Does a tumor-kinetic model add anything to a single early tumor change?

Daiki Kumakura · 2026-10-09 · Technical note · 1.0

Abstract

In a public subset of a colorectal-cancer trial, a mixed-effects model of each patient's tumor trajectory did not improve held-out prediction of survival beyond the observed change from baseline, at 16, 24 or 32 weeks. For the next tumor measurement, simply carrying the last value forward was the most accurate rule at every landmark. Within this dataset, early tumor assessment did not need a kinetic model.

Introduction

Tumor-growth-inhibition models are often proposed as a way to extract more information from early tumor measurements than a single percentage change. The question of interest here was narrower: at the same landmark, does an individual kinetic estimate from a population mixed-effects model add out-of-sample information about later overall survival, or about the next tumor measurement, beyond the observed one-point change?

The context of use is a research decision: whether early tumor assessment needs a kinetic model or whether a simple change suffices. It is not a surrogate-endpoint qualification, a treatment-effect analysis, or an individual prognosis tool. The fixed rule was to claim added value at a landmark only if the survival C-index improvement over the one-point model had a 95% interval above zero and the kinetic model predicted the next measurement better than both simple rules.

Methods

The colorectal and colorectalLongi datasets in CRAN frailtypack 3.8.1 contain a random subset of 150 participants from the FFCD 2000–05 trial of sequential versus combination chemotherapy for metastatic colorectal cancer: 906 tumor measurements, 121 deaths (1, 2, 3). Tumor size is the sum of lesion diameters after a Box–Cox transform (λ = 0.3); 34 measurements sit at a censoring floor. One participant has a measurement recorded after death and was excluded, as in the earlier landmark analysis of the same data (4).

That earlier analysis compared clinical factors with the observed early change at 16 weeks and noted that sparse sampling did not justify individual growth modeling. The measurement counts show where that limitation applies:

Figure 1 shows this comparison.

Figure 1. Follow-up measurements available by each landmark. Most participants have only one follow-up at 16 weeks, limiting individual kinetic estimation. Original analysis.

At 16 weeks, 89 of 128 patients have a single follow-up measurement; with one point after baseline, a patient's slope is the one-point change divided by time. Individual kinetics only become estimable at 24 weeks (111 of 123 with two or more) and 32 weeks (111 of 113). The primary comparisons are therefore at 24 and 32 weeks, with 16 weeks reported for continuity.

Models and validation

All models are Cox models for survival from the landmark, fitted to the same patients and folds using survival (5).

Table 1 presents the comparison.

Table 1. Survival-model predictors

P and K each add two terms to C, so the comparison concerns the information in the terms rather than their number.

The kinetic model is a linear mixed-effects model on the transformed scale with a random intercept and slope (nlme, restricted maximum likelihood estimation, REML) (6), fitted only to measurements up to the landmark. Population parameters are estimated in the training fold; each held-out patient's individual effects are best linear unbiased predictions computed with those fixed parameters. No measurement after the landmark enters any predictor. A quadratic time term was added only if, by a pre-specified rule, it was both statistically supported and reduced next-measurement error by at least 5%. That rule was met at 24 and 32 weeks, so the quadratic model is reported as an additional comparison. Mechanistic TGI models (Claret- or Stein-type) were not fitted: they need sizes on the original scale, and the floor values and point counts make back-transformed fits unstable.

Validation repeated a death-stratified five-fold split ten times with a fixed seed, averaged each patient's held-out predictions, and formed paired bootstrap intervals over patients. These intervals are conditional on the fitted models and do not include refitting uncertainty. The 16-week clinical and one-point C-indices (0.532 and 0.581) reproduce the earlier analysis exactly, and an independent re-implementation of all primary metrics agreed to within 5 × 10⁻⁹.

Results

Survival discrimination

Figure 2 shows this comparison.

Figure 2. Differences in held-out survival concordance. Patient-bootstrap intervals are conditional on the fitted cross-validated predictions and exclude refitting uncertainty. Original analysis.

Table 2 presents the comparison.

Table 2. Held-out C-index differences and conditional 95% intervals

The kinetic terms changed the C-index by at most 0.016, and every interval included zero. One-year Brier score differences were similarly centered on zero. The reason is visible in the diagnostics: across patients, the kinetic terms correlated about 0.97 with the one-point terms. The mixed model largely restates the observed change, shrunk toward the population (slope shrinkage 0.24–0.35).

