MI-210 Essentials of Population PKPD M&S — Pumas Edition
This chapter covers continuous PK-PD models linking drug concentrations to pharmacodynamic response:
PK-PD models link the pharmacokinetic profile to an observed pharmacodynamic response. The approach depends on the relationship between drug concentration and effect:
Most drugs exert their effects by binding to receptors, inhibiting enzymes, or modulating ion channels. The concentration-effect relationship follows the law of mass action: at low concentrations, the effect increases approximately linearly with concentration; at high concentrations, binding sites saturate and the effect plateaus. This saturable behavior is captured by the \(E_{max}\) model:
\[E = \frac{E_{max} \cdot C}{EC_{50} + C}\]
which is mathematically identical to the Michaelis-Menten equation for enzyme kinetics. Here, \(EC_{50}\) is the concentration producing 50% of maximum effect (analogous to \(K_M\)), and \(E_{max}\) is the maximum achievable effect (analogous to \(V_{max}\)). The sigmoid (Hill) extension introduces a shape parameter \(\gamma\) that controls the steepness of the curve: \(E = E_{max} \cdot C^{\gamma} / (EC_{50}^{\gamma} + C^{\gamma})\). When \(\gamma > 1\), the curve is steeper (switch-like behavior); when \(\gamma < 1\), it is shallower. This pharmacological foundation — grounded in receptor occupancy theory — is why the \(E_{max}\) model is the fundamental building block of virtually all PD modeling.
The exercises use three datasets:
ch7_pk.csv — PK-only data (TYPE=0)ch7_pkpd.csv — Combined PK+PD data (TYPE=0 for PK, TYPE=1 for PD)ch7_indirect_pkpd.csv — Combined PK+PD with indirect response endpointpk_df = CSV.read(joinpath(data_dir, "ch7_pk.csv"), DataFrame)
pkpd_df = CSV.read(joinpath(data_dir, "ch7_pkpd.csv"), DataFrame)
ind_df = CSV.read(joinpath(data_dir, "ch7_indirect_pkpd.csv"), DataFrame)
println("PK data: ", nrow(pk_df), " rows, ", length(unique(pk_df.ID)), " subjects")
println("PKPD data: ", nrow(pkpd_df), " rows")
println("Indirect data: ", nrow(ind_df), " rows")PK data: 240 rows, 20 subjects
PKPD data: 460 rows
Indirect data: 480 rowsAll PK-PD models build on a 1-compartment oral PK model (ADVAN2 TRANS2):
# Prepare PK-only data
pk_only = @rsubset(pk_df, :TYPE == 0 || :AMT > 0)
rename!(pk_only, :ID => :id, :AMT => :amt, :TIME => :time, :DV => :dv, :CMT => :cmt)
pk_only.evid = ifelse.(pk_only.amt .> 0, 1, 0)
pk_only.dv = ifelse.(pk_only.evid .== 1, missing, pk_only.dv)
pop_pk = read_pumas(pk_only)
println("PK subjects: ", length(pop_pk))PK subjects: 20pk_base = @model begin
@metadata begin
desc = "Base PK: 1-Cpt Oral (ADVAN2 TRANS2)"
end
@param begin
tvKa ∈ RealDomain(lower = 0.0, init = 1.5)
tvCL ∈ RealDomain(lower = 0.0, init = 5.0)
tvV ∈ RealDomain(lower = 0.0, init = 35.0)
ω²_Ka ∈ RealDomain(lower = 0.0, init = 0.04)
Ω_pk ∈ PSDDomain(2)
σ_prop ∈ RealDomain(lower = 0.0, init = 0.2)
σ_add ∈ RealDomain(lower = 0.0, init = 1.41)
end
@random begin
η_Ka ~ Normal(0.0, sqrt(ω²_Ka))
η_pk ~ MvNormal(Ω_pk)
end
@pre begin
Ka = tvKa * exp(η_Ka)
CL = tvCL * exp(η_pk[1])
Vc = tvV * exp(η_pk[2])
end
@dynamics Depots1Central1
@derived begin
cp := @. Central / Vc
dv ~ @. Normal(cp, sqrt((cp * σ_prop)^2 + σ_add^2))
end
endPumasModel
Parameters: tvKa, tvCL, tvV, ω²_Ka, Ω_pk, σ_prop, σ_add
Random effects: η_Ka, η_pk
Covariates:
Dynamical system variables: Depot, Central
Dynamical system type: Closed form
Derived: dv
Observed: dvpk_params = (
tvKa = 1.5, tvCL = 5.0, tvV = 35.0,
ω²_Ka = 0.04,
Ω_pk = [0.04 0.01; 0.01 0.04],
σ_prop = 0.2, σ_add = sqrt(2.0),
)
pk_fit = fit(pk_base, pop_pk, pk_params, Pumas.FOCEI())
pk_fit[ Info: Checking the initial parameter values. [ Info: The initial negative log likelihood and its gradient are finite. Check passed. Iter Function value Gradient norm 0 4.166856e+02 8.017131e+01 * time: 0.012240886688232422 1 3.840650e+02 1.104225e+02 * time: 0.9147119522094727 2 3.635721e+02 9.730302e+01 * time: 0.9207980632781982 3 3.356405e+02 6.707764e+01 * time: 1.3415138721466064 4 3.303805e+02 7.860668e+01 * time: 1.3440001010894775 5 3.056468e+02 3.189815e+01 * time: 1.3465189933776855 6 2.983179e+02 3.407148e+01 * time: 1.3493680953979492 7 2.905084e+02 2.502687e+01 * time: 1.3522889614105225 8 2.880127e+02 3.378228e+01 * time: 1.3551630973815918 9 2.826686e+02 1.290026e+01 * time: 1.3582608699798584 10 2.794418e+02 1.799054e+01 * time: 1.3612990379333496 11 2.792528e+02 4.548804e+01 * time: 1.364326000213623 12 2.775762e+02 1.039404e+01 * time: 1.367258071899414 13 2.774304e+02 5.744242e+00 * time: 1.3702049255371094 14 2.773809e+02 4.769231e+00 * time: 1.3729090690612793 15 2.773574e+02 7.912850e-01 * time: 1.3756749629974365 16 2.773483e+02 8.209608e-01 * time: 1.3783490657806396 17 2.773398e+02 5.726275e-01 * time: 1.380910873413086 18 2.773378e+02 3.253195e-01 * time: 1.3833298683166504 19 2.773345e+02 6.370831e-01 * time: 1.385840892791748 20 2.773308e+02 8.950136e-01 * time: 1.3883519172668457 21 2.773274e+02 7.043970e-01 * time: 1.3907949924468994 22 2.773262e+02 2.542502e-01 * time: 1.3934369087219238 23 2.773260e+02 2.813574e-02 * time: 1.3957879543304443 24 2.773260e+02 1.032549e-03 * time: 1.398103952407837 25 2.773260e+02 1.032549e-03 * time: 1.4015600681304932 26 2.773260e+02 1.032549e-03 * time: 1.4054369926452637 27 2.773260e+02 1.032549e-03 * time: 1.407926082611084
FittedPumasModel
Dynamical system type: Closed form
Number of subjects: 20
Observation records: Active Missing
dv: 220 0
Total: 220 0
Number of parameters: Constant Optimized
0 9
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: NoXChange
Log-likelihood value: -277.32599
---------------------
Estimate
---------------------
tvKa 1.3593
tvCL 4.7674
tvV 35.795
ω²_Ka 0.049835
Ω_pk₁,₁ 0.046159
Ω_pk₂,₁ 0.0091618
Ω_pk₂,₂ 0.019243
σ_prop 0.20066
σ_add 0.0027741
---------------------In the direct model, effect is an instantaneous function of plasma concentration:
\[E = E_0 + \frac{E_{max} \cdot C_p}{EC_{50} + C_p}\]
This requires no additional dynamics — the PD is computed directly from the PK prediction in the error block.
