Introduction
The coagulation network model is used to describe the relationship between an anticoagulant, or pro-coagulant, and a clotting test outcome [1]. The system consists of an in-vivo (62 ODEs and 184 parameters) and in-vitro (62 ODEs and 182 parameters) modules. These two modules are linked by a discontinuous interface with the whole function not being continuously differentiable. The model has been adapted to describe the dose-response relationship of heparin in children [2]. However, due to the large number of states and parameters, the model is not amenable to typical estimation analyses and stochastic simulations. Additionally, dimensionality and discontinuity at the interface of the two modules creates difficulty for parametric model-order reduction (MOR) methods. This work explores the use of a structure-independent non-parametric MOR approach using artificial neural networks (ANNs). The heparin dose-response input-output (I/O) relationship in children from the coagulation network model was used as a motivating example.
Methods
The MOR technique considered in this work consisted of two steps: 1) Simulation of pseudo I/O data from the full-order model; 2) Training and validation of a minimal ANN architecture that can approximate the target I/O relationship up to an arbitrary accuracy.
Three datasets were simulated from the full-order model: (1) a training dataset to estimate ANN parameters, (2) an evaluation dataset to determine when training should be stopped, and (3) a validation dataset to validate the performance of the trained ANN on previously unseen combinations of input variables. The training and evaluation datasets were simulated simultaneously by randomly generating 10,000 sets of input variables from pre-specified distributions that were then randomly split into a training (80% of the data) and evaluation (remaining 20%) datasets. A validation dataset was simulated so that it contains combinations of input variables values that are unlikely to be present in the training and evaluation datasets. Additionally, the performances of both the reduced and full-order models were compared against a clinical dataset for 31 children who received heparin infusion during extracorporeal membrane oxygenation procedure. All experiments were performed in MATLAB (Release 2018b The MathWorks, Massachusetts, US).
Results
The minimum network architecture consisted of 7 nodes, 2 hidden layers, and 43 parameters which achieved a mean squared error (MSE) of 10-3. In comparison, to achieve an MSE of 10-6 a network with 25 nodes, 4 hidden layers and 179 parameters was required. The median training time of ANNs was 17.3 seconds (range 1.9 – 59.9 seconds) per network. Re-simulation of the whole pseudo-data sets through trained neural network models with different accuracy levels took a median of 0.02 seconds compared to 5.8 hours for simulation through the full-order model. The performance of all trained ANNs on the evaluation and validation datasets was comparable to that of the training set indicating good capability of interpolating outputs for previously unseen inputs. All trained networks performed similarly well as the full-order model for prediction of both activated partial thromboplastin time and anti-Xa responses in children receiving heparin infusions.
Conclusions
In conclusion, the proposed MOR technique using ANN enables the development of efficient approximations to complex models within a desired level of accuracy. The technique is applicable to a wide variety of QSP models and provides a substantial speed boost for use of such models in simulation, estimation, and potentially control purposes. Additionally, the proposed technique did not require any experimental data. This technique is also adaptable to a parametric framework where features of interest from the full-order model are retained in the reduced-order model and become available for further exploration.
References
- Wajima T, Isbister GK, Duffull SB (2009) A comprehensive model for the humoral coagulation network in humans. Clin Pharmacol Ther 86 (3):290-298. doi:10.1038/clpt.2009.87
- Derbalah A, Duffull S, Moynihan K, Al-Sallami H (2020) The Influence of Haemostatic System Maturation on the Dose–Response Relationship of Unfractionated Heparin. Clin Pharmacokinet. doi:10.1007/s40262-020-00949-0