Calibration slopes were well below 1 for all models (0.25–0.46 for P and K), so even the best model's risk spread is overstated. This is expected with about 100 events and weak predictors, and it is another reason not to read small C-index differences as meaningful.

Predicting the next measurement

Figure 3 shows this comparison.

Figure 3. Mean absolute prediction error on the transformed tumor scale for the first measurement after the landmark. The last observed value is the strongest comparator here. Original analysis.

Table 3 presents the comparison.

Table 3. Mean absolute errors for the next tumor measurement

Values are mean absolute errors on the transformed scale. The mixed model beat a straight line drawn from baseline but was worse than carrying the last value forward, by 0.10 to 0.29 (all intervals excluded zero). Even the quadratic model remained worse than the last value, by 0.11 to 0.13.

The residual pattern suggests a structural limitation of the fitted trend. Residuals from the linear model were curved in time (correlation with squared centered time 0.22–0.27 at 24 and 32 weeks), consistent with shrinkage that levels off or reverses. A trend fitted to the early decline extrapolates further decline that does not happen. Adding curvature reduced the error but did not overcome it. In these data, the most recent measurement is a better summary of the near future than any fitted trend.

Sensitivity analyses

Excluding floor-censored measurements before the landmark, or retaining the excluded participant without the inconsistent measurement, did not change any conclusion: all kinetic-versus-one-point intervals included zero, and the last value remained the best next-measurement predictor.

Discussion

Supported: In this 150-patient subset, a population kinetic model did not add information beyond the observed one-point change, either for later survival or for the next tumor measurement, at any landmark from 16 to 32 weeks. At 16 weeks it could not have: most patients had one follow-up measurement. For early assessment with this measurement density, a simple change is sufficient, and a fitted trend can be worse than no trend.

Not supported: That mechanistic TGI models are uninformative in general (none was fitted); that the one-point change itself is a useful predictor (its improvement over clinical factors also included zero, as in the earlier analysis); or that these results transfer to other trials, tumor types or scan schedules (no external cohort was available).

Conclusion

The kinetic model did not improve held-out survival discrimination or the next-measurement prediction over the simple comparators in this subset. Original-scale lesion sizes and an external cohort were unavailable. The sparse pre-landmark records also prevented a reliable comparison of decline-and-regrowth structures. These limitations bound the negative result to the models and observation schedules tested here.

Data and code availability

The companion contains the acquisition and export scripts, the analysis and independent validation code, aggregate result tables and figure code. It does not contain participant-level data: acquire.py downloads the source package from CRAN and checks its SHA-256, and export.R writes the two datasets locally. Run python acquire.py, Rscript export.R, python audit.py, Rscript analyze.R, python validate.py and python figures.py (R 4.5.1 with survival 3.8-3 and nlme 3.1-168; Python 3.12 with matplotlib). The seed is fixed; two runs gave identical results.

Download code, aggregate results and figure scripts.

References

1. frailtypack developers. frailtypack [Software and dataset documentation]. Version 3.8.1. Datasets colorectal and colorectalLongi. CRAN package.
2. Chiou SH, Xu G, Yan J, Huang CY. Regression modeling for recurrent events possibly with an informative terminal event using R package reReg. J Stat Softw. 2023;105(5):1–34. doi:10.18637/jss.v105.i05. Section 6 describes the FFCD 2000–05 subset.
3. Ducreux M, Malka D, Mendiboure J, Etienne PL, Texereau P, Auby D, et al. Sequential versus combination chemotherapy for the treatment of advanced colorectal cancer (FFCD 2000–05): an open-label, randomised, phase 3 trial. Lancet Oncol. 2011;12(11):1032–1044. doi:10.1016/S1470-2045(11)70199-1.
4. Kumakura D. Do early tumor measurements improve prediction of subsequent survival? [Internet]. 2026 Oct 4; updated 2026 Oct 27. Earlier landmark analysis.
5. Therneau TM. survival: a package for survival analysis in R [Software]. Version 3.8-3. CRAN package.
6. Pinheiro J, Bates D, R Core Team. nlme: linear and nonlinear mixed effects models [Software]. Version 3.1-168. CRAN package.

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