(195h) Estimation of Probability Density Functions of Model Parameters for a Heterogeneous Population with Neural Network: Application to TNF? Signaling Pathway

Modeling a heterogeneous population, whose dynamics are governed by same differential equations but with different parameter’ values, has received much attention in various disciplines such as quantum physics and cellular biology [1-3]. Previous studies mostly focused on the problem of the optimal control of this heterogeneous population while methods on how to estimate such systems have begun to emerge [3-6].

Motivated by this, this study proposes a methodology to develop a heterogeneous dynamic model for a heterogeneous population. Specifically, we propose a numerical method to estimate the probability density functions (PDFs) of model parameters from experimental measurements. Here, we particularly consider the case where the output PDFs are available in the form of population snapshot data (PSD): PSD provide measurements from all members in the population at each time instance [6]. First, the PDFs of the model parameters are assumed to be normal so that the overall estimation problem become finite-dimensional. Second, Sobol’ sensitivity analysis method is performed to determine a set of model parameters whose means and variances are practically identifiable. Third, a neural network model is developed to find an empirical function mapping from the PDFs of the identifiable parameters to the output PDFs. Lastly, the PDFs of the identifiable parameters are estimated by solving an optimization problem to minimize the difference between the measured and predicted output PDFs, which are computed based on PSD and the developed neural network model, respectively.

Our proposed methodology was implemented to infer the PDFs of the parameters in a tumor necrosis factor-α (TNFα) signaling pathway. Artificial PSD of the TNFα signaling dynamics were generated by simulating the model accordingly. Next, the proposed method was implemented to determine identifiable parameters and infer their PDFs from the PSD. After inferring the PDFs of these identifiable parameters, the corresponding output PDFs were calculated and compared with the PSD to validate the inference results. The reasonable agreement between the predicted and measured output PDFs demonstrated the validity of the proposed methods.

References

[1] Li, J.-S.; Khaneja, N. Control of inhomogeneous quantum ensembles. Physical Review A 2006, 73, 030302.

[2] Waldherr, S. Estimation methods for heterogeneous cell population models in systems biology. Journal of the Royal Society Interface 2018, 15, 20180530.

[3] Zeng, S.; Waldherr, S.; Allgöwer, F. Ensemble Observability of Linear Systems. IEEE Transactions on Automatic Control 2016, 61, 1452-1465.

[4] Li, J.-S. Ensemble control of finite-dimensional time-varying linear systems. IEEE Transactions on Automatic Control 2011, 56, 345-357.

[5] Helmke, U.; Schönlein, M. Uniform ensemble controllability for one-parameter families of time-invariant linear systems. System & Control Letters 2014, 71, 69-77.

[6] Hasenauer, J.; Waldherr, S.; Doszczak, M.;Scheurich, P.; Radde, N.; Allgöwer, F. Analysis of heterogeneous cell populations: A density-based modeling and identification framework. Journal of Process Control 2011, 21, 1417-1425.