Abdallah Derbalah (1), Hesham Al-Sallami (1), Stephen Duffull (1)
(1) School of Pharmacy, University of Otago, Dunedin, New Zealand
Objectives: The coagulation network model is a QSP model that describes the coagulation process as a series of ordinary differential equations representing the interaction of positive and negative feedback and feedforward reactions that result in the formation and degradation of a fibrin clot [1]. The model, however, is complex with a large number of parameters and states and is not easily amenable to typical estimation analyses and stochastic simulations. Additionally, parameter estimation of complex non-linear models may be unreliable if the model is not formally evaluated for identifiability.
Several techniques for model-order reduction (MOR) have been proposed to address the issue of model complexity [2]. The choice of a particular method is typically restricted by the nature of the original model. The QSP model example considered here consists of two modules, an in vitro module which characterises the normal function of the coagulation network, and an in vivo module which represents a clotting time test (e.g. INR). Both modules are connected through a discontinuous high dimensional interface (62-state vector). The in vitro module is initiated with a snap shot of the in vivo concentrations of all the 62 components at a specific point in time. The dimensionality and discontinuity of the interface creates a difficulty for the application of parametric MOR techniques. Two MOR techniques have been performed on the coagulation QSP model. The first considered only the in vivo component [3]. The second example was an empirical approximation specific for warfarin and INR [4]. Neither example provides a generalised solution.
In this work, we explored the use of artificial neural networks (ANNs) as an MOR technique to approximate the multidimensional non-linear I/O relationships within QSP models.
Methods: Heparin dose-response in children was chosen as the target I/O relationship for this model reduction approach. This relationship has two features that make this a difficult problem. First, the model contains multiple sources of nonlinearity from the system itself, which has both amplification and dampening mechanisms, and secondly the presence of nonlinear maturation of the system and drug pharmacokinetics in young children. In addition, this specific I/O relationship spans both in vivo and in vitro modules. In this example, the inputs consisted of four variables that pertain to dosing information (infusion amount and duration) and patient covariates (weight and age). The output was a 2-dimensional vector representing the QSP model predicted anti-Xa activity (aXa) and activated partial thromboplastin time (aPTT).
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 different types of datasets were simulated from the full-order model. The training and evaluation datasets were simulated simultaneously by randomly generating 10,000 sets of input variables from pre-specified distributions. This was used to obtain corresponding outputs from the full-order model. The I/O pairs were split into a training set (80% of the data) which was used to estimate ANN parameters and an evaluation set (remaining 20%) used to determine when training should be stopped. The third dataset was the validation dataset which was simulated as 4 sets of 10,000 I/O pairs each with input variables being generated from different distributions than that of the training dataset. Therefore, the validation dataset is likely to contain combinations of variables that are unlikely to be seen in the training dataset. Additionally, the performances of both the reduced and full-order models in describing a clinical dataset from children receiving heparin infusion were compared.
The Levenberg-Marquardt algorithm was used for feedforward ANN training with the normalised mean squared error (MSE) as the loss function. Training was stopped when a maximum of 1000 iterations or an evaluation MSE goal is reached. All experiments were performed with four different MSE goals (10-3, 10-4, 10-5, and 10-6). A binary search-based algorithm[5] was proposed for automatable selection of neural network architecture. The algorithm generates an exhaustive list of all possible network architectures with a given maximum number of parameters then sorts that list in order of increasing approximation power. A modified binary search was then applied to find the minimum architecture that achieves a given MSE goal. The maximum architecture was specified as one where the number of parameters in the ANN was the same as the number of parameters in the full-order QSP model. All experiments were performed in MATLAB(Release 2018b The MathWorks, Massachusetts, US).
Results: The binary search algorithm required a total of 13 training experiments to find the minimum network architecture that achieved a given MSE goal. The minimum network architecture required to approximate the coagulation model was 7 nodes, 2 hidden layers, and 43 parameters which achieved an MSE of 10-3. A performance goal of 10-6 was achievable through a network with 25 nodes, 4 hidden layers and 179 parameters. 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. However, there were common areas where all ANNs performed less well compared to other areas reflecting a potential lack of adequate training data. Nonetheless, larger networks performed better than smaller ones in those training deficient areas. This suggests that larger networks might be able to learn to generalise better to areas with less training data and hence may have lesser data requirements. All trained networks performed similarly well as the full-order model for prediction of both aPTT and aXa responses in children receiving heparin infusion.
Conclusions: In conclusion, the proposed MOR technique using ANN enables the development of efficient approximations to complex models with the desired level of accuracy. The technique is applicable to a wide variety of QSP models and provides 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 parameters of interest from the full-order model are retained in the reduced-order model and become available for further exploration.
References:
[1] Wajima, T., G.K. Isbister, and S.B. Duffull, A comprehensive model for the humoral coagulation network in humans. Clin Pharmacol Ther, 2009. 86(3): p. 290-8.
[2] Snowden, T.J., P.H. van der Graaf, and M.J. Tindall, Methods of Model Reduction for Large-Scale Biological Systems: A Survey of Current Methods and Trends. Bulletin of Mathematical Biology, 2017. 79(7): p. 1449-1486.
[3] Gulati, A., et al., Application of Adaptive DP-optimality to Design a Pilot Study for a Clotting Time Test for Enoxaparin. Pharm Res, 2015. 32(10): p. 3391-402.
[4] Ooi, Q.-X., et al., A factor VII-based method for the prediction of anticoagulant response to warfarin. Scientific Reports, 2018. 8(1): p. 12041.
[5] Parmar, V.P. and C. Kumbharana, Comparing Linear Search and Binary Search Algorithms to Search an Element from a Linear List Implemented through Static Array, Dynamic Array and Linked List. International Journal of Computer Applications, 2015. 121(3).
Reference: PAGE () Abstr 9284 [www.page-meeting.org/?abstract=9284]
Poster: Oral: Methodology - New Tools