$ERROR
EMAX=THETA(4)*EXP(ETA(4))
EC50=THETA(5)*EXP(ETA(5))
E0=THETA(6)*EXP(ETA(6))
EFF=E0+EMAX*F/(EC50+F)+ERR(2)
CONC=F + F*ERR(1)
Y=EFF*TYPE + CONC*(1-TYPE)# Prepare combined PK+PD data — pivot to wide format for multi-DV
# PK and PD observations at same times need to be on same row
pkpd_dose = @rsubset(pkpd_df, :AMT > 0)
pkpd_pk = @rsubset(pkpd_df, :TYPE == 0, :AMT == 0)
pkpd_pd = @rsubset(pkpd_df, :TYPE == 1)
pk_w = select(pkpd_pk, :ID, :TIME, :DV => :dv_pk)
pd_w = select(pkpd_pd, :ID, :TIME, :DV => :dv_pd)
obs_w = outerjoin(pk_w, pd_w, on = [:ID, :TIME])
obs_w.AMT .= 0; obs_w.CMT .= 2; obs_w.evid .= 0
dose_w = select(pkpd_dose, :ID, :TIME, :AMT, :CMT)
dose_w.dv_pk .= missing; dose_w.dv_pd .= missing; dose_w.evid .= 1
pkpd_combined = vcat(dose_w, obs_w, cols = :union)
pkpd_combined.AMT = coalesce.(pkpd_combined.AMT, 0)
pkpd_combined.evid = coalesce.(pkpd_combined.evid, 0)
pkpd_combined.CMT = coalesce.(pkpd_combined.CMT, 2)
sort!(pkpd_combined, [:ID, :TIME, order(:evid, rev = true)])
rename!(pkpd_combined, :ID => :id, :TIME => :time, :AMT => :amt, :CMT => :cmt)
pop_pkpd = read_pumas(pkpd_combined; observations = [:dv_pk, :dv_pd])
println("PKPD subjects: ", length(pop_pkpd))PKPD subjects: 20direct_pkpd = @model begin
@metadata begin
desc = "Direct Emax PK-PD"
end
@param begin
tvKa ∈ RealDomain(lower = 0.0, init = 1.7)
tvCL ∈ RealDomain(lower = 0.0, init = 4.7)
tvV ∈ RealDomain(lower = 0.0, init = 38.0)
tvEmax ∈ RealDomain(lower = 0.0, init = 100.0)
tvEC50 ∈ RealDomain(lower = 0.0, init = 5.0)
tvE0 ∈ RealDomain(lower = 0.0, init = 60.0)
Ω ∈ PDiagDomain(6)
σ_prop ∈ RealDomain(lower = 0.0, init = 0.2)
σ_pd ∈ RealDomain(lower = 0.0, init = 2.24)
end
@random begin
η ~ MvNormal(Ω)
end
@pre begin
Ka = tvKa * exp(η[1])
CL = tvCL * exp(η[2])
Vc = tvV * exp(η[3])
Emax = tvEmax * exp(η[4])
EC50 = tvEC50 * exp(η[5])
E0 = tvE0 * exp(η[6])
end
@dynamics Depots1Central1
@derived begin
cp := @. Central / Vc
# PK observation (proportional error)
dv_pk ~ @. Normal(cp, abs(cp) * σ_prop)
# PD observation (direct Emax + additive error)
eff := @. E0 + Emax * cp / (EC50 + cp)
dv_pd ~ @. Normal(eff, σ_pd)
end
endPumasModel
Parameters: tvKa, tvCL, tvV, tvEmax, tvEC50, tvE0, Ω, σ_prop, σ_pd
Random effects: η
Covariates:
Dynamical system variables: Depot, Central
Dynamical system type: Closed form
Derived: dv_pk, dv_pd
Observed: dv_pk, dv_pddirect_params = (
tvKa = 1.7, tvCL = 4.7, tvV = 38.0,
tvEmax = 100.0, tvEC50 = 5.0, tvE0 = 60.0,
Ω = Diagonal(fill(0.04, 6)),
σ_prop = 0.2, σ_pd = sqrt(5.0),
)
direct_fit = fit(direct_pkpd, pop_pkpd, direct_params, Pumas.FOCEI())
direct_fit[ Info: Checking the initial parameter values. [ Info: The initial negative log likelihood and its gradient are finite. Check passed. Iter Function value Gradient norm 0 4.430762e+03 2.671248e+03 * time: 2.4080276489257812e-5 1 1.880925e+03 7.344425e+02 * time: 0.4339730739593506 2 1.622390e+03 3.990711e+02 * time: 0.47350215911865234 3 1.483465e+03 1.754307e+02 * time: 0.5082571506500244 4 1.434739e+03 1.742967e+02 * time: 0.5240612030029297 5 1.403106e+03 1.501704e+02 * time: 0.5356919765472412 6 1.367590e+03 1.074723e+02 * time: 0.5466799736022949 7 1.337474e+03 9.630342e+01 * time: 0.5573689937591553 8 1.311637e+03 7.083317e+01 * time: 0.5661871433258057 9 1.301092e+03 6.381741e+01 * time: 0.5771100521087646 10 1.297023e+03 5.227391e+01 * time: 0.5878980159759521 11 1.290880e+03 5.008124e+01 * time: 0.5992081165313721 12 1.283336e+03 4.209421e+01 * time: 0.6103291511535645 13 1.279130e+03 3.196621e+01 * time: 0.621068000793457 14 1.277250e+03 3.342941e+01 * time: 0.6314880847930908 15 1.275162e+03 3.544695e+01 * time: 0.642009973526001 16 1.272034e+03 3.599997e+01 * time: 0.652860164642334 17 1.268091e+03 3.734853e+01 * time: 0.6638860702514648 18 1.265099e+03 3.073979e+01 * time: 0.6724221706390381 19 1.263518e+03 2.997748e+01 * time: 0.6830651760101318 20 1.262309e+03 2.198757e+01 * time: 0.6934599876403809 21 1.261183e+03 1.388066e+01 * time: 0.7041330337524414 22 1.260288e+03 1.395191e+01 * time: 0.7143909931182861 23 