Volume 6, Issue 7 p. 449-457
Original Article
Open Access

Pharmacometric Modeling of Liver Metastases' Diameter, Volume, and Density and Their Relation to Clinical Outcome in Imatinib-Treated Patients With Gastrointestinal Stromal Tumors

E Schindler

E Schindler

Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden

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SM Krishnan

SM Krishnan

Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden

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RHJ Mathijssen

RHJ Mathijssen

Department of Medical Oncology, Erasmus MC Cancer Institute, Rotterdam, The Netherlands

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A Ruggiero

A Ruggiero

Department of Radiology, Papworth Hospital NHS Foundation Trust, Cambridge University Health Partners, Cambridge, CB23 3RE United Kingdom

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G Schiavon

G Schiavon

Department of Medical Oncology, Erasmus MC Cancer Institute, Rotterdam, The Netherlands

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LE Friberg

Corresponding Author

LE Friberg

Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden

Correspondence: L Friberg ([email protected])Search for more papers by this author
First published: 05 April 2017
Citations: 11


Three-dimensional and density-based tumor metrics have been suggested to better discriminate tumor response to treatment than unidimensional metrics, particularly for tumors exhibiting nonuniform size changes. In the developed pharmacometric modeling framework based on data from 77 imatinib-treated gastrointestinal patients, the time-courses of liver metastases' maximum transaxial diameters, software-calculated actual volumes (Vactual) and calculated ellipsoidal volumes were characterized by logistic growth models, in which imatinib induced a linear dose-dependent size reduction. An indirect response model best described the reduction in density. Substantial interindividual variability in the drug effect of all response assessments and additional interlesion variability in the drug effect on density were identified. The predictive ability of longitudinal tumor unidimensional and three-dimensional size and density on overall survival (OS) and progression-free survival (PFS) were compared using parametric time-to-event models. Death hazard increased with increasing Vactual. This framework may guide early clinical interventions based on three-dimensional tumor responses to enhance benefits for patients with gastrointestinal stromal tumors (GIST).

Study Highlights


☑ Conventional unidimensional size measurements may not adequately detect treatment response for tumors with nonspherical shapes or nonuniform size changes, such as liver metastases from imatinib-treated GIST. Instead, 3D measurements and structural density assessments may better reflect treatment effect.


☑ Pharmacometric models characterized the unidimensional, 3D, and density responses of imatinib-treated GIST liver metastases, and compared the predictive ability of each metrics on OS and PFS.


☑ Three-dimensional may be more sensitive than unidimensional measurements at detecting tumor response to imatinib. Three-dimensional changes predicted OS slightly better than unidimensional changes, whereas density changes had no predictive value.


☑ These results improve our understanding of the relationships among imatinib exposure, conventional and alternative tumor measurements, and long-term outcomes. They may guide early clinical decisions for patients with GIST, and encourage the incorporation of 3D measurements in prospective clinical trials to test their ability and value vs. unidimensional measurements in predicting outcomes.

Efficacy assessment in cancer clinical trials and clinical practice to evaluate tumor shrinkage or vice versa development of disease progression has been standardized through the implementation of criteria for morphological imaging measurements. Identification of responders and nonresponders early after therapy initiation is crucial to trigger treatment modification whenever needed. Highly sensitive methods for tumor response quantification that correlate with clinical outcome are, therefore, required. The most recent and widely-adopted Response Evaluation Criteria in Solid Tumors (RECIST) are based on unidimensional (1D) changes in the sum of longest diameters of target tumor lesions, assessed by magnetic resonance imaging or computed tomography.1, 2 One question raised by the RECIST Working Group when updating the guidelines was whether and when functional or volumetric assessment could supplement or potentially replace RECIST.2, 3

Choi et al.4, 5 proposed novel criteria combining tumor size and density changes that reflect morphological and structural changes, respectively, to assess response of gastrointestinal stromal tumors (GIST) to imatinib, a multitargeted tyrosine kinase inhibitor.4, 5 In imatinib-treated GIST, the Choi criteria correlated with time-to-tumor progression and disease-specific survival.6 Contrasting results have been reported for different cancer types regarding the predictive value of Choi criteria on clinical outcome, such as overall survival (OS) or progression-free survival (PFS).7-12

