Joint PK/PD Modeling with Warfarin

Fit a joint PK/PD model with multiple endpoints, ODEs, diagnostics, and VPCs using the warfarin example.
Tip

Big picture: Joint PK/PD models connect exposure and response in one model. This lesson shows how ODEs, multiple endpoints, diagnostics, and VPCs come together.

Learning Objectives

By the end of this lesson, you will be able to:

  • recognize multiple-endpoint PK/PD data
  • explain how PK and PD endpoints are linked
  • identify ODEs inside an nlmixr2 model
  • fit a joint warfarin PK/PD model
  • inspect diagnostics and VPCs for a PK/PD model

Key Ideas

  • one subject may contribute PK and PD observations
  • dvid identifies the endpoint
  • PK drives PD through concentration
  • delayed response can be modeled with ODEs
  • diagnostics still apply to PK/PD models

Setup

library(tidyverse)
library(nlmixr2)
library(nlmixr2data)
library(nlmixr2plot)
library(ggPMX)

data("warfarin", package = "nlmixr2data")

Why Joint PK/PD Modeling?

Population PK asks:

Dose → Concentration

PK/PD asks:

Dose

↓

Concentration

↓

Response

The response may not happen immediately.

For warfarin, concentration affects a biological turnover process that changes the pharmacodynamic response over time.

Worked Example 1: Inspect Multiple Endpoints

Inspect endpoint counts.

warfarin %>%
  count(dvid)
  dvid   n
1   cp 283
2  pca 232

Interpretation:

dvid Endpoint Meaning
cp PK warfarin concentration
pca PD prothrombin complex activity

Inspect the data.

glimpse(warfarin)
Rows: 515
Columns: 9
$ id   <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2,…
$ time <dbl> 0.0, 0.5, 1.0, 2.0, 3.0, 6.0, 9.0, 12.0, 24.0, 24.0, 36.0, 36.0, …
$ amt  <dbl> 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0…
$ dv   <dbl> 0.0, 0.0, 1.9, 3.3, 6.6, 9.1, 10.8, 8.6, 5.6, 44.0, 4.0, 27.0, 2.…
$ dvid <fct> cp, cp, cp, cp, cp, cp, cp, cp, cp, pca, cp, pca, cp, pca, cp, pc…
$ evid <int> 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,…
$ wt   <dbl> 66.7, 66.7, 66.7, 66.7, 66.7, 66.7, 66.7, 66.7, 66.7, 66.7, 66.7,…
$ age  <int> 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 5…
$ sex  <fct> male, male, male, male, male, male, male, male, male, male, male,…

The dataset contains:

id, time, amt, dv, dvid, evid, wt, age, sex

Important variables:

Variable Meaning
id subject identifier
time observation time
amt dose amount
dv observed value
dvid endpoint identifier
evid event type
wt weight
age age
sex sex

Earlier PK examples used one endpoint.

Here:

one subject → concentration endpoint + response endpoint

This introduces the idea of multiple endpoints.

Worked Example 1.5: Visualize PK and PD Observations

Before fitting the model, inspect the observed profiles.

warfarin %>%
  ggplot(
    aes(
      time,
      dv
    )
  ) +
geom_point(
  alpha = 0.5
) +
geom_line(
  aes(group = id),
  alpha = 0.15
) +
facet_wrap(
  ~ dvid,
  scales = "free_y"
) +
labs(
  title = "Observed PK and PD Profiles",
  x = "Time",
  y = "Observed Value"
)

Interpretation:

Compare the endpoints.

PK (cp) often shows:

Dose

↓

Peak

↓

Decline

PD (pca) may show:

Exposure

↓

Delayed Response

↓

Recovery

Question:

Does the response appear to occur immediately?

Notice that the PD response may evolve differently from concentration.

This motivates joint PK/PD modeling.

Worked Example 2: Define the Warfarin PK/PD Model

This lesson introduces a joint PK/PD model.

Compared with earlier population PK examples, this model adds pharmacodynamics.

Conceptually:

Dose

↓

Concentration

↓

Drug Effect

↓

Response

This model includes:

  • transit absorption
  • PK distribution
  • concentration-driven inhibition of production
  • indirect response dynamics
  • separate observation models

Visual Intuition: Warfarin PK/PD Model

This diagram shows the model structure before reading the implementation.

