PK/PD Model: Assessing the Impact of Covariates on a Biomarker

Here we illustrate how the ordinary differential equations (ODEs) model and the approach of varying one covariate at a time, can be expanded.
We link the same two-compartment PK model (from the PK Example vignette) to an indirect response pharmacodynamic (PD) model where the drug concentrations inhibit the rate constant of input (Kin).
The covariates model included several covariates effects on Clearance, Volume and Kin. The baseline PD value is controlled by the ratio of Kin/Kout.
In this vignette we do not go into a lot of details, as we assume that the user has read and run the code of the Introduction to coveffectsplot and PK Example vignettes and that the reader is familiar with PK/PD concepts.
At the end we show how we can add a table under a multiple parameters forest plot.

Specifying a PK/PD Model using `mrgsolve`

codepkpdmodelcov <- '
$PARAM @annotated
KA     : 0.5   : Absorption rate constant Ka (1/h)
CL     : 4     : Clearance CL (L/h)
V      : 10    : Central volume Vc (L)
Vp     : 50    : Peripheral volume Vp (L)
Qp     : 10    : Intercompartmental clearance Q (L/h)
CLALB  : -0.8  : Ablumin on CL (ref. 45 g/L)
CLSEX  : 0.2   : Sex on CL (ref. Female)
CLWT   : 1     : Weight on CL (ref. 85 kg)
VSEX   : 0.07  : Sex on Vc (ref. Female)
VWT    : 1     : Weight on Vc (ref. 85 kg)
KIN    : 3     : Zero-order Rate constant of biomarker production (amount/h)
KOUT   : 0.06  : First-order Rate constant of biomarker loss (1/h)
IC50   : 3     : Drug concentration producing 50% of maximum inhibition
IMAX   : 0.999 : Maximum Inhibition Response
gamma  : 0.55  : Sigmoidicity factor of the sigmoid Emax equation
KINWT  : 0.4   : Weight on KIN (ref. 85 kg)
KINAGE : -0.08 : Age on KIN (ref. 40 years)
KINHLTY: 1.5   : Weight on CL (ref. 85 kg)

$PARAM @annotated // reference values for covariate
WT     :  85    : Weight (kg)
SEX    :  0     : Sex (0=Female, 1=Male)
ALB    :  45    : Albumin (g/L)
AGE    :  40    : Age (years)
HEALTHY:  0     : Health Status (0=Diseased, 1=Healthy)

$CMT GUT CENT PER RESP
$GLOBAL
#define CP   (CENT/Vi)
#define CPER (PER/Vpi)
#define INH  (IMAX*pow(CP,gamma)/(pow(IC50,gamma)+pow(CP,gamma)))
#define PDRESP RESP

$MAIN
double KAi = KA;
double Vpi = Vp *pow((WT/70.0),    1);
double Qpi = Qp *pow((WT/70.0), 0.75);
double CLi = CL *
    pow((ALB/45.0), CLALB)*
    (SEX == 1.0 ? (1.0+CLSEX) : 1.0)*
    pow((WT/85.0), CLWT)*exp(ETA(1)); 
double Vi = V *
    (SEX == 1.0 ? (1.0+VSEX) : 1.0)*
    pow((WT/85.0), VWT)*exp(ETA(2));  
double KINi = KIN *
  pow((AGE/40), KINAGE)*
  (HEALTHY == 1.0 ? KINHLTY : 1.0)*
  pow((WT/85.0), KINWT)*exp(ETA(3));
double RESP_0 = KINi/KOUT;

$OMEGA
0.09 
0.01 0.09
$OMEGA
0.25

$ODE
dxdt_GUT    = -KAi *GUT;
dxdt_CENT   =  KAi *GUT  - (CLi+Qpi)*CP  + Qpi*CPER;
dxdt_PER    =                   Qpi*CP   - Qpi*CPER;
dxdt_RESP   =  KINi*(1-INH) - KOUT*RESP;