As tumors are three-dimensional (3D) structures, using a single dimension might not accurately reflect tumor burden and may underestimate shrinkage or overestimate growth, especially for lesions that are nonspherical and/or display nonuniform size changes during treatment, such as primary tumors13 and liver metastasis14, 15 from GIST or in tumors whose main changes occur in a plan other than axial. At the time of the RECIST update in 2009 (RECIST 1.1), the lack of rigorous clinical validation and of widespread access to these newer imaging approaches prevented their implementation and adoption. However, in recent years, a number of publications have demonstrated the high reproducibility and interobserver reliability of volume measurements using semiautomated tumor segmentation technique implemented in commercially available15-18 or in-house software.19 The estimated ellipsoidal volume has been suggested as an approximation for software-calculated actual volume to overcome limitations related to software availability and has shown good correlation to actual volume.15, 18 Moreover, for a number of cancers 3D assessments enable to capture finer changes in tumor size,17, 18 and better correlates to long-term clinical outcome, such as OS,18, 2, 21 compared to 1D measurements. Such results were supported by Schiavon et al.,15 showing that ellipsoidal volume provides a satisfactory approximation for the actual volume of liver metastases in imatinib-treated GIST. The 3D changes could be detected more frequently than 1D changes after 3 months of treatment. Moreover, volume-based criteria tended to be better associated to OS than RECIST. Noteworthy, longitudinal analysis of tumor volume on a continuous scale are only rarely reported21; most published results are based on landmark analyses, in which the correlation between the often-categorized change in tumor burden after a fixed treatment duration and clinical outcome is investigated, resulting in a loss of information.

Pharmacometric models have proved their value in leveraging data collected during clinical trials and clinical practice.22-24 They allow for characterization of the tumor response time-course as a function of drug exposure, quantification of interindividual variability (IIV), interlesional variability (ILV),25 and residual variability in the response, and investigation of the relationships between longitudinal tumor size and time-to-event outcome. In the present study, we developed a pharmacometric modeling framework to (i) characterize the time-course and quantify the IIV and ILV of tumor 1D, 3D, and density response of liver metastases in imatinib-treated patients with GIST and (ii) investigate the predictive ability of these size and density metrics on OS and PFS.


Study population and data

The tumor data were available from 77 patients with GIST treated with first-line imatinib and involved in two retrospective, noninterventional studies based on historical data published previously.15, 26 Imatinib was administered orally at a starting dose of 400 mg (N = 74) or 800 mg (N = 3) daily until unacceptable toxicity or disease progression (assessed as progressive disease by RECIST). Any dose escalation or reduction was based on safety and efficacy and recorded at routine clinical assessments. Information on subsequent systemic therapy (e.g., sunitinib and regorafenib) was also available. One or two liver metastases (target lesions) per patient were retrospectively selected to be followed on computed tomography scans performed from initiation of imatinib treatment (baseline) and at least once during treatment (at ∼3, 6, and 12 months), with a median time for tumor follow-up of 360 days (range, 82–495 days). Tumor 1D measurements of target lesion(s) consisted of the maximum transaxial diameter (MTD), as per RECIST version 1.1 definition. Tumor 3D measurements of the same lesions included the segmented volume (Vactual), calculated using the semi-automated segmentation by “syngo computed tomography Oncology” software version 2009E (Siemens Medical Solutions), and the calculated ellipsoidal volume ( urn:x-wiley:21638306:media:psp412195:psp412195-math-0001, where MTD, d2, and d3 are three orthogonal diameters).26 For plotting purposes, the spherical volume ( urn:x-wiley:21638306:media:psp412195:psp412195-math-0002) was calculated assuming that lesions are perfect spheres. Tumor density (in Hounsfield units [HUs]) was recorded for each lesion by measuring the computed tomography attenuation coefficient (see Schiavon et al.26 for details on the computation of these measurements). OS and PFS data were collected for a median of 4.5 years (range, 0.79–13 years) and 3.4 years (range, 0.25–13 years), respectively, after treatment initiation. PFS was defined as the time from first imatinib dose to discontinuation of treatment for any reason, including disease progression (assessed by RECIST), drug-related toxicities, or death. Censoring was defined as loss to follow-up or nonoccurrence of death or progression at the end of the follow-up period, for OS and PFS, respectively. In the analysis presented here, OS and PFS data collection was updated and closed on February 17, 2015. Table 1 provides a summary of patients' characteristics and available data. Ethical standards were followed and approvals were obtained as described earlier.26

Table 1. Summary of patients' characteristics and available data
No. of patients, total/with 2 lesions 77/60
Gender, male/female, no. (%) 47/30 (61/39)
Age at start of imatinib, years, median (range) 62 (34–83)
Imatinib starting dose, mg (n) 400/800 (74/3)
Second line of therapy and beyond (yes/no), no. (%) 35/42 (45/55)
Patients with dose escalationa, no. (%) 30 (39)
Escalated dose, mg, median (range) 800 (600–1200)
Patients with dose reductionsa, no. (%) 8 (10)
Reduced dose, mg, median (range) 300 (200–300)
Events for overall survival (censored/death), no. (%) 34/43 (45/55)
Events for progression-free survival (censored/progression), no. (%) 27/50 (35/65)
  • a Taking into account first dose alteration only.