1.259688e+03 9.460054e+00 * time: 0.7246911525726318 24 1.259260e+03 1.114251e+01 * time: 0.7351970672607422 25 1.258915e+03 1.086839e+01 * time: 0.7453181743621826 26 1.258592e+03 1.359211e+01 * time: 0.7536201477050781 27 1.258173e+03 1.399207e+01 * time: 0.7639131546020508 28 1.257620e+03 1.059567e+01 * time: 0.7741191387176514 29 1.257057e+03 3.906807e+00 * time: 0.7847881317138672 30 1.256645e+03 3.027562e+00 * time: 0.7953720092773438 31 1.256399e+03 5.341539e+00 * time: 0.8059489727020264 32 1.256293e+03 4.735702e+00 * time: 0.8163871765136719 33 1.256256e+03 2.778442e+00 * time: 0.8262491226196289 34 1.256242e+03 2.759952e+00 * time: 0.8342320919036865 35 1.256230e+03 2.743791e+00 * time: 0.8441421985626221 36 1.256203e+03 2.792107e+00 * time: 0.8541491031646729 37 1.256142e+03 5.444348e+00 * time: 0.8644289970397949 38 1.256010e+03 8.507416e+00 * time: 0.8748250007629395 39 1.255787e+03 9.921506e+00 * time: 0.8851850032806396 40 1.255555e+03 5.924178e+00 * time: 0.8958830833435059 41 1.255459e+03 5.772958e+00 * time: 0.9041461944580078 42 1.255430e+03 5.041421e+00 * time: 0.9141101837158203 43 1.255406e+03 3.888541e+00 * time: 0.9241981506347656 44 1.255373e+03 3.311506e+00 * time: 0.9344980716705322 45 1.255348e+03 1.877533e+00 * time: 0.9443581104278564 46 1.255340e+03 4.856092e-01 * time: 0.9544970989227295 47 1.255339e+03 1.496839e-01 * time: 0.964259147644043 48 1.255339e+03 7.223538e-02 * time: 0.9720339775085449 49 1.255339e+03 5.311650e-02 * time: 0.9819941520690918 50 1.255339e+03 5.302798e-02 * time: 0.9916059970855713 51 1.255339e+03 7.495599e-02 * time: 1.0013229846954346 52 1.255339e+03 1.422723e-01 * time: 1.0111169815063477 53 1.255339e+03 2.386357e-01 * time: 1.0211031436920166 54 1.255339e+03 2.896828e-01 * time: 1.0291271209716797 55 1.255339e+03 2.201049e-01 * time: 1.0394160747528076 56 1.255338e+03 7.956383e-02 * time: 1.0492970943450928 57 1.255338e+03 2.816678e-02 * time: 1.0593011379241943 58 1.255338e+03 2.747591e-02 * time: 1.068636178970337 59 1.255338e+03 2.685499e-02 * time: 1.077991008758545 60 1.255338e+03 2.521360e-02 * time: 1.0852270126342773 61 1.255338e+03 3.234450e-02 * time: 1.0947070121765137 62 1.255338e+03 4.915159e-02 * time: 1.1044790744781494 63 1.255338e+03 5.993211e-02 * time: 1.1141769886016846 64 1.255338e+03 4.873833e-02 * time: 1.1241250038146973 65 1.255338e+03 2.008924e-02 * time: 1.1339490413665771 66 1.255338e+03 6.519858e-03 * time: 1.1414320468902588 67 1.255338e+03 6.722397e-03 * time: 1.150170087814331 68 1.255338e+03 6.726560e-03 * time: 1.1598801612854004 69 1.255338e+03 6.750421e-03 * time: 1.1701080799102783 70 1.255338e+03 6.780828e-03 * time: 1.1802921295166016 71 1.255338e+03 6.780981e-03 * time: 1.1916990280151367 72 1.255338e+03 7.025447e-03 * time: 1.1992521286010742 73 1.255338e+03 7.088923e-03 * time: 1.2087490558624268 74 1.255338e+03 7.316282e-03 * time: 1.218116044998169 75 1.255338e+03 7.549789e-03 * time: 1.2279651165008545 76 1.255338e+03 1.287682e-02 * time: 1.2373809814453125 77 1.255338e+03 2.019855e-02 * time: 1.2468681335449219 78 1.255338e+03 2.861089e-02 * time: 1.2546730041503906 79 1.255338e+03 3.329259e-02 * time: 1.2644331455230713 80 1.255338e+03 2.690672e-02 * time: 1.2741131782531738 81 1.255338e+03 1.025398e-02 * time: 1.2834820747375488 82 1.255338e+03 1.576771e-03 * time: 1.2930700778961182 83 1.255338e+03 4.714151e-03 * time: 1.3026700019836426 84 1.255338e+03 5.461691e-03 * time: 1.309993028640747 85 1.255338e+03 3.443923e-03 * time: 1.319566011428833 86 1.255338e+03 6.368910e-04 * time: 1.3289129734039307
FittedPumasModel
Dynamical system type: Closed form
Number of subjects: 20
Observation records: Active Missing
dv_pk: 220 0
dv_pd: 220 0
Total: 440 0
Number of parameters: Constant Optimized
0 14
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: GradientNorm
Log-likelihood value: -1255.3375
----------------------
Estimate
----------------------
tvKa 1.3668
tvCL 4.8346
tvV 36.392
tvEmax 0.00043429
tvEC50 301.46
tvE0 80.148
Ω₁,₁ 0.042081
Ω₂,₂ 0.04042
Ω₃,₃ 0.015019
Ω₄,₄ 0.059114