$CAPTURE CP PDRESP KAi CLi Vi Vpi Qpi WT SEX ALB AGE HEALTHY
'
modpkpdsim <- mcode("codepkpdmodelcov", codepkpdmodelcov)
partab <- setDT(modpkpdsim@annot$data)[block=="PARAM", .(name, descr, unit)]
partab <- merge(partab, melt(setDT(modpkpdsim@param@data), meas=patterns("*"), var="name"))
knitr::kable(partab)

name	descr	unit	value
AGE	Age	years	40.000
ALB	Albumin	g/L	45.000
CL	Clearance CL	L/h	4.000
CLALB	Ablumin on CL	ref. 45 g/L	-0.800
CLSEX	Sex on CL	ref. Female	0.200
CLWT	Weight on CL	ref. 85 kg	1.000
HEALTHY	Health Status	0=Diseased, 1=Healthy	0.000
IC50	Drug concentration producing 50% of maximum inhibition	.	3.000
IMAX	Maximum Inhibition Response	.	0.999
KA	Absorption rate constant Ka	1/h	0.500
KIN	Zero-order Rate constant of biomarker production	amount/h	3.000
KINAGE	Age on KIN	ref. 40 years	-0.080
KINHLTY	Weight on CL	ref. 85 kg	1.500
KINWT	Weight on KIN	ref. 85 kg	0.400
KOUT	First-order Rate constant of biomarker loss	1/h	0.060
Qp	Intercompartmental clearance Q	L/h	10.000
SEX	Sex	0=Female, 1=Male	0.000
V	Central volume Vc	L	10.000
VSEX	Sex on Vc	ref. Female	0.070
VWT	Weight on Vc	ref. 85 kg	1.000
Vp	Peripheral volume Vp	L	50.000
WT	Weight	kg	85.000
gamma	Sigmoidicity factor of the sigmoid Emax equation	.	0.550

Simulate Reference Subjects with BSV

We simulate at reference covariate values with between subject variability (BSV) and then we show a plot of the PK and PD profiles of five random subjects.

idata <- data.table(ID=1:nbsvsubjects, WT=85, SEX=0, ALB=45, AGE=40, HEALTHY = 0)
ev1 <- ev(time = 0, amt = 100, cmt = 1, ii = 24, addl = 20)
data.dose <- ev(ev1)
data.dose <- setDT(as.data.frame(data.dose))
data.all <- data.table(idata, data.dose)

set.seed(678549)
outputpkpdsim <- modpkpdsim %>%
  data_set(data.all) %>%
  mrgsim(end = 28*24, delta = 0.25) %>%
  as.data.frame %>%
  as.data.table

outputpkpdsim$HEALTHY <- as.factor(outputpkpdsim$HEALTHY)

yvar_names <- c(
  'CP'="Plasma Concentrations",
  'RESP'="PD Values"
)
set.seed(678549)
outputpkpdsimlong <- outputpkpdsim[outputpkpdsim$ID %in%
sample(unique(outputpkpdsim$ID), 5), ] %>% 
  gather(key,value,CP,RESP)

ggplot(data =outputpkpdsimlong ,
       aes(time, value, group = ID)) +
  geom_line(alpha = 0.8, size = 0.3) +
  facet_grid(key ~ID,scales="free_y",switch="y",
             labeller = labeller(key=yvar_names)) +
  labs(y = "", color = "Sex", x = "Time (h)")+
  theme(strip.placement = "outside",
        axis.title.y=element_blank())

Compute PD Parameters and Summarize BSV

Here we compute the PD baseline (where we start), nadir response (minimum response achieved) and the delta (difference) between the baseline and nadir. We then summarize and report the BSV around these parameters as ranges of 50 and 90% of patients. We then show a plot of the first 10 replicates as an example of the simulated PD profiles. Since the code is similar to the PK Example vignette it is not shown.

derive.exposure <- function(time, PDRESP) {
  x <- c(
    nadir = min(PDRESP, na.rm = TRUE),
    baselinepd = PDRESP[1L],
    deltapd = PDRESP[1L]-min(PDRESP, na.rm = TRUE)
  )
  data.table(paramname=names(x), paramvalue=x)
}
refbsv <- outputpkpdsim[, derive.exposure(time, PDRESP),
                        by=.(ID, WT, SEX, ALB, AGE, HEALTHY)]

refbsv[, stdparamvalue := paramvalue/median(paramvalue), by=paramname]

bsvranges <- refbsv[,list(
    P05 = quantile(stdparamvalue, 0.05),
    P25 = quantile(stdparamvalue, 0.25),
    P50 = quantile(stdparamvalue, 0.5),
    P75 = quantile(stdparamvalue, 0.75),
    P95 = quantile(stdparamvalue, 0.95)), by = paramname]
bsvranges

paramname	P05	P25	P50	P75	P95
nadir	0.6276383	0.8514868	1	1.222234	1.701741
baselinepd	0.6230568	0.8462084	1	1.203854	1.740792
deltapd	0.6017202	0.8147581	1	1.199841	1.708660

Construct ans Simulate at Combinations of Covariate of Interest

Similarly to the PK Example vignette we generate covariate combinations of interest and we simulate with uncertainty using an invented varcov matrix.