Tumor models

MTD, Vactual, ellipsoidal volume (Vellipsoid), and density models were developed separately before being combined into a joint model to explore correlations and to simplify the model structure where appropriate. Exponential IIV and ILV were explored in all non-negative model parameters (Eq. 1).25
urn:x-wiley:21638306:media:psp412195:psp412195-math-0004 represents the parameter for the ith individual and urn:x-wiley:21638306:media:psp412195:psp412195-math-0005 the typical parameter value in the population. urn:x-wiley:21638306:media:psp412195:psp412195-math-0006 is the term describing IIV, common to all lesions but specific to each individual ( urn:x-wiley:21638306:media:psp412195:psp412195-math-0007). urn:x-wiley:21638306:media:psp412195:psp412195-math-0008 and urn:x-wiley:21638306:media:psp412195:psp412195-math-0009 are the ILV terms for lesion 1 and lesion 2, respectively, assumed to be normally distributed with mean 0 and a common variance urn:x-wiley:21638306:media:psp412195:psp412195-math-0010. Residual unexplained variability was described by additive models on a logarithmic scale.

Maximum transaxial diameter, actual volumes, and ellipsoidal volume models

Tumor models in which imatinib-induced tumor size reductions were developed describe log-transformed MTD, Vactual, and Vellipsoid data. Various tumor growth models were explored, including exponential, zero-order, Gompertz, and logistic functions.27, 28 Linear, power, and maximum effect functions of imatinib daily dose were compared to describe the dose-response relationship. A time-dependent mono-exponential decay in drug effect was evaluated to explain tumor regrowth following initial shrinkage. Addition of an effect compartment was also tested. Semiparametric distributions29 and mixture models were investigated to describe the observed bimodal distribution in baseline MTD, Vactual, and Vellipsoid.

Tumor density model

Indirect response models30 in which imatinib inhibits the input or stimulates the output of density response were investigated to describe log-transformed tumor density data, which tended to decrease during therapy. Linear and nonlinear (power, maximum effect) drug effect functions driven by imatinib daily dose were tested. A potential delay in affecting density changes was evaluated through the addition of an effect compartment. A potential drug-independent increase in density over time was assessed using time-dependent linear or nonlinear (that plateaus over time, as apparent from some observed lesion profiles) disease progression functions applied on the production rate.

Overall survival and progression-free survival models

Parametric time-to-event (TTE) models were developed to investigate the effect of potential predictors of OS and PFS hazards (h(t)). The baseline hazard (h0(t)) was explored using exponential, Weibull, Gompertz, log-normal, and log-logistic models.31 A sequential analysis similar to the population pharmacokinetic parameters and data32 approach was initially considered but led to model instability, preventing likelihood ratio testing. Therefore, an approach similar to the individual pharmacokinetic parameter approach was used,32 in which individual empirical Bayes estimates from the final joint tumor model were used to predict individual tumor size or density time-courses, which were then investigated as predictors in the TTE models. Baseline predictors included patient characteristics (age and gender) and model-predicted (log-transformed) baseline MTD, Vactual, Vellipsoid, and tumor density. Time-varying predictors tested on OS consisted of (log-transformed) model-predicted tumor time-courses (S(t)), and absolute (dS(t)) and relative (Srel(t)) change from baseline over time, where S is MTD, Vactual, or Vellipsoid. Additionally, relative change in size and density up until 1.5, 3, 6, and 12 months for OS and 1.5 and 3 months for PFS were evaluated. As the PFS definition is partly based on MTD changes, only early size-related predictors (1.5 or 3 months) were considered in the PFS analysis to minimize potential confounding between the predictor and the outcome. When patients presented with two lesions, MTD, Vactual, and Vellipsoid were summed across lesions, whereas density was averaged over lesions. Predictors were evaluated alone (univariate analysis) and in combination (multivariate analysis). Predictors related to different size metrics were not tested in combination, but size-related predictors could be combined with density-related predictors. For each endpoint, a competing TTE model described censoring.

