Direct Emax Model¶

Simple hyperbolic concentration-effect relationship where effect is directly proportional to receptor occupancy.

Model Overview¶

graph LR
    C[Plasma<br/>Concentration] -->|Direct Link| E[Effect]

    subgraph "Emax Function"
        direction TB
        E0[E0: Baseline]
        EM[Emax: Maximum]
        EC[EC50: Potency]
    end

Clinical Applications¶

Simple receptor-mediated effects
Anticoagulants (warfarin effect on INR)
Beta-blockers (heart rate reduction)
Enzyme inhibitors
Rapid-equilibrium systems

When to Use¶

Use When	Don't Use When
Effect tracks concentration	Temporal delay observed
Rapid equilibration	Hysteresis present
Simple dose-response	Tolerance develops
Reversible binding	Effect persists after drug

Mathematical Formulation¶

Parameters¶

Parameter	Symbol	Units	Description	Constraints
Baseline effect	E0	varies	Effect with no drug	any real
Maximum effect	Emax	varies	Maximum drug-induced change	typically > 0
Half-maximal concentration	EC50	mg/L	Concentration at 50% Emax	EC50 > 0

Equation¶

\[E(C) = E_0 + \frac{E_{max} \cdot C}{EC_{50} + C}\]

Fraction of Maximum Effect¶

At any concentration:

\[\frac{E - E_0}{E_{max}} = \frac{C}{EC_{50} + C}\]

Key Concentrations¶

Fraction of Emax	Concentration Required
20%	C = EC50/4
50%	C = EC50
80%	C = EC50 × 4
90%	C = EC50 × 9
99%	C = EC50 × 99

Model Behavior¶

Concentration-Effect Curve¶

Effect
   │
Emax├─────────────────────────────
    │                     ┌──────
    │                  ┌──┘
    │               ┌──┘
    │            ┌──┘
    │         ┌──┘
E0+½├ ─ ─ ─ ┬┘
    │       │
 E0 ├───────┴─────────────────────
    └───────┼─────────────────────→ C
          EC50

Key Properties¶

Hyperbolic shape: Effect increases rapidly at low C, plateaus at high C
Saturable: Effect cannot exceed E0 + Emax
First-order at low C: When C << EC50, E ≈ E0 + (Emax/EC50) × C
Zero-order at high C: When C >> EC50, E ≈ E0 + Emax

Julia API¶

Type Definitions¶

# Model kind
struct DirectEmax <: PDModelKind end

# Parameters
struct DirectEmaxParams
    E0::Float64      # Baseline effect
    Emax::Float64    # Maximum effect
    EC50::Float64    # Half-maximal concentration (mg/L)
end

Basic Usage¶

using NeoPKPD

# Define PD parameters
# E0 = 70 (baseline), Emax = 40 (reduction), EC50 = 1.5 mg/L
pd_params = DirectEmaxParams(70.0, -40.0, 1.5)

# Create PD specification
pd_spec = PDSpec(DirectEmax(), "heart_rate", pd_params)

# Evaluate at specific concentrations
concentrations = [0.0, 0.5, 1.0, 1.5, 2.0, 3.0, 5.0]
effects = evaluate(pd_spec, concentrations)

for (c, e) in zip(concentrations, effects)
    println("C = $c mg/L: Effect = $(round(e, digits=1))")
end

Expected Output:

C = 0.0 mg/L: Effect = 70.0
C = 0.5 mg/L: Effect = 60.0
C = 1.0 mg/L: Effect = 54.0
C = 1.5 mg/L: Effect = 50.0
C = 2.0 mg/L: Effect = 47.1
C = 3.0 mg/L: Effect = 43.3
C = 5.0 mg/L: Effect = 39.2

Coupled PK-PD Simulation¶

using NeoPKPD

# PK model: One-compartment oral
pk_params = OneCompOralFirstOrderParams(1.5, 5.0, 30.0)  # Ka, CL, V
pk_spec = ModelSpec(OneCompOralFirstOrder(), "pk", pk_params, [DoseEvent(0.0, 100.0)])