reference.values <- data.frame(WT = 85, ALB = 45, AGE = 40, SEX = 0, HEALTHY = 0)   
covcomb <- expand.modelframe(
  WT  = c(56,128), 
  AGE = c(20,60),
  ALB = c(40,50),
  SEX = c(1),#Refernce is for SEX =0
  HEALTHY = c(1),#Refernce is for HEALTHY =0
  rv = reference.values)

# Add the reference
covcomb <- rbind(covcomb, data.table(reference.values, covname="REF"))
covcomb$ID <- 1:nrow(covcomb)

covcomb

WT	AGE	ALB	SEX	HEALTHY	covname	ID
56	40	45	0	0	WT	1
128	40	45	0	0	WT	2
85	40	40	0	0	ALB	3
85	40	50	0	0	ALB	4
85	20	45	0	0	AGE	5
85	60	45	0	0	AGE	6
85	40	45	1	0	SEX	7
85	40	45	0	1	HEALTHY	8
85	40	45	0	0	REF	9

albumin.labs <- c("albumin: 40 ng/mL","albumin: 45 ng/mL","albumin: 50 ng/mL")
names(albumin.labs) <- c("40","45","50")
wt.labs <- c("weight: 85 kg","weight: 56 kg","weight: 128 kg")
names(wt.labs) <- c("85","56","128")

age.labs <- c("age: 20 years","age: 40 years","age: 60 years")
names(age.labs) <- c("20","40","60")

pdprofiles <- ggplot(iter_sims[iter_sims$rep<=10,], aes(time/24,PDRESP,col=factor(WT),linetype=factor(HEALTHY) ) )+
  geom_line(aes(group=interaction(ID,rep)),alpha=0.3,size=0.3)+
  geom_line(data=outcovcomb,aes(group=interaction(ID)),color="black")+
  facet_nested(ALB+SEX~ AGE+WT,  labeller = 
                 labeller( WT = wt.labs,
                           ALB = albumin.labs,
                           AGE = age.labs))+
  labs(linetype="Black Lines\nNo Uncertainty\nHealthy Status",
       colour="Colored Lines\nUncertainty\nReplicates\n(1 to 10)\nWeight (kg)",
       caption ="Simulation\nwith Uncertainty without BSV" ,
       x="Days", y = "PD Values")+
  guides(colour = guide_legend(override.aes = list(alpha = 1)))

pdprofiles

Compute PD Parameters and Distributions Plots

Similar to the above we compute the PD parameters, standardize by the median and provide a plot. Since the code is similar to the PK Example vignette it is not shown.

boxplotpd <- ggplot(boxplotdat,
       aes(x=covvalue ,y=paramvalue))+
  facet_grid(paramname ~covname2,scales="free",switch="both",
             labeller = label_parsed)+
    geom_boxplot()+
  theme(axis.title = element_blank(),strip.placement = "outside")+
  labs(y="PD Parameter Values",x="Covariate Value")
boxplotpd

pdggridges<- ggplot(out.df.univariatecov.nca,
       aes(x=paramvaluestd,y=covvalue,fill=factor(..quantile..),height=..ndensity..))+
  facet_grid(covname2~paramname,scales="free_y",space="free")+
  annotate( "rect",
            xmin = 0.5,
            xmax = 2,
            ymin = -Inf,
            ymax = Inf,
            fill = "gray",alpha=0.4
  )+
  stat_density_ridges(
    geom = "density_ridges_gradient", calc_ecdf = TRUE,
    quantile_lines = TRUE, rel_min_height = 0.001,scale=0.9,
    quantiles = c(0.05,0.5, 0.95)) +
  scale_fill_manual(
    name = "Probability", values = c("white", "#0000FFA0","#0000FFA0", "white"),
    labels = c("(0, 0.05]", "(0.05, 0.5]","(0.5, 0.95]", "(0.95, 1]")
  )+
  geom_vline( aes(xintercept = 1),size = 1)+
  theme_bw()+
  labs(x="Effects Relative to Parameter Reference value",y="")+
  scale_x_continuous(breaks=c(0.25,0.5,0.8,1/0.8,1/0.5,1/0.25))+
  scale_x_log10()
pdggridges

Summarize, add the BSV Ranges and Putting it all Together Using `forest_plot`

Here we show how a multiple parameters, multiple covariates and table can be done.