Model development and evaluation

Models were developed using the nonlinear mixed-effects modeling software (NONMEM version 7.3).33 The first-order conditional estimation method with interaction was used for parameter estimation. R (version 3.1) was used for data management and graphical analysis. Model development and evaluation was aided by Perl-speaks-NONMEM (PsN) toolkit (version 4.5), the R-based package Xpose (version 4), and Pirana (version 2.9.2).34

Model performance was evaluated based on objective function value (OFV; −2·log-likelihood) and graphical diagnostics. Upon addition of one parameter (1 degree of freedom (df)), an OFV decrease of at least 6.63 (P < 0.01) for tumor models (repeated measures within subjects), and an OFV of at least 3.84 (P < 0.05) for TTE models (sparser data – single event per individual), was considered as statistically significant. Relative standard errors (RSEs) of parameter estimates were obtained from the NONMEM Sandwich matrix for continuous data and from the R matrix for TTE models. The predictive performance of continuous models was evaluated by visual predictive checks (VPCs), in which 95% confidence intervals (CIs) of the 10th, 50th, and 90th percentiles from 1,000 simulated datasets were compared to the corresponding percentiles from the observed data. TTE models' predictive performance was assessed by Kaplan–Meier VPCs, in which the 95% confidence interval based on 200 simulated datasets was compared to the observed Kaplan–Meier curve.



Tumor data consisted of 502 observations for each size metrics and 496 density observations, collected from 136 liver lesions in 77 patients (maximum 2 per patient). Lesions were numbered so that lesion 1 has the largest MTD at baseline.

Joint tumor model for maximum transaxial diameter, actual volumes, ellipsoidal volume, and density

A tumor model with a logistic growth function in which imatinib induces tumor size reduction (Eq. 2) best characterized the individual lesion data from all three tumor sized metrics. Logistic growth fitted the data better than exponential growth (objective function value difference [dOFV]; −10.3, −13.4, and −10.2 for MTD, Vactual, and Vellipsoid, respectively). A linear effect with slope Kdrug,S driven by imatinib daily dose, normalized by the median dose (400 mg), best described the dose-response relationship. Moreover, a time-dependent mono-exponential function with rate constant urn:x-wiley:21638306:media:psp412195:psp412195-math-0011 described a drug effect washout. Addition of an effect compartment did not improve model fit.

S(t) represents the lesion size (MTD, Vactual, or Vellipsoid) at time t. KG is the tumor growth rate constant and Smax the maximal size above which the lesion cannot grow (i.e., carrying capacity). A mixture model with two subpopulations of patients described the bimodal distribution of baseline lesion size (S0), with subpopulation 1 (estimated probability of 0.348) having lesions with larger baseline Vactual, Vellipsoid, and MTD than subpopulation 2. Of note, 22% of patients with one target lesion were assigned to subpopulation 1 vs. 39% for patients with two target lesions. For a typical patient with two lesions, the estimated total S0 was 3.4, 41, and 42 times larger in subpopulation 1 than in subpopulation 2, for MTD, Vactual, and Vellipsoid, respectively. For each mixture and size metrics, S0 was associated with an ILV term that was specific to each lesion but shared a common variance for the two lesions. The typical total Smax estimate for an individual with two lesions was 296 mm for MTD, 1,730 mL for Vactual, and 1,818 mL for Vellipsoid. For Vactual and Vellipsoid, an alternative parameterization for Smax was considered, in which the carrying capacity for the total volume of lesion 1 and 2 ( urn:x-wiley:21638306:media:psp412195:psp412195-math-0013) was estimated together with the fraction of urn:x-wiley:21638306:media:psp412195:psp412195-math-0014 for lesion 1; however, this resulted in model instability and large parameter uncertainty and was not retained. All Vactual and Vellipsoid estimates were similar, with Vellipsoid estimates within 100–116% of Vactual estimates (Table 2). Kdrug,S estimates were fourfold to fivefold larger in Vactual and Vellipsoid models compared to the MTD model; IIV but no ILV was identified in KG and Kdrug,S.