Notice:

  • PK determines concentration (cp)
  • concentration drives drug effect
  • drug effect inhibits production
  • the response (pca) changes gradually over time

Response does not occur immediately.

This delayed behavior is why this is an indirect response model.

NoteWhy Is There a Transit Compartment?

You may notice:

Depot → Gut → Central

Earlier population PK examples often used:

Depot → Central

This model adds a transit step.

Conceptually:

Dose
↓
Delay before absorption
↓
Systemic concentration

The transit compartment allows absorption to occur more gradually.

For this lesson, do not focus on the mathematics.

Focus on the bigger idea:

PK creates concentration
↓
Concentration drives PD

Later courses revisit transit and more advanced absorption models in greater detail.

Now inspect how this structure appears in nlmixr2.

pk_turnover_emax <- function(){

    ini({

        tktr <- log(1)
        tka <- log(1)
        tcl <- log(0.1)
        tv <- log(10)

        eta.ktr ~ 1
        eta.ka ~ 1
        eta.cl ~ 2
        eta.v ~ 1

        prop.err <- 0.1
        pkadd.err <- 0.1

        temax <- logit(0.8)
        tec50 <- log(0.5)
        tkout <- log(0.05)
        te0 <- log(100)

        eta.emax ~ 0.5
        eta.ec50 ~ 0.5
        eta.kout ~ 0.5
        eta.e0 ~ 0.5

        pdadd.err <- 10

    })

    model({

        ktr <- exp(tktr + eta.ktr)
        ka <- exp(tka + eta.ka)
        cl <- exp(tcl + eta.cl)
        v <- exp(tv + eta.v)

        emax <- expit(temax + eta.emax)
        ec50 <- exp(tec50 + eta.ec50)
        kout <- exp(tkout + eta.kout)
        e0 <- exp(te0 + eta.e0)

        dcp <- center / v

        pd <- 1 - emax * dcp / (ec50 + dcp)

        effect(0) = e0
        kin <- e0 * kout

        d/dt(depot) = -ktr * depot
        d/dt(gut) = ktr * depot - ka * gut
        d/dt(center) = ka * gut - cl / v * center
        d/dt(effect) = kin * pd - kout * effect

        cp <- center / v

        cp ~ prop(prop.err) + add(pkadd.err)
        effect ~ add(pdadd.err) | pca

    })

}

Notice two important parts.

First:

pd <- 1 - emax * dcp / (ec50 + dcp)

Interpretation:

Higher concentration

↓

Stronger inhibition

↓

Lower production

Second:

d/dt(effect) = kin * pd - kout * effect

Interpretation:

Rate of Change = Production − Loss

Finally, inspect the endpoint definitions.

cp ~ prop(prop.err) + add(pkadd.err)

effect ~ add(pdadd.err) | pca

Interpretation:

cp → PK observations

effect → PD observations

Each endpoint receives its own observation model.

This allows PK and PD behavior to be evaluated separately while remaining connected through concentration.

The | pca portion maps the model variable effect to observations whose endpoint identifier (dvid) is pca.

Conceptually:

effect state variable
        ↓
mapped to
        ↓
pca observations

Worked Example 3: Inspect the Parsed Model

Compile the model.

ui <- pk_turnover_emax()

ui

\[\begin{align*} {ktr} & = \exp\left({tktr}+{eta.ktr}\right) \\ {ka} & = \exp\left({tka}+{eta.ka}\right) \\ {cl} & = \exp\left({tcl}+{eta.cl}\right) \\ {v} & = \exp\left({tv}+{eta.v}\right) \\ {emax} & = expit({temax}+{eta.emax}, {0}, {1}) \\ {ec50} & = \exp\left({tec50}+{eta.ec50}\right) \\ {kout} & = \exp\left({tkout}+{eta.kout}\right) \\ {e0} & = \exp\left({te0}+{eta.e0}\right) \\ {dcp} & = \frac{{center}}{{v}} \\ {pd} & = {1}-\frac{{emax} {\times} {dcp}}{\left({ec50}+{dcp}\right)} \\ effect({0}) & = {e0} \\ {kin} & = {e0} {\times} {kout} \\ \frac{d \: depot}{dt} & = -{ktr} {\times} {depot} \\ \frac{d \: gut}{dt} & = {ktr} {\times} {depot}-{ka} {\times} {gut} \\ \frac{d \: center}{dt} & = {ka} {\times} {gut}-\frac{{cl}}{{v}} {\times} {center} \\ \frac{d \: effect}{dt} & = {kin} {\times} {pd}-{kout} {\times} {effect} \\ {cp} & = \frac{{center}}{{v}} \\ {cp} & \sim prop({prop.err})+add({pkadd.err}) \\ {effect} & \sim add({pdadd.err}){\lor}{pca} \end{align*}\]