coveffectsdatacovrep <- out.df.univariatecov.nca %>% 
  dplyr::group_by(paramname,covname,covvalue) %>% 
  dplyr::summarize(
    mid= median(paramvaluestd),
    lower= quantile(paramvaluestd,0.05),
    upper = quantile(paramvaluestd,0.95))

coveffectsdatacovreplabel<-   coveffectsdatacovrep %>%
  mutate(
    label= covvalue,
    LABEL = paste0(format(round(mid,2), nsmall = 2),
                   " [", format(round(lower,2), nsmall = 2), "-",
                   format(round(upper,2), nsmall = 2), "]"))

setkey(bsvranges, paramname)
coveffectsdatacovrepbsv <- coveffectsdatacovrep[coveffectsdatacovrep$covname=="REF",]
coveffectsdatacovrepbsv$covname <- "BSV"
coveffectsdatacovrepbsv$covvalue <- "50% of patients"
coveffectsdatacovrepbsv$label <-    "50% of patients"
coveffectsdatacovrepbsv$lower <- bsvranges$P25
coveffectsdatacovrepbsv$upper <- bsvranges$P75

coveffectsdatacovrepbsv2 <- coveffectsdatacovrep[coveffectsdatacovrep$covname=="REF",]
coveffectsdatacovrepbsv2$covname <- "BSV"
coveffectsdatacovrepbsv2$covvalue <- "90% of patients"
coveffectsdatacovrepbsv2$label <-    "90% of patients"
coveffectsdatacovrepbsv2$lower <- bsvranges$P05
coveffectsdatacovrepbsv2$upper <- bsvranges$P95
coveffectsdatacovrepbsv<- rbind(coveffectsdatacovrep,coveffectsdatacovrepbsv,coveffectsdatacovrepbsv2)
coveffectsdatacovrepbsv <- coveffectsdatacovrepbsv %>% 
  mutate(
    label= covvalue,
    LABEL = paste0(format(round(mid,2), nsmall = 2),
                   " [", format(round(lower,2), nsmall = 2), "-",
                   format(round(upper,2), nsmall = 2), "]"))
coveffectsdatacovrepbsv<- as.data.frame(coveffectsdatacovrepbsv)
coveffectsdatacovrepbsv$label <- as.factor(coveffectsdatacovrepbsv$covvalue )
coveffectsdatacovrepbsv$label <- reorder(coveffectsdatacovrepbsv$label,
                                         coveffectsdatacovrepbsv$lower)


coveffectsdatacovrepbsv$covname <-factor(as.factor(coveffectsdatacovrepbsv$covname ),levels =c("WT","SEX","ALB","AGE","HEALTHY", "REF", "BSV"),
labels=  c("Weight","Sex","Albumin","Age","Healthy", "Reference", "BSV")
)

interval_legend_text <- "Median (points)\n90% CI (horizontal lines)"
interval_bsv_text <- "BSV (points)\nPrediction Intervals (horizontal lines)"
ref_legend_text <- "Reference\n(vertical line)\nClinically relevant limits\n(gray area)"
area_legend_text <- "Reference\n(vertical line)\nClinically relevant limits\n(gray area)"
png("./Figure_S_PD_4.png",width =12 ,height = 9,units = "in",res=72)
coveffectsplot::forest_plot(coveffectsdatacovrepbsv,
                            ref_area = c(0.5, 1/0.5),
                            x_range = c(0.25,4),
                            strip_placement = "outside",
                            base_size = 16,
                            y_label_text_size = 12,
                            y_label_text_width = 50,
                            xlabel = "Fold Change Relative to Reference",
                            ref_legend_text = ref_legend_text,
                            area_legend_text =area_legend_text,
                            interval_legend_text = interval_legend_text,
                            interval_bsv_text = interval_bsv_text,
                            facet_formula = "covname~paramname",
                            facet_switch = "y",
                            facet_scales = "free_y",
                            facet_space = "fixed",
                            paramname_shape = FALSE,
                            table_position = "below",
                            table_text_size=4,
                            plot_table_ratio = 1,
                            table_facet_switch = "both",
                            show_table_facet_strip = "both",
                            show_table_yaxis_tick_label = TRUE,
                            logxscale = TRUE,
                            major_x_ticks          = c(0.5, 1,  1/0.5),
                            major_x_labels         = c("1/2", "1", "2"),
                            table_margin = c(0,5.5,0,0),
                            plot_margin =c(0,5.5,0,0),
                            reserve_table_xaxis_label_space = FALSE,
                            return_list = FALSE)
dev.off()
#> png 
#>   2
# consider returning a list and editing the y axis label line breaks height
# theme(axis.text.y = element_text(lineheight = ))

Covariate Effects Plot.