Table 2. Parameter estimates and their uncertainty in the final tumor models for maximum transaxial diameter, actual volumes, and ellipsoidal volume
Parameter (unit) MTD Vactual Vellipsoid
Typical value (RSE%) IIV, CV% (RSE%) Typical value (RSE%) IIV, CV% (RSE%) Typical value (RSE%) IIV, CV% (RSE%)
S0, pop 1, mm or mLa Lesion 1 76.6 (12) 47 (11)b 161 (34) 140 (12)b 187 (35) 140 (12)b
Lesion 2 41.9 (15) 47 (11)b 29.7 (46) 140 (12)b 33.4 (46) 140 (12)b
S0, pop 2, mm or mLa Lesion 1 20.9 (6.3) 29 (13)c 3.45 (16) 76 (13)c 3.93 (17) 78 (12)c
Lesion 2 14.2 (6.7) 29 (13)b 1.21 (18) 76 (13)b 1.27 (19) 78 (12)b
Smax, mm or mLa Lesion 1 171 (8.4) - 1190 (43) - 1230 (57)d -
Lesion 2 125 (3.7) - 540 (12) - 588 (18) -
KG (week−1) 0.00176 (47) 170 (18) 0.00861 (38) 135 (20) 0.00882 (43) 140 (27)
λ (week−1) 0.0475 (35) - 0.0469 (28) - 0.0508 (37) -
Kdrug,S (week−1) 0.0124 (36) 77 (20) 0.0547 (21) 56 (29) 0.0610 (24) 42 (36)
Ppop 1 0.348 (18)c - 0.348 (18)c - 0.348 (18)c -
RUV (%) 14.0 (6.6) - 36.8 (8.0) - 43.3 (7.5) -
  • CV, coefficient of variation; λ, resistance development rate constant; IIV, interindividual variability; Kdrug,S, slope of the linear drug effect; KG, tumor growth rate constant; MTD, mean transaxial diameter; Ppop 1, probability of belonging to mixture subpopulation 1; RSE, relative standard error; RUV, residual unexplained variability Smax, carrying capacity; S0, pop n, baseline size in mixture subpopulation n; Vactual, actual volume; Vellipsoid, ellipsoidal volume.
  • Lesions were numbered such that lesion 1 corresponds to the one with the largest MTD at baseline.
  • a Millimeters for MTD, mL for Vactual and Vellipsoid.
  • b IIV term variance shared between lesions, within each subpopulation and model.
  • c Common parameter to MTD, Vactual, and Vellipsoid models.
  • d Ninety-five percent confidence intervals obtained from log-likelihood profiling was (685;3260).
An indirect response model in which imatinib stimulates the output of response through a linear drug effect with slope Kdrug,D driven by imatinib daily dose, normalized by the median dose (400 mg), best described tumor density time-course (D(t)) (Eq. 3). Addition of an effect compartment did not improve model fit (dOFV = −1.14). Accounting for disease progression using an increase in the production rate that levels off over time statistically significantly improved the OFV (dOFV = −13.3); however, because of that the OFV decrease was highly dependent on two individuals, model instability and negligible improvement in VPCs, disease progression was not retained. A Box-Cox transformation with an estimated shape parameter of −1.06 was applied to baseline density (D0) sum of urn:x-wiley:21638306:media:psp412195:psp412195-math-0015 and urn:x-wiley:21638306:media:psp412195:psp412195-math-0016 to account for the skewed random effects distribution. The IIV magnitude of Kdrug,D was larger than the ILV magnitude (120 vs. 53% CV). No IIV or ILV was identified in the rate constant for loss of response (kout).

The rate constant for production of response was parameterized as urn:x-wiley:21638306:media:psp412195:psp412195-math-0018. The derived mean turnover time ( urn:x-wiley:21638306:media:psp412195:psp412195-math-0019) was 75 days. Estimated correlations between density model parameters and MTD, Vactual, or Vellipsoid model parameters were <20%.

VPCs indicated appropriate predictive performance of the joint model (Figure 1 and Supplementary Figures S1–S4). Tables 2 and 3 present the joint tumor model parameter estimates and their uncertainty. Parameter uncertainty was within acceptable range (≤47% RSE), except for Smax,lesion 1 in the Vellipsoid model (56% RSE), for which 95% confidence interval from log-likelihood profiling35 was 685–3,260 mL.

Details are in the caption following the image

Visual predictive checks of the joint tumor model for actual volume (Vactual), ellipsoidal volume (Vellipsoid), maximum transaxial diameter (MTD), and density models. Median (solid line), 10th and 90th percentiles (dashed lines) of the observed data are compared to the 95% confidence intervals (shaded areas) for the median (blue), 10th and 90th percentiles (gray) of the simulated data, based on 1,000 simulations.

Table 3. Parameter estimates and their uncertainty in the final tumor density model.
Parameter (unit) Typical value (RSE%) IIV – CV% (RSE%) ILV – CV% (RSE%)
D0 (HU) 59.0 (5.7) 30 (10) 18 (19)
Box-Cox D0 −1.06 (47)a - -
kout (week−1) 0.0935 (32) - -
Kdrug,D 0.154 (29) 120 (17) 53 (38)
RUV (%) 20.6 (6.8) -
  • CV, coefficient of variation; D0, baseline tumor density; HU, Hounsfield unit; IIV, interindividual variability; ILV, interlesion variability; Kdrug, density reduction constant; kout, rate constant for loss of tumor density; RSE, relative standard error; RUV, residual variability.
  • a The 95% confidence intervals obtained from log-likelihood profiling was (-2.01;-0.397) for Box-Cox D0.