Inspect endpoint mapping.

ui$multipleEndpoint
    variable                cmt                dvid*
1     cp ~ …  cmt='cp' or cmt=5  dvid='cp' or dvid=1
2 effect ~ … cmt='pca' or cmt=6 dvid='pca' or dvid=2

Interpretation:

cp → PK endpoint

effect → PD endpoint

Inspect the ODE structure.

Focus on:

d/dt(effect) = kin * pd - kout * effect

Interpretation:

Rate of Change = Production − Loss

Drug concentration modifies the production term.

Worked Example 4: Fit the Joint PK/PD Model

Fit the model.

fit_warf <-
  nlmixr2(
    pk_turnover_emax,
    warfarin,
    est = "saem",
    control = list(
      print = 0
    ),
    table = list(
      cwres = TRUE,
      npde = TRUE
    )
  )

Notice that we changed estimation methods.

Earlier modules primarily used:

est = "focei"

This lesson uses:

est = "saem"

where SAEM stands for:

Stochastic Approximation Expectation Maximization

because this model combines:

  • multiple endpoints
  • ODEs
  • nonlinear relationships
  • more random effects

Conceptually:

Simpler Population PK → FOCEi often works well

More Complex PK/PD → SAEM is commonly preferred

Inspect the fit.

print(fit_warf)
── nlmixr² SAEM OBJF by FOCEi approximation ──

          OBJF      AIC      BIC Log-likelihood Condition#(Cov) Condition#(Cor)
FOCEi 1384.359 2310.053 2389.474      -1136.027        3337.016        18.81139

── Time (sec $time): ──

           setup optimize covariance   saem table compress
elapsed 0.002618    3e-06    0.04601 68.728 6.076        0

── Population Parameters ($parFixed or $parFixedDf): ──

             Est.     SE  %RSE Back-transformed(95%CI) BSV(CV% or SD)
tktr        0.441  0.544   124      1.55 (0.535, 4.51)           111.
tka        -0.262  0.269   103      0.769 (0.454, 1.3)           13.2
tcl         -1.97 0.0511   2.6     0.14 (0.127, 0.155)           26.7
tv           2.01 0.0481   2.4       7.43 (6.77, 8.17)           21.7
prop.err    0.121                                0.121               
pkadd.err   0.803                                0.803               
temax        3.44  0.642  18.7    0.969 (0.899, 0.991)          0.248
tec50     -0.0934  0.136   146     0.911 (0.698, 1.19)           45.8
tkout       -2.94  0.037  1.26  0.0531 (0.0493, 0.057)           6.01
te0          4.57 0.0115 0.251       96.6 (94.4, 98.8)           5.25
pdadd.err     3.6                                  3.6               
          Shrink(SD)%
tktr           45.0% 
tka            81.2% 
tcl            6.66% 
tv             15.9% 
prop.err             
pkadd.err            
temax          81.7% 
tec50          9.45% 
tkout          44.6% 
te0            16.9% 
pdadd.err            
 
  Covariance Type ($covMethod): linFim
  No correlations in between subject variability (BSV) matrix
  Full BSV covariance ($omega) or correlation ($omegaR; diagonals=SDs) 
  Distribution stats (mean/skewness/kurtosis/p-value) available in $shrink 
  Censoring ($censInformation): No censoring

