Indirect Response Models (IRM)¶

Mechanism-based PD models where drug affects the production (Kin) or elimination (Kout) of a response variable, rather than directly modulating the response.

Model Overview¶

graph LR
    subgraph "Turnover System"
        K1[Kin<br/>Production] --> R[Response R]
        R --> K2[Kout<br/>Elimination]
    end

    C[Drug<br/>Concentration] -.->|Inhibits or<br/>Stimulates| K1
    C -.->|Inhibits or<br/>Stimulates| K2

The Four IRM Types¶

Model	Target	Direction	Effect on R
IRM-I	Kin (production)	Inhibition	Decreases R
IRM-II	Kin (production)	Stimulation	Increases R
IRM-III	Kout (elimination)	Inhibition	Increases R
IRM-IV	Kout (elimination)	Stimulation	Decreases R

Clinical Applications¶

IRM Type	Example Applications
IRM-I	Corticosteroids (cortisol), Statins (cholesterol), Warfarin (clotting factors)
IRM-II	EPO (RBC production), G-CSF (neutrophils), Growth factors
IRM-III	Diuretics (renal function), Thyroid hormones (metabolism)
IRM-IV	Laxatives (bowel motility), Some immunosuppressants

Mathematical Formulation¶

Baseline Turnover¶

At baseline (no drug):

\[\frac{dR}{dt} = K_{in} - K_{out} \cdot R = 0\]

Therefore: \(R_0 = K_{in} / K_{out}\)

Drug Functions¶

Inhibition function (for IRM-I, IRM-III):

\[I(C) = \frac{I_{max} \cdot C}{IC_{50} + C}\]

Stimulation function (for IRM-II, IRM-IV):

\[S(C) = \frac{S_{max} \cdot C}{SC_{50} + C}\]

IRM Equations¶

Model	Differential Equation
IRM-I	\(\frac{dR}{dt} = K_{in} \cdot (1 - I(C)) - K_{out} \cdot R\)
IRM-II	\(\frac{dR}{dt} = K_{in} \cdot (1 + S(C)) - K_{out} \cdot R\)
IRM-III	\(\frac{dR}{dt} = K_{in} - K_{out} \cdot (1 - I(C)) \cdot R\)
IRM-IV	\(\frac{dR}{dt} = K_{in} - K_{out} \cdot (1 + S(C)) \cdot R\)

Parameters¶

Common Parameters¶

Parameter	Symbol	Units	Description	Constraints
Production rate	Kin	units/h	Zero-order production	Kin > 0
Elimination rate	Kout	1/h	First-order elimination	Kout > 0
Baseline response	R0	units	Steady-state baseline	R0 = Kin/Kout

Inhibition Parameters (IRM-I, IRM-III)¶

Parameter	Symbol	Units	Description	Constraints
Maximum inhibition	Imax	-	Fraction at saturation	0 < Imax ≤ 1
Inhibition potency	IC50	mg/L	Concentration for 50% Imax	IC50 > 0

Stimulation Parameters (IRM-II, IRM-IV)¶

Parameter	Symbol	Units	Description	Constraints
Maximum stimulation	Smax	-	Fold-increase at saturation	Smax > 0
Stimulation potency	SC50	mg/L	Concentration for 50% Smax	SC50 > 0

Julia API¶

Type Definitions¶

# IRM-I: Inhibition of Kin
struct IndirectResponseIRM1 <: PDModelKind end
struct IndirectResponseIRM1Params
    Kin::Float64      # Production rate
    Kout::Float64     # Elimination rate constant
    R0::Float64       # Baseline response
    Imax::Float64     # Maximum inhibition [0, 1]
    IC50::Float64     # Inhibition potency
end

# IRM-II: Stimulation of Kin
struct IndirectResponseIRM2 <: PDModelKind end
struct IndirectResponseIRM2Params
    Kin::Float64
    Kout::Float64
    R0::Float64
    Smax::Float64     # Maximum stimulation (can exceed 1)
    SC50::Float64     # Stimulation potency
end

# IRM-III: Inhibition of Kout (also known as IndirectResponseTurnover)
struct IndirectResponseTurnover <: PDModelKind end
struct IndirectResponseTurnoverParams
    Kin::Float64
    Kout::Float64
    R0::Float64
    Imax::Float64
    IC50::Float64
end

# IRM-IV: Stimulation of Kout
struct IndirectResponseIRM4 <: PDModelKind end
struct IndirectResponseIRM4Params
    Kin::Float64
    Kout::Float64
    R0::Float64
    Smax::Float64
    SC50::Float64
end

IRM-I: Inhibition of Production¶

Mechanism¶

Drug inhibits the production of the response variable.

\[\frac{dR}{dt} = K_{in} \cdot \left(1 - \frac{I_{max} \cdot C}{IC_{50} + C}\right) - K_{out} \cdot R\]

Clinical Example: Corticosteroid Effect on Cortisol¶

using NeoPKPD

# Cortisol dynamics
# Baseline cortisol: 15 mcg/dL
# Turnover half-life: ~1.5 hours
Kout = log(2) / 1.5  # 0.46/h
R0 = 15.0
Kin = Kout * R0  # 6.9 mcg/dL/h