Overall survival and progression-free survival models

A lognormal distribution with estimated mean urn:x-wiley:21638306:media:psp412195:psp412195-math-0020 and urn:x-wiley:21638306:media:psp412195:psp412195-math-0021 and SD urn:x-wiley:21638306:media:psp412195:psp412195-math-0022 and urn:x-wiley:21638306:media:psp412195:psp412195-math-0023, for OS and PFS respectively, best described the baseline hazard (h0(t)).

In the univariate OS analysis, the most statistically significant predictor was log-transformed Vactual(t) (dOFV = −8.09). Vactual(t) (dOFV = −7.09) and dVactual(t) (dOFV = −4.26) were also significant but to a lower degree. Other statistically significant predictors identified in the univariate analysis are summarized in Supplementary Material Table S1. None of the density-related variables or patient demographics predicted OS. In the multivariate OS analysis, no other predictor improved the fit further when log-transformed Vactual(t) had been added. This model was concluded in the final OS model (Eq. 4).

urn:x-wiley:21638306:media:psp412195:psp412195-math-0025 is the standard normal cumulative distribution function. The coefficient for the effect of log(Vactual(t)) on the hazard, urn:x-wiley:21638306:media:psp412195:psp412195-math-0026, was estimated to 0.190, denoting an increased death hazard with increasing Vactual. Figure 2 (left panel) illustrates the hazard ratio (HR) over the range of observed Vactual, using the typical total S0 (2 lesions, subpopulation 1) as reference. For instance, the HR is estimated to 1.52 at the typical total Vactual Smax (1,730 mL) in subpopulation 1, and to 0.494 at the typical total Vactual S0 value (4.66 mL) in subpopulation 2.

Noteworthy, in the OS multivariate analysis, a lower OFV was achieved by combining log-transformed baseline and relative change from baseline up to 3 months in Vactual (dOFV = −13.7) or Vellipsoid (dOFV = −13.3), whereas for MTD the corresponding combination resulted in a smaller improvement in the model fit (dOFV = −9.31); however, these multivariate models requested one extra model parameter compared to the final model.

In the univariate PFS analysis, the most statistically significant predictor was Vactual relative change from baseline up to 3 months (Vactual,rel,3m, dOFV = −13.9). All other investigated size-related predictors were also significant, with Vactual and Vellipsoid predictors always being better than corresponding MTD predictors. None of the density-related variables could predict PFS. The model fit improved further when log-transformed Vactual baseline (log[Vactual,0], dOFV = −12.0) was added (Eq. 5).

The estimated coefficients for Vactual,rel,3m ( urn:x-wiley:21638306:media:psp412195:psp412195-math-0028) and log(Vactual,0) ( urn:x-wiley:21638306:media:psp412195:psp412195-math-0029) effects indicate that hazard increases with increasing Vactual and larger baseline (Table 4). The estimated HR was 0.920 (95% confidence interval = 0.892–0.948) for every 10% decrease in Vactual.

Table 4. Parameter estimates and their uncertainty in the final overall survival and progression-free survival models.
OS model PFS model
Parameter Estimate (% RSE) Parameter Estimate (% RSE)
µOS 1.42 (15) µPFS 1.19 (12)
σOS 8.46 (5.0) σPFS 7.79 (3.4)
θVactual 0.190 (36) θVactual,rel,3m 0.836 (19)
θVactual,0 0.239 (28)
  • OS, overall survival; PFS, progression-free survival; RSE, relative standard error; µOS, µPFS, mean of the log-normal hazard model for OS and PFS, respectively; σOS, σPFS, SD of the log-normal hazard model for OS and PFS; θVactual, coefficient of the effect of log-transformed actual volume time-course on the hazard of OS; θVactual,rel,3m, coefficient of the effect of actual volume relative change from baseline up to 3 month on the hazard of PFS; θVactual,0, coefficient of the effect of log-transformed actual volume baseline on the hazard of PFS.

Kaplan–Meier VPCs (Figure 3) show satisfactory predictive performance of the OS and PFS models, even though the PFS model tended to underpredict the survival rate toward the end of the study period. Parameter estimates and their uncertainty for both models are reported in Table 4.