── Fit Data (object is a modified tibble): ──
# A tibble: 483 × 44
  ID     TIME CMT      DV EPRED  ERES  NPDE    NPD    PDE     PD  PRED   RES
  <fct> <dbl> <fct> <dbl> <dbl> <dbl> <dbl>  <dbl>  <dbl>  <dbl> <dbl> <dbl>
1 1       0.5 cp      0    1.77 -1.77 -1.93 -1.53  0.0267 0.0633  1.38 -1.38
2 1       1   cp      1.9  4.07 -2.17  1.71 -0.954 0.957  0.17    3.87 -1.97
3 1       2   cp      3.3  7.94 -4.64 -1.93 -1.71  0.0267 0.0433  8.18 -4.88
# ℹ 480 more rows
# ℹ 32 more variables: WRES <dbl>, IPRED <dbl>, IRES <dbl>, IWRES <dbl>,
#   CPRED <dbl>, CRES <dbl>, CWRES <dbl>, eta.ktr <dbl>, eta.ka <dbl>,
#   eta.cl <dbl>, eta.v <dbl>, eta.emax <dbl>, eta.ec50 <dbl>, eta.kout <dbl>,
#   eta.e0 <dbl>, depot <dbl>, gut <dbl>, center <dbl>, effect <dbl>,
#   ktr <dbl>, ka <dbl>, cl <dbl>, v <dbl>, emax <dbl>, ec50 <dbl>, kout <dbl>,
#   e0 <dbl>, dcp <dbl>, pd <dbl>, kin <dbl>, tad <dbl>, dosenum <dbl>

This model may take longer than earlier PK examples.

Reasons:

  • ODE solving
  • multiple endpoints
  • SAEM estimation

Worked Example 5: Inspect Model Output

Convert the fit output.

fit_tbl <-
  fit_warf %>%
  as_tibble()

Inspect available columns.

names(fit_tbl)
 [1] "ID"       "TIME"     "CMT"      "DV"       "EPRED"    "ERES"    
 [7] "NPDE"     "NPD"      "PDE"      "PD"       "PRED"     "RES"     
[13] "WRES"     "IPRED"    "IRES"     "IWRES"    "CPRED"    "CRES"    
[19] "CWRES"    "eta.ktr"  "eta.ka"   "eta.cl"   "eta.v"    "eta.emax"
[25] "eta.ec50" "eta.kout" "eta.e0"   "depot"    "gut"      "center"  
[31] "effect"   "ktr"      "ka"       "cl"       "v"        "emax"    
[37] "ec50"     "kout"     "e0"       "dcp"      "pd"       "kin"     
[43] "tad"      "dosenum"

Inspect the first rows.

fit_tbl %>%
  select(
    ID,
    TIME,
    CMT,
    DV,
    PRED,
    IPRED,
    CWRES
  ) %>%
  head()
# A tibble: 6 × 7
  ID     TIME CMT      DV  PRED IPRED  CWRES
  <fct> <dbl> <fct> <dbl> <dbl> <dbl>  <dbl>
1 1       0.5 cp      0    1.38 0.504 -1.14 
2 1       1   cp      1.9  3.87 1.63  -0.880
3 1       2   cp      3.3  8.18 4.32  -1.58 
4 1       3   cp      6.6 10.6  6.60  -1.26 
5 1       6   cp      9.1 12.2  9.65  -1.42 
6 1       9   cp     10.8 11.8  9.72  -0.548

Notice that endpoint information is now stored in:

CMT

rather than dvid.

The fitted object now contains:

Predictions + Residuals + States + Parameters

This output becomes the foundation for diagnostics and model evaluation.

Worked Example 6: Explore Built-In Diagnostic Plots

nlmixr2 can generate diagnostic plots directly from the fitted object.

Instead of printing all plots at once, store them in an object.

pl <-
  plot(
    fit_warf
  )

Inspect the top-level structure.

names(pl)
[1] "traceplot"         "Endpoint:  depot"  "Endpoint:  gut"   
[4] "Endpoint:  center" "Endpoint:  effect" "Endpoint:  cp"    
[7] "Endpoint:  pca"

Interpretation:

The object separates diagnostics by endpoint.

You should see endpoint-specific groups such as:

Endpoint:  cp
Endpoint:  pca

This is useful for joint PK/PD models because PK and PD endpoints should be interpreted separately.

