Modeling Signal Transduction in Pharmacodynamics

Time-Dependent Transduction

Mediator or Receptor

Response

Transduction Processes can be rapid to slow, single to multiple, linear or nonlinear. Donald E. Mager, Pharm.D., Ph.D. Department of Pharmaceutical Sciences

Approaches:

•

Modeling Time-Dependent Transduction

Gamma Distribution Function (Poisson) N1

slow

R

ln C

t Basic question: How to capture a response profile where transduction steps cause time delays? Problem: No intermediary measurements.

Probability Density Function

fast

τ

N2

τ

N3

R

DR

D

• Transit Compartment • Complete Mechanisms

Operational Model of Agonism

0.4

Ni

k N ⋅t N −1 − k ⋅t ⋅e (N - 1)! 1 where k = τ gN ( t ) =

N=2

N=3

τ = Transduction Time

N=5

0.2

N = 10

t = 1 hr

0.0 0

10 Time (Hour)

20

Assume: Unit dose input into N1

Flexibility of Gamma Distribution Function

N small τ Mod.

Stochastic Events Underlie Ca2+ Signaling in Neutrophils Hallett and Pettit, J Theor Biol. 186:1-6 (1997)

N large τ small

N↑ τ ↑

R

Time

Use of Parallel Erlang Density Functions First-Pass Pulmonary Uptake of Multiple Indicators in Dogs

PBPK Model for Methotrexate

Krejcie, JPB, 24(6) 1996

Bischoff et al., J Pharm Sci 60:1129 (1971)

Transit Compartments for Time-Delays Input

C1

τ

C2

dC1 1 = Input − ⋅ C1 dt τ dC 2 1 1 = ⋅ C1 − ⋅ C 2 dt τ τ

τ

τ

C3 τ=

Signal Transduction Process Using Transit Compartments

dM 1 1 1 = ⋅ DR − ⋅ M1 dt τ τ

1 k

(first-order constant)

dC 3 1 1 1 = ⋅ C 2 − ⋅ C 3 = ⋅ (C2 − C 3 ) dt τ τ τ

Modeling Signal Transduction: Parameters N, τ, γ

Sun and Jusko, J Pharm Sci 87:732 (1998)

Signal Amplification or Dampening: Use of γ

τ = 2.0 hr

PK/PD Profiles of Time-Dependent Transduction

Modeling Signal Transduction: One Step

+ R

C

τ

Er

τ

E*

M1

τ

τ

M2

τ

M3

τ

Response

E

100

E ⋅C Er = max EC50 + C

Pharmacological Response

80

dE 1 = (Er − E ) dt τ Emax = 0.38,

⎞ dE 1 ⎛ Emax ⋅ C − E ⎟⎟ = ⎜⎜ dt τ ⎝ EC50 + C ⎠

EC50 = 6.6 ng/µl,

τ = 0.18 h,

10000

10 00 0 10 00

1000

10000

100

10 0 60

10

1000

10

1

100

40

0.1

Drug Concentration

Receptor

RC

0.01

20

10

E0 = 0.39

0.001 0 0.0001 0

10

20

30

Mager D and Jusko WJ, CPT 70: 210, 2001.

⎞ dM1 1 ⎛ E max C = ⎜⎜ − M1 ⎟⎟ dt τ ⎝ EC50 + C ⎠ dM 2 1 = (M1 − M 2 ) dt τ dM 3 1 = (M 2 − M 3 ) dt τ

τ

M2

M3

τ

15

10000

Emax = 6.92 EC50 = 9.38 U/mL τ = 5.06 hours

12

1000

9 100 6 10 3 1 0

0

20

40

60

Time (h)

Mager and Jusko, Clin Pharmacol Ther. 70:210 (2001)

80

0.1 100

60

Mager and Jusko (2007)

WinNonlin Model for IFNα (Linear S) MODEL COMMANDS NFUN 1 NDER 5 NPARAMETERS 3 PNAMES ‘S','tau','Eo' END TEMPORARY T=X Dose=10000000 Tinf=10/60 k12 = 6.24 k21 = 0.693 10 kel = 2.61 Vc = 7234 8 IF T