# Corticosteroid effect
pd_params = IndirectResponseIRM1Params(
    Kin,
    Kout,
    R0,
    0.9,    # Imax: 90% suppression possible
    0.01    # IC50: 0.01 mg/L (very potent)
)

# Validate
spec = PDSpec(IndirectResponseIRM1(), "cortisol", pd_params)
validate(spec)

println("Baseline cortisol: $(R0) mcg/dL")
println("Turnover t1/2: $(round(log(2)/Kout, digits=2)) h")

Response Profile¶

# PK: Single oral dose
pk_params = OneCompOralFirstOrderParams(1.5, 2.0, 50.0)
doses = [DoseEvent(0.0, 10.0)]  # 10 mg dose

pk_spec = ModelSpec(OneCompOralFirstOrder(), "steroid", pk_params, doses)

grid = SimGrid(0.0, 48.0, collect(0.0:0.25:48.0))
solver = SolverSpec(:Tsit5, 1e-10, 1e-12, 10_000_000)

# Simulate coupled PKPD
result = simulate_pkpd(pk_spec, spec, grid, solver)

# Key features of IRM-I:
# - Effect is delayed relative to concentration
# - Maximum suppression occurs AFTER Cmax
# - Slow return to baseline (governed by Kout)

IRM-II: Stimulation of Production¶

Mechanism¶

Drug increases the production rate.

\[\frac{dR}{dt} = K_{in} \cdot \left(1 + \frac{S_{max} \cdot C}{SC_{50} + C}\right) - K_{out} \cdot R\]

Clinical Example: EPO on Red Blood Cells¶

using NeoPKPD

# RBC dynamics
# Baseline RBC: 5 million/mcL
# Turnover half-life: ~30 days
Kout = log(2) / (30 * 24)  # 0.001/h
R0 = 5.0  # million/mcL
Kin = Kout * R0

# EPO stimulation
pd_params = IndirectResponseIRM2Params(
    Kin,
    Kout,
    R0,
    3.0,    # Smax: up to 4× production
    0.1     # SC50: 0.1 mIU/mL
)

spec = PDSpec(IndirectResponseIRM2(), "rbc", pd_params)

# Maximum possible response at infinite C:
# R_max = Kin × (1 + Smax) / Kout = R0 × (1 + Smax) = 20 million/mcL
println("Maximum possible RBC: $(R0 * (1 + pd_params.Smax)) million/mcL")

IRM-III: Inhibition of Elimination¶

Mechanism¶

Drug inhibits the elimination of the response variable.

\[\frac{dR}{dt} = K_{in} - K_{out} \cdot \left(1 - \frac{I_{max} \cdot C}{IC_{50} + C}\right) \cdot R\]

Clinical Example: Warfarin Effect on Clotting Factors¶

using NeoPKPD

# Clotting factor dynamics
# Baseline: 100% of normal
# Turnover half-life: ~36 hours
Kout = log(2) / 36.0
R0 = 100.0  # % of normal
Kin = Kout * R0

# Warfarin effect (inhibits synthesis, modeled as IRM-III)
pd_params = IndirectResponseTurnoverParams(
    Kin,
    Kout,
    R0,
    0.95,   # Imax: 95% inhibition possible
    1.5     # IC50: 1.5 mg/L
)

spec = PDSpec(IndirectResponseTurnover(), "clotting", pd_params)

# Minimum possible response (at complete inhibition):
# When I(C) → Imax, Kout_eff → Kout × (1 - Imax)
# New steady state: R_min = Kin / (Kout × (1 - Imax))
# For Imax = 0.95: R_min = R0 / 0.05 = 2000 (unrealistic)
# In practice, Imax < 1 ensures R remains finite
println("With complete inhibition, R could increase to: ",
        round(R0 / (1 - pd_params.Imax), digits=1), "%")

IRM-IV: Stimulation of Elimination¶

Mechanism¶

Drug increases the elimination rate.

\[\frac{dR}{dt} = K_{in} - K_{out} \cdot \left(1 + \frac{S_{max} \cdot C}{SC_{50} + C}\right) \cdot R\]

Clinical Example: Laxative Effect¶

using NeoPKPD

# Bowel content dynamics
Kout = log(2) / 12.0  # ~12 hour transit
R0 = 100.0  # arbitrary units
Kin = Kout * R0

# Laxative stimulates elimination
pd_params = IndirectResponseIRM4Params(
    Kin,
    Kout,
    R0,
    5.0,    # Smax: up to 6× elimination rate
    0.5     # SC50
)

spec = PDSpec(IndirectResponseIRM4(), "bowel", pd_params)

# Minimum response at high drug concentration:
# R_min = Kin / (Kout × (1 + Smax)) = R0 / (1 + Smax)
println("Minimum response: $(round(R0 / (1 + pd_params.Smax), digits=1)) units")

