describe the recovery of fibrinogen concentrations post‑snake bite. Available data includes timed fibrinogen concentrations in patients with complete venom‑induced consumption coagulopathy resulting from Brown snake envenomation [2]. The patients (N=61) were recruited to the Australian Snakebite Project from over 100 hospitals in Australia between January 2004 and May 2008.

**Aims**

The overall aim of this work was to explore a simplification of a coagulation systems pharmacology model for use in modelling pharmacokinetic-pharmacodynamic (PKPD) data. Four specific objectives were identified: (1) to create a simplified model for exploring fibrinogen recovery after envenomation that mechanistically aligns with the coagulation systems pharmacology model, (2) to extract the simplified model for use for estimation purposes, (3) to assess structural identifiability of the simplified model based on the inputs and outputs available in the dataset and (4) to develop a population PKPD model for fibrinogen concentration-time data based on the mechanisms apparent in the simplified model.

**Methods**

(1) __Simplification of the coagulation systems pharmacology model:__ The technique of proper lumping, based on a previously published method [3], was used to simplify the 62 compartment ("original") model. Fibrinogen and Brown snake venom absorption and plasma compartments were left unlumped. For each of the remaining 59 lumpable compartments, the compartments were lumped randomly and a lumping matrix constructed. This lumping matrix was used to transform the full state parameter vector to the lumped state vector ("lumped" model). The simulated time courses of fibrinogen post Brown snake bite were compared among the lumped and original models to assess for loss of predictive performance. Simulations were carried out using MATLAB^{®} R2011a. (2) __Extraction of the simplified model:__ ODEs of the lumped model were "extracted" from the ODEs of the original model by eliminating the "unwanted" reactions that did not have any influence on the fibrinogen profile. ODEs of the lumped compartments that were formed as a result of merging of various compartments from the original model had to be explicitly written as if they had been unlumped compartments. The clotting factor that was most relevant to the Brown snake venom‑fibrinogen relationship represented its respective lumped compartment. (3) __Identifiability of the simplified model:__ The structural identifiability of the extracted model was assessed using an Information Theoretic Approach [4]. A criterion that consisted of two pre‑defined conditions, as per [4], had to be met for a model to be structurally identifiable. __P__opulation __OPT__imal (POPT) design software was used for the analysis. (4) __Modelling the fibrinogen concentration time data using the simplified model:__ A full population approach was carried out to analyze the fibrinogen data using NONMEM^{®} v7.2. The extracted model was used as the structural model and no further changes were made to the structure of the model. The unidentifiable parameters obtained from Methods (3) were fixed. BSV was considered for parameters that were identifiable. The models with BSV on one or more structural parameters were assessed for significance using the likelihood ratio test that required a decrease in the objective function value of at least 3.84. A visual predictive check (VPC) to evaluate the final model was performed by simulating 1000 replicates from the model and comparing the observed data and the prediction intervals derived from the simulated data graphically.

**Results**

(1) __Simplification of the coagulation systems pharmaco____logy model:__ The original 62 compartment model was lumped to a 5 compartment model that described the Brown snake venom‑fibrinogen relationship. An *in silico* Brown snake venom bite followed by an *in silico* antivenom administration at 4 hours resulted in a similar consumption-recovery profile for fibrinogen using the lumped and original models. Lumping the compartments further significantly reduced the predictive performance of the lumped model. (2) __Extraction of the simplified model:__ Extraction of the ODEs of the lumped model resulted in reduction of the total number of parameters to 11 compared to 178 in the original model. A Brown snake bite using the extracted model resulted in the nadir of fibrinogen depletion to 0.025 g/L compared to 0.018 g/L with the original model. (3) __Identifiability of the simplified model:__ Assessment of identifiability of the extracted model using POPT found that 9 parameters out of the total 11 parameters were identifiable. The remaining two parameters were fixed. (4) __Modelling the fibrinogen concentration time data using the simplified model:__ The decline and eventual recovery of fibrinogen after Brown snake envenomation was described by the 5 compartment model. A VPC showed that the model explained the observed data well. The half-life of fibrinogen was estimated to be 40 hrs (1.5 days) post Brown snake envenomation which was close to a half-life of 1 day observed in patients post Taipan snake bites [5]. The half-life of Brown snake venom was estimated to be equal to 55 minutes and refers to the procoagulant toxin in the venom and not the venom itself.

**Conclusions**

The technique of proper lumping was able to simplify a complicated systems pharmacology model to a much simpler model that retained a clear physical interpretation of the input-output relationship as seen in the original model. Coagulation factors - prothrombin and thrombin seemed to play the most important role in the Brown snake venom-fibrinogen relationship. The technique of structural identifiability analysis identified the parameters that could be estimated precisely after fixing the unidentifiable parameters. The techniques used in this study can be applied to other multi-scale pharmacology models.

**References**

[1] Gulati A, Isbister GK, Duffull SB. Effect of Australian elapid venoms on blood coagulation: Australian Snakebite Project (ASP-17). Toxicon. 2013;61:94-104.

[2] Isbister GK, Scorgie FE, O'Leary MA, Seldon M, Brown SG, Lincz LF. Factor deficiencies in venom-induced consumption coagulopathy resulting from Australian elapid envenomation: Australian Snakebite Project (ASP-10). J Thromb Haemost. 2010;8(11):2504-13.

[3] Dokoumetzidis A, Aarons L. Proper lumping in systems biology models. IET systems biology. 2009;3(1):40-51.

[4] Shivva V, Korell J, Tucker IG, Duffull SB. Identifiability of Population Pharmacokinetic-Pharmacodynamic Models. Population Approach Group in Australia and New Zealand (PAGANZ); University of Queensland, Brisbane, Australia. 2013.

[5] Tanos PP, Isbister GK, Lalloo DG, Kirkpatrick CM, Duffull SB. A model for venom-induced consumptive coagulopathy in snake bite. Toxicon. 2008;52(7):769-80.