Inspect the available plots for the PK endpoint.

names(
  pl$`Endpoint:  cp`
)
 [1] "dv_pred_ipred_linear" "dv_pred_ipred_log"    "dv_cpred_linear"     
 [4] "dv_cpred_log"         "dv_epred_linear"      "dv_epred_log"        
 [7] "NPD_TIME_linear"      "NPD_TIME_log"         "RES_TIME_linear"     
[10] "RES_TIME_log"         "IRES_TIME_linear"     "IRES_TIME_log"       
[13] "IWRES_TIME_linear"    "IWRES_TIME_log"       "CWRES_TIME_linear"   
[16] "CWRES_TIME_log"       "NPD_EPRED_linear"     "NPD_EPRED_log"       
[19] "RES_EPRED_linear"     "RES_EPRED_log"        "IRES_EPRED_linear"   
[22] "IRES_EPRED_log"       "IWRES_EPRED_linear"   "IWRES_EPRED_log"     
[25] "CWRES_EPRED_linear"   "CWRES_EPRED_log"      "NPD_PRED_linear"     
[28] "NPD_PRED_log"         "RES_PRED_linear"      "RES_PRED_log"        
[31] "IRES_PRED_linear"     "IRES_PRED_log"        "IWRES_PRED_linear"   
[34] "IWRES_PRED_log"       "CWRES_PRED_linear"    "CWRES_PRED_log"      
[37] "NPD_IPRED_linear"     "NPD_IPRED_log"        "RES_IPRED_linear"    
[40] "RES_IPRED_log"        "IRES_IPRED_linear"    "IRES_IPRED_log"      
[43] "IWRES_IPRED_linear"   "IWRES_IPRED_log"      "CWRES_IPRED_linear"  
[46] "CWRES_IPRED_log"      "NPD_CPRED_linear"     "NPD_CPRED_log"       
[49] "RES_CPRED_linear"     "RES_CPRED_log"        "IRES_CPRED_linear"   
[52] "IRES_CPRED_log"       "IWRES_CPRED_linear"   "IWRES_CPRED_log"     
[55] "CWRES_CPRED_linear"   "CWRES_CPRED_log"      "NPD_tad_linear"      
[58] "NPD_tad_log"          "RES_tad_linear"       "RES_tad_log"         
[61] "IRES_tad_linear"      "IRES_tad_log"         "IWRES_tad_linear"    
[64] "IWRES_tad_log"        "CWRES_tad_linear"     "CWRES_tad_log"       
[67] "individual_1"         "individual_17"

You will see many available diagnostic plots, including:

  • observed vs predicted plots
  • residuals vs time
  • residuals vs predictions
  • individual profile plots

Examples include:

dv_pred_ipred_linear
CWRES_TIME_linear
CWRES_PRED_linear
individual_1

Interpretation:

We will not use every plot.

Instead, we will select a few diagnostic plots that answer the most important questions.

For this introductory PK/PD lesson, focus on:

Question Example plot
Do observations agree with predictions? dv_pred_ipred_linear
Are residuals centered over time? CWRES_TIME_linear
Are residuals centered across predictions? CWRES_PRED_linear
Do individual profiles look reasonable? individual_1

This keeps the lesson focused while still showing that the fitted object contains a rich diagnostic set.

NoteWhy Use plot(fit_warf) Instead of ggPMX Here?

Earlier diagnostic lessons emphasized ggPMX.

That remains an excellent workflow for many population PK models.

This lesson introduces something new:

Multiple Endpoints + Joint PK/PD Models

For these models, plot(fit) provides a convenient advantage:

  • diagnostics are already organized by endpoint
  • PK and PD outputs stay separated
  • many diagnostic plots are generated automatically

Conceptually:

Single Endpoint → ggPMX often works naturally

Multiple Endpoints → built-in endpoint diagnostics become convenient

This is not a rule.

Use the tool that makes interpretation clearer.

Worked Example 7: Observed versus Predicted by Endpoint

Inspect observed versus predicted behavior for the PK endpoint.

pl$`Endpoint:  cp`$dv_pred_ipred_linear

Inspect observed versus predicted behavior for the PD endpoint.

pl$`Endpoint:  pca`$dv_pred_ipred_linear

Interpretation:

Ask separately for cp and pca:

  • are observations centered around predictions?
  • does IPRED improve agreement compared with PRED?
  • is there endpoint-specific bias?
  • does one endpoint fit better than the other?

This is important because a model may describe PK well but PD poorly, or the reverse.

Worked Example 8: Residuals versus Prediction by Endpoint

Inspect residual behavior for the PK endpoint.

pl$`Endpoint:  cp`$CWRES_PRED_linear

Inspect residual behavior for the PD endpoint.

pl$`Endpoint:  pca`$CWRES_PRED_linear

Interpretation:

Look for:

  • residuals centered around zero
  • absence of strong trends
  • endpoint-specific residual spread
  • extreme residuals

Question:

Are prediction errors random for both endpoints?