Comparing IRM Types¶

using NeoPKPD

# Same baseline and turnover for all
Kout = 0.1  # 1/h
R0 = 100.0
Kin = Kout * R0

# Same drug concentration profile
pk_params = OneCompIVBolusParams(2.0, 20.0)
doses = [DoseEvent(0.0, 100.0)]
pk_spec = ModelSpec(OneCompIVBolus(), "drug", pk_params, doses)

grid = SimGrid(0.0, 48.0, collect(0.0:0.25:48.0))
solver = SolverSpec(:Tsit5, 1e-10, 1e-12, 10_000_000)

pk_result = simulate(pk_spec, grid, solver)

# IRM-I and IRM-IV: Response DECREASES
params_irm1 = IndirectResponseIRM1Params(Kin, Kout, R0, 0.8, 2.0)
params_irm4 = IndirectResponseIRM4Params(Kin, Kout, R0, 4.0, 2.0)

# IRM-II and IRM-III: Response INCREASES
params_irm2 = IndirectResponseIRM2Params(Kin, Kout, R0, 3.0, 2.0)
params_irm3 = IndirectResponseTurnoverParams(Kin, Kout, R0, 0.8, 2.0)

println("At high drug concentrations:")
println("  IRM-I (inhib Kin): R decreases toward $(R0 * (1 - 0.8))")
println("  IRM-II (stim Kin): R increases toward $(R0 * (1 + 3.0))")
println("  IRM-III (inhib Kout): R increases (limited by Imax < 1)")
println("  IRM-IV (stim Kout): R decreases toward $(round(R0 / (1 + 4.0), digits=1))")

Response Dynamics¶

Time to Maximum/Minimum Effect¶

Unlike direct effect models, IRM effects are delayed:

Factor	Effect on Delay
Lower Kout	Longer delay (slower turnover)
Higher IC50/SC50	Longer time at effective concentration
Longer drug exposure	Greater effect accumulation

Return to Baseline¶

After drug washout:

\[t_{90\%\ recovery} \approx \frac{2.3}{K_{out}}\]

Population Simulation¶

using NeoPKPD

# Typical IRM-I parameters
typical_params = IndirectResponseIRM1Params(10.0, 0.1, 100.0, 0.8, 2.0)

# IIV on turnover and potency
omega = OmegaMatrix([
    0.09 0.0  0.0;    # ω²_Kout (30% CV on turnover)
    0.0  0.16 0.0;    # ω²_Imax (40% CV on max effect)
    0.0  0.0  0.25    # ω²_IC50 (50% CV on potency)
])

# Note: R0 and Kin are usually fixed or derived
# since R0 = Kin/Kout at baseline

base_spec = PDSpec(IndirectResponseIRM1(), "pop_irm", typical_params)
pop_spec = PopulationSpec(base_spec, 50, omega, 12345)

Model Selection Guide¶

Observation	Suggested Model
Drug decreases biomarker	IRM-I or IRM-IV
Drug increases biomarker	IRM-II or IRM-III
Effect persists after drug washout	Any IRM (vs direct effect)
Biomarker is a production rate	IRM-I (inhibit) or IRM-II (stimulate)
Biomarker is eliminated/degraded	IRM-III (inhibit) or IRM-IV (stimulate)
Known mechanism of action	Select based on biology

Equations Summary¶

Model	dR/dt	Steady State at High C
IRM-I	\(K_{in}(1-I(C)) - K_{out}R\)	\(R_0(1-I_{max})\)
IRM-II	\(K_{in}(1+S(C)) - K_{out}R\)	\(R_0(1+S_{max})\)
IRM-III	\(K_{in} - K_{out}(1-I(C))R\)	\(R_0/(1-I_{max})\)
IRM-IV	\(K_{in} - K_{out}(1+S(C))R\)	\(R_0/(1+S_{max})\)

Common to all: - Baseline: \(R_0 = K_{in}/K_{out}\) - Recovery t1/2: \(\ln(2)/K_{out}\)

Indirect Response Models (IRM)¶

Model Overview¶

The Four IRM Types¶

Clinical Applications¶

Mathematical Formulation¶

Baseline Turnover¶

Drug Functions¶

IRM Equations¶

Parameters¶

Common Parameters¶

Inhibition Parameters (IRM-I, IRM-III)¶

Stimulation Parameters (IRM-II, IRM-IV)¶

Julia API¶

Type Definitions¶

IRM-I: Inhibition of Production¶

Mechanism¶

Clinical Example: Corticosteroid Effect on Cortisol¶

Response Profile¶

IRM-II: Stimulation of Production¶

Mechanism¶

Clinical Example: EPO on Red Blood Cells¶

IRM-III: Inhibition of Elimination¶

Mechanism¶

Clinical Example: Warfarin Effect on Clotting Factors¶

IRM-IV: Stimulation of Elimination¶

Mechanism¶

Clinical Example: Laxative Effect¶

Comparing IRM Types¶

Response Dynamics¶

Time to Maximum/Minimum Effect¶

Return to Baseline¶

Population Simulation¶

Model Selection Guide¶

Equations Summary¶

See Also¶