Frequency‐Domain Response Analysis for Quantitative Systems Pharmacology Models

Figures

• 

  Figure 1. Frequency‐domain response analysis (FdRA) workflow. The sequence of steps needed to perform FdRA for a given mathematical model.

• 

  Figure 2. Indirect response model. (a) Structures of four model flavors (1 = stimulation of production, 2 = stimulation of loss, 3 = inhibition of production, and 4 = inhibition of loss) of the indirect response model. Blue arrows represent stimulation or inhibition. (b) Time course simulations of linearized model flavor 1. Input sinusoid (black) of two different frequencies ( 124 h−1 and 4 h−1) and response (pink) are shown. (c) Frequency response of all four linearized model flavors is given by the amplitude ratio for various frequencies of the input sinusoids. Schematically, low‐frequency gain, cutoff frequency, threshold frequency, and roll‐off are depicted. (d) Time course stimulations of model flavor 1. Plasma concentration as derived from pharmacokinetic (PK) model (black) for two drugs with different elimination rates ( 124 h−1 and 4 h−1) and the response (pink) are shown. (e) Frequency response of all four PK‐driven model flavors is given by the amplitude ratio for various elimination rates. Note that model flavor 1 (stimulation of production) results in the same frequency response as model flavor 3 (inhibition of production). Model parameters used in all simulations: kin=1…

• 

  Figure 3. Autoregulation model. (a) Structures of four model flavors (1 = stimulation with positive feedback, 2 = stimulation with negative feedback, 3 = inhibition with positive feedback, and 4 = inhibition with negative feedback). Blue arrows represent stimulation or inhibition. (b) Time course simulations of linearized model flavor 1. Input sinusoid (black) of two different frequencies ( 124 h−1 and 4 h−1) and response (pink) are shown. (c) Frequency response of all four linearized model flavors is given by the amplitude ratio for various frequencies of the input sinusoids. Note that model flavors 1 and 3 as well as 2 and 4 results in the same frequency response. (d) Time course stimulations of model flavor 1. Plasma concentration as derived from pharmacokinetic (PK) model (black) for two drugs with different elimination rates ( 124 h−1 and 4 h−1) and the response (pink) are shown. (e) Frequency response of all four PK‐driven model flavors is given by the amplitude ratio for various elimination rates. Model parameters used in all simulations: kin=1 ml⋅h−1, kout=1 h−1, K=0.25 μM, Emax=1, and EC50=0.25 μM.

• 

  Figure 4. Precursor‐pool model. (a) Structures of two model flavors (1 = stimulation, and 2 = inhibition). Blue arrows represent stimulation or inhibition. (b) Time course simulations of linearized model flavor 1. Input sinusoid (black) of three different frequencies ( 124 h−1, 16 h−1, and 4 h−1) and response (pink) are shown. (c) Frequency response of both linearized model flavors is given by the amplitude ratio for various frequencies of the input sinusoids. Note that both model flavors result in the same frequency response. (d) Time course stimulations of model flavor 2. Plasma concentration as derived from pharmacokinetic (PK) model (black) for three drugs with different elimination rates ( 1720 h−1, 16 h−1, and 4 h−1) and the response (pink) are shown. (e) Frequency response of both PK‐driven model flavors is given by the amplitude ratio for various elimination rates. Model parameters used in all simulations: kin=1 ml⋅h−1, kout=1 h−1, Emax=1, and EC50=0.25 μM.

• 

  Figure 5. Moderator‐mediated feedback model. (a) Structures of two model flavors (1 = stimulation with negative feedback and 2 = inhibition with negative feedback). Green, red, and blue arrows represent stimulation, inhibition, and stimulation or inhibition, respectively. (b) Time course simulations of linearized model flavor 1. Input sinusoid (black) of three different frequencies ( 1168 h−1, 112 h−1, and 4 h−1) and response (pink) are shown. (c) Frequency response of both linearized model flavors is given by the amplitude ratio for various frequencies of the input sinusoids. Note that all model flavors result in the same frequency response. (d) Time course stimulations of model flavor 1. Plasma concentrations as derived from pharmacokinetic (PK) models (black) for three drugs three different elimination rates ( 11440 h−1, 112 h−1, and 4 h−1) and the response (pink) are shown. (e) Frequency response of both PK‐driven model flavors is given by the amplitude ratio for various elimination rates. Model parameters used in all simulations: kin=1 ml⋅h−1, kout=1 h−1, ktol=0.25 h−1, Emax=1, and EC50=0.25 μM.

• 

  Figure 6. Double moderator‐mediated feedback model. (a) Structures of two model flavors (1 = stimulation and 2 = inhibition). Green, red, and blue arrows represent stimulation, inhibition, and stimulation or inhibition, respectively. (b) Time course simulations of linearized model flavor 1. Input sinusoid (black) of three different frequencies ( 1168 h−1, 112 h−1, and 4 h−1) and response (pink) are shown. (c) Frequency response of both linearized model flavors is given by the amplitude ratio for various frequencies of the input sinusoids. Note that all model flavors result in the same frequency response. (d) Time course stimulations of model flavor 1. Plasma concentrations as derived from pharmacokinetic (PK) models (black) for three drugs with different elimination rates ( 1168 h−1, 112 h−1, and 4 h−1) and the response (pink) are shown. (e) Frequency response of both PK‐driven model flavors is given by the amplitude ratio for various drug elimination rates. Model parameters used in all simulations: kin=1 ml⋅h−1, kout=1 h−1, ktol=0.25 h−1, Emax=1, andEC50=0.25 μM.

• 

  Figure 7. Frequency response possibilities. (a) Frequency responses of all possible filters (low‐pass, band‐pass, high‐pass, and band‐stop). (b) Block diagram of the state‐space representation of a linear time invariant system. Each block represents a matrix of the state‐space, whereas the block in the center is an integrator. The gray‐shaded part of the diagram highlights the model structures that were discussed in this tutorial.