Worked Example 9: Residuals Over Time by Endpoint

Inspect residuals over time for the PK endpoint.

pl$`Endpoint:  cp`$CWRES_TIME_linear

Inspect residuals over time for the PD endpoint.

pl$`Endpoint:  pca`$CWRES_TIME_linear

Interpretation:

Ask:

  • is there time-dependent bias?
  • does bias differ between PK and PD?
  • does the delayed response appear adequately captured?
  • are residuals worse at early or late times?

This helps evaluate whether the ODE structure captures response timing.

Worked Example 10: Individual Profiles by Endpoint

Inspect individual profile behavior for the PK endpoint.

pl$`Endpoint:  cp`$individual_1

Inspect individual profile behavior for the PD endpoint.

pl$`Endpoint:  pca`$individual_1

Interpretation:

Individual profile plots answer a different question than observed versus predicted plots.

Observed vs Predicted → point agreement

Individual Profiles → trajectory agreement

Ask:

  • does the model reproduce individual profile shapes?
  • are peaks and declines captured?
  • does the PD response timing look reasonable?
  • do some subjects show systematic mismatch?

This is especially useful in PK/PD models where timing and delayed response matter.

Worked Example 11: Visual Predictive Check for the Joint Model

VPCs remain useful for joint PK/PD models.

Question:

If the PK/PD model were true,
would simulated data resemble observations?

Generate a VPC.

vpcPlot(
  fit_warf,
  n = 100,
  bins = "jenks",
  scales = "free"
)

Interpretation:

Unlike earlier modules, interpret this VPC carefully.

This model contains:

PK Endpoint + PD Endpoint

Look for:

  • agreement in central tendency
  • agreement in variability
  • delayed response behavior
  • endpoint-specific deviations

Question:

Does the model reproduce both exposure and response?

VPCs for multiple-endpoint PK/PD models are powerful, but they require careful interpretation because endpoints may have different scales and biological meanings.

What Changed Compared with Population PK?

Population PK:

Dose → Concentration

Joint PK/PD:

Dose

↓

Concentration

↓

Response

The main new pieces are:

  • multiple endpoints
  • ODE implementation
  • PD turnover
  • endpoint-specific diagnostics

Looking Ahead

The next lesson focuses on interpretation.

We will interpret:

  • \(E_{max}\)
  • \(EC_{50}\)
  • \(k_{in}\)
  • \(k_{out}\)
  • baseline response
  • delayed response behavior

Strategies

  • inspect endpoint counts
  • interpret ODEs biologically
  • evaluate PK and PD behavior separately
  • use diagnostics and VPCs together

Common Mistakes

  • assuming PD follows concentration immediately
  • ignoring endpoint-specific residual error
  • treating ODEs as abstract math
  • assuming complexity means adequacy

Practice Problems

  1. What does dvid represent in the warfarin dataset?

  2. Why is the warfarin model an indirect response model?

  3. What does this line represent?

d/dt(effect) = kin * pd - kout * effect

  1. Why does the model need both cp and effect endpoints?

  2. Why should diagnostics be interpreted by endpoint?

Problem 1

dvid identifies the endpoint.

In this dataset:

cp → PK concentration
pca → PD response

Problem 2

The drug does not directly equal the response.

Concentration modifies a turnover process.

That turnover process changes the response over time.

Problem 3

This is the ODE for the PD response.

It means:

rate of response change = production/input - loss

The drug changes the input term through pd.

Problem 4

cp represents concentration.

effect represents response.

PK/PD modeling needs both because exposure and response are modeled together.

Problem 5

A model may describe PK well but PD poorly, or the reverse.

Endpoint-specific diagnostics help identify where the model is adequate and where it may need improvement.

Summary

  • warfarin introduces joint PK/PD modeling
  • dvid identifies PK and PD endpoints
  • ODEs are implemented directly in nlmixr2
  • warfarin uses an indirect response structure
  • diagnostics and VPCs still matter
  • cp = PK endpoint
  • pca = PD endpoint
  • ODE = change over time
  • PK drives PD
  • Diagnostics still matter