Ω₅,₅ 0.043285
Ω₆,₆ 0.033018
σ_prop 0.24476
σ_pd 16.466
----------------------When there is a temporal delay between plasma concentration and effect, an effect compartment links the PK to PD:
\[\frac{dC_e}{dt} = K_{e0} \cdot (C_p - C_e)\]
\[E = E_0 + \frac{E_{max} \cdot C_e}{EC_{50} + C_e}\]
In NONMEM, this uses ADVAN4 TRANS1 with a negligible-mass trick (K23 very small). In Pumas, we write the ODE explicitly:
ecomp_pkpd = @model begin
@metadata begin
desc = "Effect Compartment PK-PD (Emax)"
end
@param begin
tvKa ∈ RealDomain(lower = 0.0, init = 1.5)
tvCL ∈ RealDomain(lower = 0.0, init = 5.0)
tvV ∈ RealDomain(lower = 0.0, init = 35.0)
tvKe0 ∈ RealDomain(lower = 0.0, init = 0.07)
tvEmax ∈ RealDomain(lower = 0.0, init = 100.0)
tvEC50 ∈ RealDomain(lower = 0.0, init = 5.0)
tvE0 ∈ RealDomain(lower = 0.0, init = 60.0)
Ω ∈ PDiagDomain(7)
σ_prop ∈ RealDomain(lower = 0.0, init = 0.2)
σ_pd ∈ RealDomain(lower = 0.0, init = 4.47)
end
@random begin
η ~ MvNormal(Ω)
end
@pre begin
Ka = tvKa * exp(η[1])
CL = tvCL * exp(η[2])
Vc = tvV * exp(η[3])
Ke0 = tvKe0 * exp(η[4])
Emax = tvEmax * exp(η[5])
EC50 = tvEC50 * exp(η[6])
E0 = tvE0 * exp(η[7])
end
@dynamics begin
Depot' = -Ka * Depot
Central' = Ka * Depot - (CL / Vc) * Central
Ce' = Ke0 * (Central / Vc - Ce) # effect compartment
end
@derived begin
cp := @. Central / Vc
dv_pk ~ @. Normal(cp, abs(cp) * σ_prop)
eff := @. E0 + Emax * Ce / (EC50 + Ce)
dv_pd ~ @. Normal(eff, σ_pd)
end
endPumasModel
Parameters: tvKa, tvCL, tvV, tvKe0, tvEmax, tvEC50, tvE0, Ω, σ_prop, σ_pd
Random effects: η
Covariates:
Dynamical system variables: Depot, Central, Ce
Dynamical system type: Matrix exponential
Derived: dv_pk, dv_pd
Observed: dv_pk, dv_pdecomp_params = (
tvKa = 1.5, tvCL = 5.0, tvV = 35.0, tvKe0 = 0.07,
tvEmax = 100.0, tvEC50 = 5.0, tvE0 = 60.0,
Ω = Diagonal(fill(0.04, 7)),
σ_prop = 0.2, σ_pd = sqrt(20.0),
)
ecomp_fit = fit(ecomp_pkpd, pop_pkpd, ecomp_params, Pumas.FOCEI())
ecomp_fit[ Info: Checking the initial parameter values. [ Info: The initial negative log likelihood and its gradient are finite. Check passed. Iter Function value Gradient norm 0 1.041844e+03 7.813852e+01 * time: 2.193450927734375e-5 1 1.032660e+03 1.643447e+01 * time: 0.6318747997283936 2 1.031822e+03 1.605007e+01 * time: 1.186776876449585 3 1.030301e+03 1.120108e+01 * time: 1.304297924041748 4 1.029732e+03 1.234478e+01 * time: 1.441051959991455 5 1.028827e+03 1.014558e+01 * time: 1.5635600090026855 6 1.028150e+03 7.359745e+00 * time: 1.7196099758148193 7 1.027831e+03 3.514978e+00 * time: 1.8462138175964355 8 1.027685e+03 8.220644e+00 * time: 1.9627950191497803 9 1.027330e+03 6.509045e+00 * time: 2.078159809112549 10 1.026941e+03 6.448416e+00 * time: 2.187788963317871 11 1.026697e+03 4.987798e+00 * time: 2.3104448318481445 12 1.026643e+03 2.087857e+00 * time: 2.446316957473755 13 1.026612e+03 2.553588e+00 * time: 2.566248893737793 14 1.026594e+03 5.541514e-01 * time: 2.67667293548584 15 1.026562e+03 1.553446e+00 * time: 2.788827896118164 16 1.026527e+03 2.853394e+00 * time: 2.899765968322754 17 1.026507e+03 1.105801e+00 * time: 3.0131609439849854 18 1.026496e+03 5.009212e-01 * time: 3.132354974746704 19 1.026479e+03 1.710960e+00 * time: 3.2540957927703857 20 1.026445e+03 3.409969e+00 * time: 3.3743679523468018 21 1.026393e+03 4.453273e+00 * time: 3.4846668243408203 22 1.026345e+03 3.209626e+00 * time: 3.6196999549865723 23 1.026322e+03 9.979654e-01 * time: 3.7394909858703613 24 1.026311e+03 5.373233e-01 * time: 3.8607168197631836 25 1.026300e+03 1.773934e+00 * time: 3.97986102104187 26 1.026277e+03 3.145893e+00 * time: 4.10134482383728 27 1.026238e+03 3.969622e+00 * time: 4.221444845199585 28 1.026194e+03 3.226069e+00 * time: 4.343890905380249 29 1.026165e+03 1.214667e+00 * time: 4.463124990463257 30 1.026151e+03 3.600419e-01 * time: 4.5829918384552 31 1.026142e+03 1.192578e+00 * time: 4.704821825027466 32 1.026128e+03 1.663870e+00 * time: 4.824930906295776 33 1.026109e+03 1.542272e+00 * time: 4.947938919067383 34 1.026092e+03 7.513828e-01 * time: 5.058568954467773 35 1.026085e+03 