Details are in the caption following the image

Model predictions of the hazard ratio (HR) from the final overall survival (OS) model vs. the actual volume (Vactual) (left), and of the ratio to baseline (middle) and volumes (right) vs. time from the final joint tumor model. The HR (left) is plotted for the range of observed Vactual (0.14–2586 mL), taking the typical baseline Vactual in the mixture subpopulation with larger lesions at baseline (190.7 mL) as a reference. The gray shaded area shows the 95% confidence interval based on parameter uncertainty in the final OS model. Model-predicted ratio to baseline (middle) and volumes (right) vs. time are plotted for the largest lesion in a typical individual belonging to the mixture subpopulation with larger lesions at baseline and treated with 400 mg imatinib daily. The spherical volume (Vspherical) time-course was derived from the maximum transaxial diameter (MTD) time-course (Vspherical = π/6;·MTD/103) assuming that lesions are perfect spheres. Vellipsoid: ellipsoidal volume.


In this pharmacometric modeling analysis of patients with GIST, we characterized the time-course of liver metastases response to first-line imatinib treatment, including three size metrics – the unidimensional MTD, the software-calculated volume Vactual, and the estimated ellipsoidal volume Vellipsoid – as well as tumor density, which informs on the tumor structural status. Log-transformed Vactual time-course best predicted OS. Interestingly, log(Vactual(t)) provided a better fit than Vactual(t), denoting a nonlinear relationship between the HR and Vactual (Figure 2 (left panel)). The Vactual relative change from baseline up to 3 months together with log-transformed Vactual baseline best predicted PFS.

Details are in the caption following the image

Kaplan–Meier visual predictive checks for the final overall survival (left) and progression-free survival (right) models. The observed Kaplan–Meier curve (black line) is compared to the 95% confidence interval (shaded area) derived from model simulations (200 samples). Vertical black lines represent censored observations.

The same structural tumor model could be applied to all three sized metrics. The binomial distribution of the observed baseline was accounted for by a mixture model, in which the probability of belonging to a subpopulation with typically larger tumors than in the other subpopulation was estimated to 34.8%. The assignment to one or the other subpopulation was not related to any of the available demographic variables (age, gender, or clinical center). A logistic growth model, in which the tumor growth rate decreases as the tumor size approaches the carrying capacity, described the data better than an exponential growth (as expected, presumably due to the dependency of the tumor growth from a limited vascular supply). The typical total carrying capacity (both lesions accounted for) was estimated to 1,730 and 1,818 mL for Vactual and Vellipsoid, respectively; these values are similar to literature values of liver volume in living donors (c.a. 1,500 mL)36 and to the baseline liver volume in a similar GIST patient population14 (median = 1,755; range = 1,112–3,354 mL), of which some of the patients reported in this study were part, and, therefore, clinically plausible. When tumor size is very small compared to the carrying capacity, logistic growth can be approximated by an exponential growth with rate constant KG. The KG estimate for MTD corresponds to a typical doubling time of small tumors of 7.4 years, vs. 1.5 years for Vactual and Vellipsoid. The latter is comparable with the value of 1.0 year reported for untreated GIST before surgical resection.37 The slope of the drug effect was estimated to be lower for MTD than for Vactual and Vellipsoid; 3D measurements may, therefore, be more sensitive to detect imatinib-induced changes compared with 1D measurements. This is illustrated in Figure 2 (middle panel), in which the predicted typical ratio to baseline decreases less for MTD than for 3D measurements. Large IIV (range, 165–170% CV) was quantified on the growth rate constant of all size metrics, and moderate to large IIV on the slope of the imatinib dose effect (42–120% CV). Part of the latter variability may have been explained by differences in pharmacokinetics, if available. ILV was not found to be statistically significant in any of the model parameters, except baseline. Accounting for ILV would enable to test individual lesion response on survival, and could result in more accurate predictions of tumor dynamics, which may be of particular importance when extrapolating the prediction beyond the tumor assessment period.25

Tumor density time-course was described by an indirect response model with stimulation of the response output. Tumor density decrease reflects structural changes secondary to treatment (such as inflammation, tumor necrosis, and reduced vascularization) that can induce paradoxical size increase, which may precede size reduction. The weak correlations between density and size-related parameters indicate that structural changes may occur independently of size changes and that both measurements may provide distinct information on tumor response. Substantial ILV (53% CV) was identified on the dose effect parameter in the density model, indicating differences in tumor sensitivity to imatinib and inter-tumor heterogeneity (e.g., due to secondary mutations or different degree of necrosis post-treatment impacting tumor density).38

In the joint tumor model, exploratory analysis showed, as expected, that large correlations (up to 99%) existed between individual model parameters in the MTD, Vactual, and Vellipsoid models. However, as the aim of our analysis was to evaluate the predictive ability of each size metrics for OS and PFS independently, no further model simplification was taken in order to retain flexibility in tumor size predictions.

