Founder and Principal Consultant Cardiorenal QSP, LLC, Georgia, United States
Disclosure(s):
Melissa Hallow, PhD: No financial relationships to disclose
Objectives: QSP models often include feedback mechanisms whose structural forms are not known from first principles. These relationships are commonly determined by preassuming different structural forms (e.g. linear, sigmoidal, exponential), and determining the best fit to experimental data. However, biology is not constrained to these predefined relationships, and there could be infinite possible forms, and inaccurately representing the true relationship could introduce bias. We explored the use of neural ordinary differential equations (Neural ODEs) to infer feedback relationships directly from data, without prior assumptions about their mathematical form. With neural ODEs, rather than assuming a functional form for a particular term in an ODE equation, the term can be replaced with a simple neural net that predicts the term as a function of input(s)[1].
Methods: We developed a workflow integrating Julia and R to embed neural networks (NNs) within ODE-based QSP models [1]. As an initial validation, we trained a NN embedded in a simple ODE model on simulated data to determine whether it could recover a known relationship. We then applied this approach to a previously published model of the renin-angiotensin-aldosterone system, training the NN on experimental data to determine the structure of a key feedback term. Finally, we replaced multiple empirically fitted feedback mechanisms in a comprehensive cardiorenal QSP model with neural networks and refit the model to experimental data describing acute and chronic responses to sodium intake.
Results: In the validation step, the NN accurately recapitulated known relationships from simulated data. In the renin-angiotensin-aldosterone model, the NN-inferred feedback structure aligned closely with our previously derived empirical function, supporting the validity of the method. When applied to the full cardiorenal model, the neural networks produced feedback terms that were generally in agreement with prior assumptions, while also revealing novel features such as asymmetric sigmoidal behavior not captured by traditional functional forms. The NN-based fitting process also proved more efficient and less labor-intensive than manual structural selection and testing. In all cases, feedback terms were modeled as direct functions of system states, without inclusion of time delays or integrative signals—an important area for future work. Notably, the inclusion of neural networks did not significantly increase simulation time in Julia, though computational cost may become more relevant in slower models.
Conclusions: Neural ODEs offer a promising approach to infer the structure of feedback mechanisms in QSP models without requiring strong a priori assumptions. This method enhances modeling efficiency and may reveal functional forms—such as asymmetric sigmoidal behavior—that better reflect biological reality.
Citations: Chris Rackauckas, Mike Innes, Yingbo Ma, Jesse Bettencourt, Lyndon White, Vaibhav Dixit. DiffEqFlux.jl - A Julia Library for Neural Differential Equations. https://doi.org/10.48550/arXiv.1902.02376