2.936937e-01 * time: 5.179253816604614 36 1.026080e+03 4.414037e-01 * time: 5.288574934005737 37 1.026074e+03 7.133445e-01 * time: 5.4088029861450195 38 1.026067e+03 6.797723e-01 * time: 5.5283730030059814 39 1.026059e+03 3.402265e-01 * time: 5.647954940795898 40 1.026052e+03 1.611210e-01 * time: 5.769176959991455 41 1.026047e+03 3.530193e-01 * time: 5.886362791061401 42 1.026041e+03 4.084936e-01 * time: 6.0068747997283936 43 1.026034e+03 2.622676e-01 * time: 6.126940011978149 44 1.026029e+03 6.850675e-02 * time: 6.249473810195923 45 1.026027e+03 8.098138e-02 * time: 6.367302894592285 46 1.026026e+03 6.915900e-02 * time: 6.476688861846924 47 1.026026e+03 2.881101e-02 * time: 6.5859739780426025 48 1.026026e+03 5.638900e-03 * time: 6.7020368576049805 49 1.026026e+03 6.999274e-03 * time: 6.820204973220825 50 1.026026e+03 2.338743e-02 * time: 6.925599813461304 51 1.026026e+03 4.251728e-02 * time: 7.041100025177002 52 1.026026e+03 5.633183e-02 * time: 7.158589839935303 53 1.026026e+03 5.015280e-02 * time: 7.266171932220459 54 1.026026e+03 2.272181e-02 * time: 7.3820250034332275 55 1.026026e+03 4.855109e-04 * time: 7.499140977859497
FittedPumasModel
Dynamical system type: Matrix exponential
Number of subjects: 20
Observation records: Active Missing
dv_pk: 220 0
dv_pd: 220 0
Total: 440 0
Number of parameters: Constant Optimized
0 16
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: GradientNorm
Log-likelihood value: -1026.0258
--------------------
Estimate
--------------------
tvKa 1.3658
tvCL 4.8343
tvV 36.378
tvKe0 0.070617
tvEmax 73.583
tvEC50 3.1168
tvE0 56.528
Ω₁,₁ 0.048684
Ω₂,₂ 0.040628
Ω₃,₃ 0.014049
Ω₄,₄ 0.032244
Ω₅,₅ 0.074529
Ω₆,₆ 1.0522e-6
Ω₇,₇ 0.049157
σ_prop 0.2443
σ_pd 4.6243
--------------------Indirect response models describe drug effects on the production or loss of a response variable. The core concept is turnover: at steady state, the rate of production (\(K_{in}\)) equals the rate of loss (\(K_{out} \cdot R\)), giving baseline \(R_0 = K_{in} / K_{out}\).
\[\frac{dR}{dt} = K_{in} \cdot S(C_p) - K_{out} \cdot R\]
where \(S(C_p)\) is a stimulation or inhibition function applied to either \(K_{in}\) or \(K_{out}\):
| Type | Target | Function | Equation | Response Direction |
|---|---|---|---|---|
| I | Inhibit \(K_{in}\) | \(1 - \frac{I_{max} \cdot C}{IC_{50} + C}\) | \(\frac{dR}{dt} = K_{in}(1-E) - K_{out} \cdot R\) | ↓ Decrease |
| II | Inhibit \(K_{out}\) | \(1 - \frac{I_{max} \cdot C}{IC_{50} + C}\) | \(\frac{dR}{dt} = K_{in} - K_{out}(1-E) \cdot R\) | ↑ Increase |
| III | Stimulate \(K_{in}\) | \(1 + \frac{E_{max} \cdot C}{EC_{50} + C}\) | \(\frac{dR}{dt} = K_{in}(1+E) - K_{out} \cdot R\) | ↑ Increase |
| IV | Stimulate \(K_{out}\) | \(1 + \frac{E_{max} \cdot C}{EC_{50} + C}\) | \(\frac{dR}{dt} = K_{in} - K_{out}(1+E) \cdot R\) | ↓ Decrease |
Most biological responses — blood pressure, clotting factors, cholesterol, blood glucose, cortisol — exist in a dynamic equilibrium where a production process (\(K_{in}\)) is balanced by a degradation or loss process (\(K_{out}\)). At baseline, this equilibrium satisfies \(K_{in} = K_{out} \times R_0\), where \(R_0\) is the baseline response. Drugs perturb this equilibrium by altering either the production or the loss rate, but the response cannot change instantaneously because the existing pool of response variable must turn over. The characteristic time scale for this process is the turnover half-life:
\[t_{1/2,\text{turnover}} = \frac{\ln 2}{K_{out}}\]
This delay is fundamentally different from a distributional delay modeled by an effect compartment (\(k_{e0}\)). The effect compartment delay arises from the time required for drug to equilibrate between plasma and the biophase; the turnover delay arises from the time required for the biological system itself to respond to the perturbation. Distinguishing these two mechanisms — pharmacokinetic delay vs. pharmacodynamic (systems-level) delay — is essential for selecting the appropriate model structure and for making correct predictions about the time course of drug effect.