Semi-automated measurements of Vactual require software for postprocessing of images, and staff to verify and possibly adjust tumor delineation. In clinical settings, this may be impractical due to software cost and restricted availability, and lack of user experience and time resources. In the future, with the introduction in the clinical arena of deep learning-based software, this challenge can be easily overcome. For well-defined tumors (whose borders can be easily identified), Vellipsoid may be easily and more efficiently obtained than Vactual and offer an alternative to account for nonspherical shapes or nonuniform changes in size. The small differences between Vactual and Vellipsoid model parameter estimates are consistent with earlier findings in which Vellipsoid was highly correlated to Vactual and only overestimated by 9% on average.15 Figure 2 (right panel) further illustrates the similarity between model-predicted Vactual and Vellipsoid time-courses. Moreover, consistent with previous findings, this figure highlights the fact that assuming a spherical volume (Vspherical) calculated based on MTD would tend to overestimate the actual volume. Figure 2 (middle panel) also shows that the models predict that the typical tumor size nadir occurs ∼80 days earlier for the volume measurements compared to MTD-based tumor sizes, which could be clinically relevant.

To the authors' knowledge, this is the first time that the predictive ability of longitudinal tumor 1D and 3D size and density on clinical outcome has been compared. Despite the differences in goodness-of-fit criteria or predictive ability were marginal, the 3D measurements resulted in a better fit to the OS and PFS data than the 1D measurements. This is in line with the results from the standard statistical analysis by Schiavon et al.,26 which showed that 3D criteria (based on a categorization of tumor response) seemed to overall better correlate to OS than 1D criteria when assessed after ∼6 or 12 months of treatment. Contrasting results have been reported regarding the predictive ability of Choi's criteria, a combination of size and density responses, on clinical outcome.5, 8-12 Here, none of the density-related metrics could predict OS or PFS when tested alone or in addition to size-related predictors. An alternative to the final OS model, which includes log(Vactual(t)) as predictor, is to use a combination of Vactual baseline and its relative change from baseline over time up to 3 months. This could help predicting long-term clinical outcome (in this study, up to 13 years of treatment) using early tumor assessments; however, these results should be interpreted with caution as the study population (77 patients) was small and OS modeling did not account for subsequent therapies and tumor heterogeneity, which may potentially confound the estimated changes in size. Finally, a maximum of two liver metastases were available in this analysis and clinical outcome may be affected by other (or new) tumor lesions in the liver or other organs (as per RECIST version 1.1).

In summary, the developed models adequately described the longitudinal 1D, 3D, and density data of liver metastases in imatinib-treated patients with GIST. All tumor responses were associated with large IIV. The ILV was identified in imatinib effect on density but not size metrics. This analysis provides an insight into the value of 3D and density measurements in response evaluation of GIST to imatinib therapy. The 3D measurements showed promising predictive value of OS, although the results should be validated in a larger independent study population. The present results encourage the incorporation of 3D measurements in prospective clinical trials to evaluate their ability and value vs. unidimensional measurements (RECIST) in predicting clinical outcomes. The developed volume and density models may be used to leverage data from other cancer types and drug therapies, especially where changes in size are believed to be nonuniform. The developed modeling framework provides better understanding of the relationship among drug exposure, short-term tumor response, and long-term clinical outcome, and may guide early clinical decisions and interventions to enhance benefits for patients with GIST.


This work was supported by the Swedish Cancer Society.

    Conflict of Interest

    As Deputy Editor-in-Chief of CPT: Pharmacometrics & Systems Pharmacology, L.E.F. was not involved in the review or decision process for this article. G.S. is employee of AstraZeneca. E.S., S.M.K., R.H.J.M., and A.R. declared no conflict of interest.

      Author Contributions

      Emilie Schindler and Sreenath M. Krishnan share first authorship. E.S., S.M.K., R.H.J.M., A.R., G.S., and L.E.F. wrote the manuscript. E.S., S.M.K., R.H.J.M., A.R., G.S., and L.E.F. designed the research. E.S., S.M.K., R.H.J.M., A.R., G.S., and L.E.F. performed the research. E.S., S.M.K., and L.E.F. analyzed the data.