# PD model: Direct Emax (inhibitory)
pd_params = DirectEmaxParams(80.0, -30.0, 2.0)  # Baseline HR, max reduction, EC50
pd_spec = PDSpec(DirectEmax(), "heart_rate", pd_params)

# Simulation grid
grid = SimGrid(0.0, 24.0, collect(0.0:0.25:24.0))
solver = SolverSpec(:Tsit5, 1e-10, 1e-12, 10_000_000)

# Simulate PK
pk_result = simulate(pk_spec, grid, solver)
conc = pk_result.observations[:conc]

# Calculate PD effect
effects = evaluate(pd_spec, conc)

# Find minimum heart rate (maximum effect)
min_hr, idx = findmin(effects)
time_of_min = grid.times[idx]

println("Maximum effect at t = $(round(time_of_min, digits=1)) h")
println("Heart rate reduced to $(round(min_hr, digits=1)) bpm")

Parameter Interpretation¶

Potency vs Efficacy¶

Parameter	Meaning	Clinical Relevance
EC50	Potency - concentration for 50% effect	Lower EC50 = more potent drug
Emax	Efficacy - maximum possible effect	Higher Emax = more efficacious

Comparing Drugs¶

# Drug A: High potency, moderate efficacy
params_A = DirectEmaxParams(0.0, 80.0, 0.5)

# Drug B: Low potency, high efficacy
params_B = DirectEmaxParams(0.0, 100.0, 5.0)

# At low concentration (1 mg/L):
# Drug A: E = 80 × 1/(0.5+1) = 53.3
# Drug B: E = 100 × 1/(5+1) = 16.7
# Drug A wins at low concentrations

# At high concentration (10 mg/L):
# Drug A: E = 80 × 10/(0.5+10) ≈ 76.2
# Drug B: E = 100 × 10/(5+10) = 66.7
# Drug A still higher but closer

Clinical Example: Beta-Blocker¶

# Propranolol effect on heart rate
# Baseline HR: 80 bpm
# Max reduction: 30 bpm
# EC50: 50 ng/mL

pd_params = DirectEmaxParams(80.0, -30.0, 0.050)  # EC50 in mg/L

# Target: HR reduction of 15 bpm (half of max effect)
# Required C = EC50 = 0.050 mg/L = 50 ng/mL

# Verify
spec = PDSpec(DirectEmax(), "hr", pd_params)
effect_at_ec50 = evaluate(spec, [0.050])[1]
println("HR at EC50: $(round(effect_at_ec50, digits=1)) bpm")
# Expected: 80 - 15 = 65 bpm

Stimulatory vs Inhibitory Effects¶

Inhibitory Effect (Emax < 0)¶

# Drug reduces blood pressure
# E0 = baseline, Emax = negative (reduction)
pd_inhibit = DirectEmaxParams(140.0, -40.0, 2.0)

# Effect range: 140 → 100 mmHg

Stimulatory Effect (Emax > 0)¶

# Drug increases enzyme activity
# E0 = baseline, Emax = positive (increase)
pd_stimulate = DirectEmaxParams(100.0, 200.0, 5.0)

# Effect range: 100 → 300 units

Model Limitations¶

Limitation	Alternative
No delay between C and E	Use Effect Compartment model
Fixed steepness	Use Sigmoid Emax (add Hill coefficient)
No tolerance	Use Indirect Response models
Symmetric curve	Use more complex models

Derived Quantities¶

Sensitivity (Slope at Origin)¶

\[\text{Sensitivity} = \frac{dE}{dC}\bigg|_{C=0} = \frac{E_{max}}{EC_{50}}\]

Therapeutic Window¶

If target effect range is E1 to E2:

\[C_1 = EC_{50} \cdot \frac{E_1 - E_0}{E_{max} - (E_1 - E_0)}\]

\[C_2 = EC_{50} \cdot \frac{E_2 - E_0}{E_{max} - (E_2 - E_0)}\]

Equations Summary¶

Quantity	Formula
Effect	\(E_0 + E_{max} \cdot C / (EC_{50} + C)\)
Fraction of Emax	\(C / (EC_{50} + C)\)
C for target effect	\(EC_{50} \cdot (E - E_0) / (E_{max} - (E - E_0))\)
Sensitivity	\(E_{max} / EC_{50}\)