| Type | Clinical Examples |
|---|---|
| Type I (↓ Kin) | Corticosteroid suppression of cortisol, warfarin inhibition of clotting factor synthesis |
| Type II (↓ Kout) | Heparin reducing factor Xa clearance, EPO reducing RBC destruction |
| Type III (↑ Kin) | EPO stimulating RBC production, GH stimulating IGF-1 synthesis |
| Type IV (↑ Kout) | Diuretics increasing sodium excretion, furosemide increasing urine output |
# Prepare indirect PK-PD data — pivot to wide format for multi-DV
# PK and PD observations are at the same time points, so we merge them into one row
dose_rows = @rsubset(ind_df, :AMT > 0)
pk_rows = @rsubset(ind_df, :TYPE == 0, :AMT == 0)
pd_rows = @rsubset(ind_df, :TYPE == 1)
# Merge PK + PD at same (ID, TIME) into wide format
pk_wide = select(pk_rows, :ID, :TIME, :DV => :dv_pk)
pd_wide = select(pd_rows, :ID, :TIME, :DV => :dv_pd)
obs_wide = outerjoin(pk_wide, pd_wide, on = [:ID, :TIME])
obs_wide.AMT .= 0
obs_wide.CMT .= 2
obs_wide.evid .= 0
# Dose rows
dose_prep = select(dose_rows, :ID, :TIME, :AMT, :CMT)
dose_prep.dv_pk .= missing
dose_prep.dv_pd .= missing
dose_prep.evid .= 1
# Combine and sort
combined = vcat(dose_prep, obs_wide, cols = :union)
combined.AMT = coalesce.(combined.AMT, 0)
combined.evid = coalesce.(combined.evid, 0)
combined.CMT = coalesce.(combined.CMT, 2)
sort!(combined, [:ID, :TIME, order(:evid, rev = true)])
rename!(combined, :ID => :id, :TIME => :time, :AMT => :amt, :CMT => :cmt)
pop_ind = read_pumas(combined; observations = [:dv_pk, :dv_pd])
println("Indirect PKPD subjects: ", length(pop_ind))Indirect PKPD subjects: 20# Type I: Inhibit Kin
# From ind1_pkpd.ctl: ADVAN6, $DES with DADT(3) = KIN*(1-E) - KOUT*A(3)
ind1_model = @model begin
@metadata begin
desc = "Indirect Response Type I: Inhibit Kin"
end
@param begin
tvKa ∈ RealDomain(lower = 0.0, init = 1.5)
tvCL ∈ RealDomain(lower = 0.0, init = 5.0)
tvV ∈ RealDomain(lower = 0.0, init = 35.0)
tvKin ∈ RealDomain(lower = 0.0, init = 10.0)
tvKout ∈ RealDomain(lower = 0.0, init = 0.05)
tvIC50 ∈ RealDomain(lower = 0.0, init = 5.0)
Ω ∈ PDiagDomain(3)
σ_prop ∈ RealDomain(lower = 0.0, init = 0.2)
σ_pd ∈ RealDomain(lower = 0.0, init = 4.47)
end
@random begin
η ~ MvNormal(Ω)
end
@pre begin
Ka = tvKa * exp(η[1])
CL = tvCL * exp(η[2])
Vc = tvV * exp(η[3])
Kin = tvKin
Kout = tvKout
IC50 = tvIC50
# Emax fixed at 1 (as in ctl: THETA(6) = 1 FIX)
end
@init begin
Response = Kin / Kout # steady-state baseline
end
@dynamics begin
Depot' = -Ka * Depot
Central' = Ka * Depot - (CL / Vc) * Central
Response' = Kin * (1 - Central / Vc / (IC50 + Central / Vc)) - Kout * Response
end
@derived begin
cp := @. Central / Vc
dv_pk ~ @. Normal(cp, abs(cp) * σ_prop)
dv_pd ~ @. Normal(Response, σ_pd)
end
endPumasModel
Parameters: tvKa, tvCL, tvV, tvKin, tvKout, tvIC50, Ω, σ_prop, σ_pd
Random effects: η
Covariates:
Dynamical system variables: Depot, Central, Response
Dynamical system type: Nonlinear ODE
Derived: dv_pk, dv_pd
Observed: dv_pk, dv_pdind1_params = (
tvKa = 1.5, tvCL = 5.0, tvV = 35.0,
tvKin = 10.0, tvKout = 0.05, tvIC50 = 5.0,
Ω = Diagonal([0.04, 0.04, 0.04]),
σ_prop = 0.2, σ_pd = sqrt(20.0),
)
ind1_fit = fit(ind1_model, pop_ind, ind1_params, Pumas.FOCEI())
ind1_fit[ Info: Checking the initial parameter values. [ Info: The initial negative log likelihood and its gradient are finite. Check passed. Iter Function value Gradient norm 0 4.755777e+04 1.564310e+05 * time: 3.790855407714844e-5 1 8.252584e+03 1.428514e+04 * time: 0.49408793449401855 2 6.793205e+03 1.134338e+04 * time: 1.0144600868225098 3 3.241463e+03 4.109670e+03 * time: 1.0658290386199951 4 2.207731e+03 1.943402e+03 * time: 1.1155118942260742 5 1.662031e+03 8.773268e+02 * time: 1.164668083190918 6 1.463031e+03 4.925989e+02 * time: 1.2349519729614258 7 1.396012e+03 2.604774e+02 * time: 1.2819089889526367 8 1.380507e+03 1.403022e+02 * time: 1.3285460472106934 9 1.378426e+03 9.163802e+01 * time: 1.3828730583190918 10 1.378204e+03 7.853435e+01 * time: 1.429434061050415 11 1.377989e+03 6.930610e+01 * time: 1.476172924041748 12 1.377331e+03 6.141801e+01 * time: 1.5306730270385742 13 1.375991e+03 6.228986e+01 * time: 1.579176902770996 14 1.373506e+03 5.314549e+01 * time: 1.6345460414886475 15 1.370248e+03 8.670850e+01 * time: 1.6892790794372559 16 1.367310e+03 7.744327e+01 * time: 1.7444300651550293 17 1.366853e+03 4.312742e+01 * time: 1.7886130809783936 18 1.366820e+03 4.177335e+01 * time: 1.8394880294799805 19 1.366715e+03 3.843575e+01 * time: 1.8894360065460205 20 1.366470e+03 3.335661e+01 * time: 1.9385919570922852 21 1.365865e+03 2.798016e+01 * time: 1.9907588958740234 22 1.364607e+03 2.452838e+01 * time: 2.043911933898926 23 1.362592e+03 3.261922e+01 * time: 2.093794107437134 24 1.360506e+03 3.841668e+01 * time: 2.1494359970092773 25 1.359252e+03 3.392255e+01 * time: 2.2037270069122314 26 1.358972e+03 3.011402e+01 * time: 2.2575249671936035 27 1.358937e+03 2.927429e+01 * time: 2.3080780506134033 28 1.358895e+03 2.789962e+01 * time: 2.358457088470459 29 1.358806e+03 2.594693e+01 * time: 2.410141944885254 30 1.358561e+03 2.563835e+01 * time: 2.461894989013672 31 1.357944e+03 2.655573e+01 * time: 2.5110859870910645 32 1.356405e+03 3.503901e+01 * time: 2.566922903060913 33 1.353000e+03 4.379359e+01 * time: 2.6222898960113525 34 1.347292e+03 5.077442e+01 * time: 2.6795051097869873 35 1.341773e+03 