James Lu, Dan Lu, Logan Brooks, Jin Y. Jin
Modeling & Simulation, Clinical Pharmacology, Genentech
Introduction: The task of estimating parameters in QSP and PMx models can be challenging. The existing algorithms such as expectation-maximization, genetic algorithms and scatter search are optimization-based, requiring many function and gradient evaluations, necessitates specialized software tools, and may entail significant estimation time and trial-and-error. Recent advances in deep learning [1] have given rise to the development of “software 2.0” [2], whereby the underlying algorithms are not explicitly designed by humans. Instead, machines themselves develop the appropriate algorithms by having been trained on extensive data sets. Inspired by the broad use of data visualization and graphical assessment techniques in the modelling community [3], we propose that convolutional neural networks (CNNs) with architectures similar to those used in visual processing applications [1] can be utilized to perform parameter estimation for QSP and PMx models.
Objectives: We investigate how machine learning can be used to generate efficient parameter estimation algorithms. In particular, we develop deep CNNs that learn in a supervised manner to perform the mapping from experimental data-to-parameters, from having been trained with QSP and PMx model simulations which provide extensive instances of parameters-to-data. As test examples, we consider the following two tasks of estimating myelosuppressive drug effects: from in-vitro data using a QSP model [4] and from in-vivo data using the Friberg model [5].
Methods: Given a model of interest, we first sample model parameters to generate simulated data to train the CNN. For the in-vitro heme toxicity QSP model (consisting of a system of 15 Ordinary Differntial Equations (ODEs) and 13×2 Emax and EC50 parameters [3]), we randomly sampled 10^6 parameter sets within the pre-specified bounds. For each parameter set drawn, we simulated the dose-response for 6 different drug concentrations (ranging from 0.2 to 2500 nM) and performed the resulting 6×10^6 of model simulations on a high performance computing cluster using Matlab Simbiology R2019a [6]. We trained a CNN that consists of an “encoder” [7] from data-to-parameter followed by a “decoder” [7] that maps parameter-to-data reconstruction (see [8] for a representative illustration), with the architectures of the two subnetworks in mirror image. For the in-vivo Friberg model [5] (with 5 ODEs and 4 parameters), we similarly generated a simulated dataset based on randomly sampled model parameters (system parameters “circ0”, “ktr” and “gamma” as well as the drug effect parameter “slope”) using Mathematica v12 [9]. Sparsely sampled PK and PD time series data are used as training dataset for CNNs to estimate the “slope” parameter of the Friberg model, for both the virtual population as well as individual patients. The computational speed and accuracy of the CNN derived algorithm are first compared against the existing algorithms using a set of simulated validation data. The proposed CNN is subsequently applied to new experimental data and the results are compared against standard estimation results from traditional QSP and PMx methods.
Results: For the QSP model, we show that on an unseen simulated data the CNN estimated parameters resulted in a small data mismatch of ~3%. L2 regularization [10] in the neural network formulation could ensure stability against simulated data noise. We also tested the CNN on a set of 6 new experimental datasets from various molecules and demonstrated a reconstruction error of ~4% compared to a global optimization approach [4] implemented in Matlab [6]. The neural network algorithm shows significant computational efficiency compared to optimization algorithms that require > 10^5 ODE evaluations for the analysis of each drug data. For the in-vivo Friberg model, we demonstrated using the validation data set that that the resulting neural network can estimate the “slope” parameter with good accuracy, showing a correlation to the ground truth with r>0.95 for both the population and individual parameters.
Conclusions: Our findings suggest that CNNs can aid parameter estimation in QSP and PMx modelling by providing highly efficient and accurate approximations. These promising results indicate that the use of CNNs warrants further development, including in applications where graphical assessment techniques are currently utilized.
References:
[1] LeCun Y, Bengio Y, Hinton G. Deep learning. Nature. 2015 May;521(7553):436-44. [2] https://medium.com/@karpathy/software-2-0-a64152b37c35
[3] Nguyen TH, Mouksassi MS, Holford N, Al-Huniti N, Freedman I, Hooker AC, John J, Karlsson MO, Mould DR, Ruixo JP, Plan EL. Model evaluation of continuous data pharmacometric models: metrics and graphics. CPT: pharmacometrics & systems pharmacology. 2017 Feb 1;6(2):87-109.
[4] Wilson J, Lu D, Corr N, Fullerton A, Lu J. An in vitro quantitative systems pharmacology approach for deconvolving mechanisms of drug-induced, multilineage cytopenias. bioRxiv. 2019 Jan 1.
[5] Friberg LE, Henningsson A, Maas H, Nguyen L, Karlsson MO. Model of chemotherapy-induced myelosuppression with parameter consistency across drugs. Journal of clinical oncology. 2002;20(24):4713-21.
[6] https://www.mathworks.com/products/matlab.html
[7] Goodfellow I, Bengio Y, Courville A. Deep learning. MIT press; 2016 Nov 10.
[8] https://deepnotes.io/public/images/AE-based.jpg
[9] https://www.wolfram.com/mathematica/
[10] Tarantola A. Inverse problem theory and methods for model parameter estimation. SIAM; 2005.
Reference: PAGE () Abstr 9391 [www.page-meeting.org/?abstract=9391]
Poster: Oral: Methodology - New Modelling Approaches