Survival Extrapolation from Published Curves • ParCC

The Problem

Clinical trials typically follow patients for 2-5 years, but health economic models often require lifetime projections (20-40 years). When Individual Patient Data (IPD) is unavailable, you must extract information from published Kaplan-Meier curves and fit parametric distributions.

Method 1: Exponential (From Median Survival)

When to Use

The exponential distribution assumes a constant hazard – the risk of the event is the same at every time point. This is rarely true biologically, but can be a reasonable approximation when:

You only have median survival reported
The Kaplan-Meier curve appears roughly linear on a log scale

The Scenario – Metastatic Colorectal Cancer

A Phase III trial of Bevacizumab + FOLFOX reports median Overall Survival of 21.3 months in the control arm (Hurwitz et al., NEJM 2004).

Worked Example

median_os <- 21.3  # months

# Hazard rate from median
lambda <- log(2) / median_os
cat("Hazard rate (lambda):", round(lambda, 5), "per month\n")
#> Hazard rate (lambda): 0.03254 per month

# Survival function: S(t) = exp(-lambda * t)
t_seq <- c(6, 12, 24, 36, 60)
surv <- exp(-lambda * t_seq)

data.frame(
  Month = t_seq,
  Survival = round(surv, 3),
  Event_Prob = round(1 - surv, 3)
)
#>   Month Survival Event_Prob
#> 1     6    0.823      0.177
#> 2    12    0.677      0.323
#> 3    24    0.458      0.542
#> 4    36    0.310      0.690
#> 5    60    0.142      0.858

Method 2: Weibull (From Two KM Points)

When to Use

The Weibull distribution allows the hazard to increase or decrease over time (monotonically). This is more flexible than exponential and is frequently used in oncology HTA.

The Scenario – NSCLC (CheckMate 017)

A published Kaplan-Meier curve for nivolumab in squamous NSCLC shows:

At 12 months, survival is approximately 42%
At 24 months, survival is approximately 23%

The Formula

The Weibull survival function is:

$S(t) = e^{-\lambda t^{\gamma}}$

Using the log-log transformation:

$\ln(-\ln(S(t))) = \ln(\lambda) + \gamma \ln(t)$

With two points, we solve a system of two linear equations.

Worked Example

# Two points from the KM curve
t1 <- 12; s1 <- 0.42
t2 <- 24; s2 <- 0.23

# Log-log transformation
y1 <- log(-log(s1)); x1 <- log(t1)
y2 <- log(-log(s2)); x2 <- log(t2)

# Solve for shape (gamma) and scale (lambda)
gamma <- (y2 - y1) / (x2 - x1)
lambda <- exp(y1 - gamma * x1)

cat("Weibull Shape (gamma):", round(gamma, 4), "\n")
#> Weibull Shape (gamma): 0.7606
cat("Weibull Scale (lambda):", format(lambda, scientific = TRUE, digits = 4), "\n")
#> Weibull Scale (lambda): 1.311e-01

# Generate survival table
t_seq <- seq(0, 60, by = 6)
surv <- exp(-lambda * t_seq^gamma)

# Transition probabilities (monthly)
t_monthly <- 0:60
s_monthly <- exp(-lambda * t_monthly^gamma)
tp <- c(1, s_monthly[-1] / s_monthly[-length(s_monthly)])

cat("\nSurvival projections:\n")
#> 
#> Survival projections:
data.frame(
  Month = seq(0, 60, by = 6),
  Survival = round(exp(-lambda * seq(0, 60, by = 6)^gamma), 3)
)
#>    Month Survival
#> 1      0    1.000
#> 2      6    0.599
#> 3     12    0.420
#> 4     18    0.307
#> 5     24    0.230
#> 6     30    0.175
#> 7     36    0.135
#> 8     42    0.105
#> 9     48    0.083
#> 10    54    0.066
#> 11    60    0.052

Interpretation

If $\gamma > 1$ : hazard is increasing over time (common in cancer)
If $\gamma = 1$ : constant hazard (reduces to exponential)
If $\gamma < 1$ : hazard is decreasing over time (common post-surgery)

Generating Markov Trace

Both methods produce cycle-specific transition probabilities for your Markov model:

# Generate annual transition probabilities from Weibull
cycles <- 0:10
s_t <- exp(-lambda * (cycles * 12)^gamma)  # Convert years to months for calculation
tp <- c(1, s_t[-1] / s_t[-length(s_t)])

trace <- data.frame(
  Year = cycles,
  Survival = round(s_t, 4),
  Annual_TP = round(tp, 4),
  Cumulative_Death = round(1 - s_t, 4)
)
print(trace)
#>    Year Survival Annual_TP Cumulative_Death
#> 1     0   1.0000    1.0000           0.0000
#> 2     1   0.4200    0.4200           0.5800
#> 3     2   0.2300    0.5476           0.7700
#> 4     3   0.1353    0.5881           0.8647
#> 5     4   0.0829    0.6131           0.9171
#> 6     5   0.0523    0.6309           0.9477
#> 7     6   0.0337    0.6448           0.9663
#> 8     7   0.0221    0.6560           0.9779
#> 9     8   0.0147    0.6654           0.9853
#> 10    9   0.0099    0.6735           0.9901
#> 11   10   0.0067    0.6805           0.9933

References

Collett D. Modelling Survival Data in Medical Research. 3rd ed. Chapman and Hall/CRC; 2015.
Latimer NR. Survival analysis for economic evaluations alongside clinical trials. Med Decis Making. 2013;33(6):743-754.
NICE Decision Support Unit. TSD 14: Survival analysis for economic evaluations. 2013.