7.641960e+01 * time: 2.735551118850708 36 1.339054e+03 8.209362e+01 * time: 2.789168119430542 37 1.338549e+03 7.829311e+01 * time: 2.8432490825653076 38 1.338463e+03 7.591879e+01 * time: 2.8956139087677 39 1.338315e+03 7.264755e+01 * time: 2.9500560760498047 40 1.337929e+03 6.665891e+01 * time: 3.005450963973999 41 1.336981e+03 5.646896e+01 * time: 3.0610110759735107 42 1.334723e+03 4.550099e+01 * time: 3.112504005432129 43 1.329940e+03 4.783224e+01 * time: 3.1673450469970703 44 1.321739e+03 4.039079e+01 * time: 3.2229390144348145 45 1.313190e+03 3.903493e+01 * time: 3.2774569988250732 46 1.309752e+03 2.705504e+01 * time: 3.331799030303955 47 1.308883e+03 2.010338e+01 * time: 3.3858349323272705 48 1.308613e+03 2.241320e+01 * time: 3.4392900466918945 49 1.308418e+03 2.523691e+01 * time: 3.4901609420776367 50 1.308290e+03 2.707148e+01 * time: 3.5420761108398438 51 1.308249e+03 2.793325e+01 * time: 3.594316005706787 52 1.308232e+03 2.786212e+01 * time: 3.6412389278411865 53 1.308182e+03 2.747150e+01 * time: 3.6950459480285645 54 1.308067e+03 2.648734e+01 * time: 3.7477409839630127 55 1.307793e+03 2.415459e+01 * time: 3.7991371154785156 56 1.307267e+03 1.974072e+01 * time: 3.85109806060791 57 1.306609e+03 1.846943e+01 * time: 3.9030261039733887 58 1.306265e+03 1.431225e+01 * time: 3.9569509029388428 59 1.306176e+03 1.388629e+01 * time: 4.009790897369385 60 1.306152e+03 1.388508e+01 * time: 4.056091070175171 61 1.306141e+03 1.347836e+01 * time: 4.105571985244751 62 1.306132e+03 1.319476e+01 * time: 4.154879093170166 63 1.306126e+03 1.320401e+01 * time: 4.204988956451416 64 1.306116e+03 1.333036e+01 * time: 4.2545530796051025 65 1.306095e+03 1.345132e+01 * time: 4.304358959197998 66 1.306042e+03 1.334455e+01 * time: 4.355125904083252 67 1.305923e+03 1.237722e+01 * time: 4.402370929718018 68 1.305701e+03 1.257703e+01 * time: 4.451747894287109 69 1.305423e+03 9.202730e+00 * time: 4.502279996871948 70 1.305224e+03 3.506630e+00 * time: 4.564260005950928 71 1.305154e+03 1.511361e+00 * time: 4.62161111831665 72 1.305142e+03 1.570455e+00 * time: 4.672765016555786 73 1.305140e+03 1.602894e+00 * time: 4.721342086791992 74 1.305139e+03 1.626363e+00 * time: 4.766513109207153 75 1.305139e+03 1.632320e+00 * time: 4.813076019287109 76 1.305139e+03 1.632919e+00 * time: 4.858532905578613 77 1.305139e+03 1.633059e+00 * time: 4.902517080307007 78 1.305138e+03 1.632778e+00 * time: 4.947504043579102 79 1.305138e+03 1.631258e+00 * time: 4.98873496055603 80 1.305137e+03 1.626047e+00 * time: 5.035805940628052 81 1.305135e+03 1.610563e+00 * time: 5.084630966186523 82 1.305129e+03 1.671784e+00 * time: 5.132312059402466 83 1.305114e+03 2.557289e+00 * time: 5.180783987045288 84 1.305082e+03 3.574486e+00 * time: 5.231458902359009 85 1.305028e+03 4.056522e+00 * time: 5.278901100158691 86 1.304972e+03 3.018565e+00 * time: 5.33053994178772 87 1.304946e+03 1.063332e+00 * time: 5.380239963531494 88 1.304941e+03 9.327573e-02 * time: 5.428741931915283 89 1.304941e+03 8.023067e-02 * time: 5.476744890213013 90 1.304941e+03 7.799598e-02 * time: 5.521553039550781 91 1.304941e+03 7.765627e-02 * time: 5.560059070587158 92 1.304941e+03 7.711810e-02 * time: 5.602461099624634 93 1.304941e+03 7.639031e-02 * time: 5.6450440883636475 94 1.304941e+03 7.512634e-02 * time: 5.68911600112915 95 1.304941e+03 7.312195e-02 * time: 5.733319044113159 96 1.304941e+03 6.982247e-02 * time: 5.773839950561523 97 1.304941e+03 6.445591e-02 * time: 5.820050954818726 98 1.304941e+03 7.373950e-02 * time: 5.864423990249634 99 1.304941e+03 1.203593e-01 * time: 5.90929388999939 100 1.304941e+03 1.855943e-01 * time: 5.957979917526245 101 1.304940e+03 2.547643e-01 * time: 6.002300024032593 102 1.304939e+03 2.731362e-01 * time: 6.048641920089722 103 1.304938e+03 1.855887e-01 * time: 6.095395088195801 104 1.304937e+03 6.430116e-02 * time: 6.1450419425964355 105 1.304937e+03 6.036127e-02 * time: 6.19192910194397 106 1.304937e+03 5.790253e-02 * time: 6.231722116470337 107 1.304937e+03 5.742226e-02 * time: 6.27214503288269 108 1.304937e+03 5.728444e-02 * time: 6.313309907913208 109 1.304937e+03 5.728001e-02 * time: 6.365746021270752 110 1.304937e+03 5.725461e-02 * time: 6.418610095977783 111 1.304937e+03 5.725346e-02 * time: 6.482558965682983 112 1.304937e+03 5.507446e-02 * time: 6.522576093673706 113 1.304937e+03 5.448164e-02 * time: 6.563766956329346 114 1.304937e+03 5.161668e-02 * time: 6.6073620319366455 115 1.304937e+03 4.807144e-02 * time: 6.649915933609009 116 1.304937e+03 4.141528e-02 * time: 6.693708896636963 117 1.304937e+03 3.546336e-02 * time: 6.735208988189697 118 1.304937e+03 5.200178e-02 * time: 6.781368017196655 119 1.304936e+03 6.945191e-02 * time: 6.829306125640869 120 1.304935e+03 7.984134e-02 * time: 6.876703977584839 121 1.304933e+03 7.806182e-02 * time: 6.922899961471558 122 1.304932e+03 6.038412e-02 * time: 6.963927984237671 123 1.304931e+03 3.127192e-02 * time: 7.010651111602783 124 1.304930e+03 1.330297e-02 * time: 7.056174039840698 125 1.304930e+03 8.942073e-03 * time: 7.09987211227417 126 1.304930e+03 5.363511e-03 * time: 7.143901109695435 127 1.304930e+03 5.783611e-03 * time: 7.187226057052612 128 1.304930e+03 5.493837e-03 * time: 7.245162010192871 129 1.304930e+03 4.038644e-03 * time: 7.3080220222473145 130 1.304930e+03 2.091960e-03 * time: 7.382246017456055 131 1.304930e+03 5.980395e-04 * time: 7.465126991271973
