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CMINNs: Compartment model informed neural networks—Unlocking drug dynamics
Nazanin Ahmadi
Brown University
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Paper
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CMINNs: Compartment model Informed Neural Networks
Compartmental Modeling
PINNs: Physics-Informed Neural Networks
Scientific Machine Learning
PK/PD Modeling
Outline
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Pharmacokinetics and Pharmacodynamics Models
The relationship between dose, concentration, and effect is fundamental in clinical pharmacology.
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Plasma Drug Concentration
Time
Distribution phase
Elimination Phase
a
b
Plasma
C1 , V1
Peripheral
C2 , V2
K12
K21
K10
Traditional Compartmental PK/PD Modeling
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In Pharmacometrics, "compartments" are abstract mathematical constructs designed to simplify the representation of drug distribution within the human body. These compartments do not correspond to specific anatomical or physiological spaces; rather, they serve as valuable tools for modeling the behavior of substances introduced into the body.
A. Talevi and C. L. Bellera, Compartmental Pharmacokinetic Models, p. Chapter 8. Springer, Cham,2024.
We can use a different model if the calculations are feasible.
We will not lose information if we reduce the number of compartments.
If we have a better tool, we can model them using a different approach.
Traditional Compartmental PK/PD Modeling
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Limitations of Traditional Compartmental Modeling
Fractional Calculus
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Ahmadi, N. , Wang, S., Karniadakis, G. (2024). CMINNs: Compartment Model Informed Neural Networks-Unlocking Drug Dynamics. arXiv preprint arXiv:2409.12998.
Time-varying parameter
Time-varying parameter
Fractional Calculus
Fractional Calculus
Solver
CMINNs Workflow
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AI-Powered Solver:
Physics-Informed Neural Networks(PINNs)
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Mathematical Equations
DATA
Partial Physics
Interactions
Neural Networks
How do we blend our understanding of physics and the interactions within the system into the workings of a neural network?
Physics-Informed Neural networks (PINNs)
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The model hasn’t ‘understood’ the scientific problem.
u
x
Compare to training Data
Output
Input
A damped harmonic oscillator
Neural networks
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Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686-707.
u
x
Compare to training Data
Output
Input
Compute derivatives and minimize underlying equation residual
Loss Function
Physics-Informed Neural networks (PINNs)
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PINNs vs Neural ODEs
NODEs
t
Dynamics
ODE solver
Minimizing Loss
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Data
Data +
Physical Laws
Physics-Informed Neural networks (PINNs)
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378, 686-707.
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Physics-Informed Neural networks (PINNs)
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378, 686-707.
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Fractional Calculus in Pharmacokinetics and Pharmacodynamics Modeling
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Fractional Calculus in Pharmacokinetics and Pharmacodynamics Modeling
Caputo fractional derivative:
Caputo, Michele (1967). "Linear model of dissipation whose Q is almost frequency independent. II". Geophysical Journal International. 13 (5): 529–539. doi:10.1111/j.1365-246x.1967.tb02303.x
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Huang, Y., & Kim, H. K. (2022). Fractional transit compartment model for describing drug delayed response to tumors using Mittag-Leffler distribution on age-structured PKPD model. PLOS ONE, 17(11), e0276654. https://doi.org/10.1371/journal.pone.0276654
Fractional Calculus for Pharmacodynamics Modeling
Total tumor Cells are given by W = u + y1 + y2 + … + yn
Total tumor Cells are given by W = u + y
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Fractional Physics-Informed Neural networks (fPINNs)
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CMINNs: Compartment model Informed Neural Networks
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Ahmadi, N. , Wang, S., & Karniadakis, G. (2024). CMINNs: Compartment Model Informed Neural Networks--Unlocking Drug Dynamics. arXiv preprint arXiv:2409.12998.
CMINNs Workflow
Time-varying parameter
Time-varying parameter
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PINNs Block in CMINNs method with time-varying parameter
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Published Real-World Data
Two-Compartment Model for Gentamicin Pharmacokinetics
Two-Compartment Model for Amiodarone Pharmacokinetics
Three-Compartment Model for Talaporfin Sodium Pharmacokinetics
PK-PD Model
A PK-PD Model for Tumor Growth and Treatment in Animal Models
PK Models
Two-Compartment Model for Amiodarone Pharmacokinetics
A PK-PD Model for Tumor Growth and Treatment in Animal Models
CMINNs’ case study
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or
The pharmacokinetics of a single dose of amiodarone, an antiarrhythmic drug known for its non-exponential pharmacokinetics have been analyzed using a general two-compartment model as follows:
Two-Compartment Model for Amiodarone Pharmacokinetics
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PINNs with Time-varying parameter
fPINNs results
Data source: M. D. Freedman and J. C. Somberg, “Pharmacology and pharmacokinetics of amiodarone.,” Journal of clinical pharmacology, vol. 31, no. 11, pp. 1061–1069, 1991.
Two-Compartment Model for Amiodarone Pharmacokinetics - Results
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Pharmacokinetics
Pharmacodynamics
A PK-PD Model for Tumor Growth and Treatment in Animal Models
Data and Model: M. Simeoni, P. Magni, C. Cammia, G. D. Nicolao, V. Croci, E. Pesenti, M. Germani, I. Poggesi, and M. Rocchetti, “Predictive pharmacokinetic pharmacodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents,” Cancer Research, vol. 64, no. 3, pp. 1094– 1101, 2004.
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Pharmacokinetics
Pharmacodynamics
Data and Model: M. Simeoni, P. Magni, C. Cammia, G. D. Nicolao, V. Croci, E. Pesenti, M. Germani, I. Poggesi, and M. Rocchetti, “Predictive pharmacokinetic pharmacodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents,” Cancer Research, vol. 64, no. 3, pp. 1094– 1101, 2004.
A PK-PD Model for Tumor Growth and Treatment in Animal Models
Data
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Time-varying parameter
Fractional Compartment
Data : M. Simeoni, P. Magni, C. Cammia, G. D. Nicolao, V. Croci, E. Pesenti, M. Germani, I. Poggesi, and M. Rocchetti, “Predictive pharmacokinetic pharmacodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents,” Cancer Research, vol. 64, no. 3, pp. 1094– 1101, 2004.
A PK-PD Model for Tumor Growth and Treatment in Animal Models
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Model Parameters
Rate of cell death k1(t)
Drug potency k2
1st dose
2nd dose
3rd dose
1st dose
2nd dose
3rd dose
A PK-PD Model for Tumor Growth and Treatment in Animal Models - Results
PINNs with Time-varying parameter
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1st dose
2nd dose
3rd dose
A PK-PD Model for Tumor Growth and Treatment in Animal Models - Results
PINNs with Time-varying parameter
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A PK-PD Model for Tumor Growth and Treatment in Animal Models - Results
fPINNs results
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Summary
Challenges
Framework Introduced
Applications Demonstrated
Outcomes
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Acknowledgement
Prof. George Karniadakis
Dr. Shupeng Wang
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Thank you for listening
https://www.linkedin.com/in/ahmadinaz/
Nazanin@Brown.edu
https://www.researchgate.net/profile/Nazanin-Ahmadi-Daryakenari
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3-Compartment Model
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2-Compartment Model