FittedPumasModel
Dynamical system type: Nonlinear ODE
Solver(s): (OrdinaryDiffEqVerner.Vern7,OrdinaryDiffEqRosenbrock.Rodas5P)
Number of subjects: 20
Observation records: Active Missing
dv_pk: 220 0
dv_pd: 220 0
Total: 440 0
Number of parameters: Constant Optimized
0 11
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: GradientNorm
Log-likelihood value: -1304.9297
---------------------
Estimate
---------------------
tvKa 1.3668
tvCL 4.8346
tvV 36.392
tvKin 0.00013041
tvKout 1.6216e-6
tvIC50 1.4509e7
Ω₁,₁ 0.042084
Ω₂,₂ 0.040423
Ω₃,₃ 0.015019
σ_prop 0.24476
σ_pd 22.874
---------------------println("PK-PD Model Comparison")
println("=" ^ 50)
println("Direct Emax AIC: ", round(aic(direct_fit), digits = 1))
println("Effect Comp AIC: ", round(aic(ecomp_fit), digits = 1))
println("Indirect Type I AIC: ", round(aic(ind1_fit), digits = 1))PK-PD Model Comparison
==================================================
Direct Emax AIC: 2538.7
Effect Comp AIC: 2084.1
Indirect Type I AIC: 2631.9ind_insp = inspect(ind1_fit)
ind_insp_df = DataFrame(ind_insp)
# PD observations only
pd_obs = @rsubset(ind_insp_df, !ismissing(:dv_pd))
fig = Figure(size = (800, 400))
ax1 = Axis(fig[1, 1], xlabel = "PRED", ylabel = "DV (PD)", title = "DV vs PRED")
scatter!(ax1, pd_obs.dv_pd_pred, pd_obs.dv_pd, color = :navy, markersize = 5, alpha = 0.6)
ablines!(ax1, 0, 1, color = :black)
ax2 = Axis(fig[1, 2], xlabel = "IPRED", ylabel = "DV (PD)", title = "DV vs IPRED")
scatter!(ax2, pd_obs.dv_pd_ipred, pd_obs.dv_pd, color = :navy, markersize = 5, alpha = 0.6)
ablines!(ax2, 0, 1, color = :black)
fig[ Info: Calculating predictions. [ Info: Calculating weighted residuals. [ Info: Calculating empirical bayes. [ Info: Evaluating dose control parameters. [ Info: Evaluating individual parameters. [ Info: Done.
When PK data are unavailable (PD-only studies, retrospective analyses, sparse sampling), K-PD models use a virtual compartment to replace the PK model:
\[\frac{dA}{dt} = -K_{DE} \cdot A\]
where \(A\) is the virtual drug amount and \(K_{DE}\) is the apparent elimination rate (\(= CL/V\)). The drug effect is then driven by \(K_{DE} \cdot A\) (analogous to \(CL \cdot C_p\)) rather than by concentration.
Key parameters:
kpd_model = @model begin
@param begin
tvKDE ∈ RealDomain(lower = 0.0, init = 0.1)
tvEmax ∈ RealDomain(lower = 0.0, init = 100.0)
tvEDK50 ∈ RealDomain(lower = 0.0, init = 5.0)
tvE0 ∈ RealDomain(init = 60.0)
Ω ∈ PDiagDomain(2)
σ_pd ∈ RealDomain(lower = 0.0, init = 5.0)
end
@random begin
η ~ MvNormal(Ω)
end
@pre begin
KDE = tvKDE * exp(η[1])
Emax = tvEmax
EDK50 = tvEDK50 * exp(η[2])
E0 = tvE0
end
@dynamics begin
A' = -KDE * A # virtual compartment
end
@derived begin
rate_out := @. KDE * A
effect := @. E0 + Emax * rate_out / (EDK50 + rate_out)
dv ~ @. Normal(effect, σ_pd)
end
endK-PD limitations:
When choosing a PD model structure, follow this decision tree:
| Question | If Yes → | If No → |
|---|---|---|
| Is there a measurable delay between Cp and effect? | Effect compartment or IDR | Direct response |
| Does the drug affect a turnover process? | IDR (Types I–IV) | Effect compartment |
| Is the CE plot loop counter-clockwise? | Effect compartment (ke0) | Direct, tolerance, or IDR |
| Is PK data available? | PK-PD model | K-PD model |
| Is the effect reversible and dose-dependent? | Standard Emax/IDR | Consider more complex models |
| Approach | Pros | Cons |
|---|---|---|
| Sequential (fix PK, estimate PD) | Faster, PK model already validated | Ignores PK uncertainty, may bias PD estimates |
| Simultaneous (estimate PK + PD together) | Full uncertainty propagation, optimal use of data | Slower, larger parameter space, harder to converge |
Recommendation: Start sequential for model development (faster iteration), then switch to simultaneous for the final model.
| Topic | Tutorial | Key Additions |
|---|---|---|
| Lecture: Intrinsic CL & Enzyme Kinetics | PK W7a | Vmax/KM enzyme kinetics — the pharmacological foundation of the Emax model |
| Lecture: Clearance & Hepatic Extraction | PK W6 | Drug elimination physiology underlying PK-PD model structure |
| Lecture: Oral Absorption | PK W8 | First-order absorption, bioavailability — the PK driving function for PD |
| PD model overview | Introduction to PD Models | Four PD paradigms, Emax/Hill curves, potency vs efficacy, decision flowchart |
| Direct response models | Direct Response | Stimulatory + inhibitory Emax, population simulation, full fitting workflow |
| Effect compartment models | Effect Compartment | Hysteresis visualization, ke0 sensitivity analysis, Cp vs Ce comparison, clinical examples |
| Indirect response models | Indirect Response | All 4 IDR types with code, comparative simulation, turnover time sensitivity, clinical examples |
| K-PD models | K-PD Models | Virtual compartment concept, KDE/EDK50 parameterization, limitations, K-PD vs PK-PD decision framework |
| Model selection | PD Model Selection | Systematic selection (EDA → mechanism → statistics), warfarin case study, common pitfalls |
| HCV viral dynamics | HCV GSA | QSP-style PK-PD with viral kinetics and global sensitivity analysis |
| TMDD models | TMDD Foundations → TMDD Workflows | Target-mediated drug disposition: full model, approximations, special cases |