Introduction

Clinical data on the natural growth of solid tumors are intrinsically limited (1). Indeed, anticancer therapies are generally started soon after diagnosis and, thus, measurements of untreated tumors are available only for short time intervals. During this limited observation period, tumor mass seems to follow an exponential growth (2). Therefore, the tumor volume doubling time (TVDT), i.e., the time required by a tumor to reach a two-fold volume, has been introduced in the clinical practice to summarize tumor growth dynamics (3). In the majority of the cases, due to the limited availability of tumor mass assessments, TVDT is computed from only two measurements of tumor volume, \({TV}_{1}\) and \({TV}_{2},\) in two subsequent time instants, \({t}_{1}\) and \({t}_{2},\) through the Schwartz’s formula (1):

$$\mathbf{T}\mathbf{V}\mathbf{D}\mathbf{T}=\frac{\left({{\varvec{t}}}_{2}-{{\varvec{t}}}_{1}\right)\mathbf{l}\mathbf{n}(2)}{\mathbf{l}\mathbf{n}\left({{\varvec{T}}{\varvec{V}}}_{2}/{{\varvec{T}}{\varvec{V}}}_{1}\right)}.$$
(1)

This implies that, if the observation period or the tumor volume increase between the two measurements are limited, the TVDT estimate could be inaccurate. In addition, tumor volume measurements are typically derived from computed tomography (CT) or magnetic resonance imaging (MRI) that could be affected by some degree of uncertainty (4, 5). For these reasons, alternative methods are under investigation. They aim to provide an estimate of TVDT from patient characteristics, such as gene signature (6, 7) or CT radiomic features at baseline (8), without requiring sequential tumor volume measurements. Despite its challenging computation, TVDT represents a useful tool for the clinical management of cancer with important implications for a multitude of medical decisions (2, 9). For example, it could allow to determine the optimal follow-up times for screening patients (10), to assess the impact of delaying diagnosis and/or treatment, to accurately estimate the period of risk and the time of metastasis, to correctly select between pharmaceutical or surgical options (11), and to optimize treatment protocols. TVDT was also shown to be strongly correlated with tumor response to therapy and, therefore, to tumor progression and patient survival (12,13,14). Recently, a quantitative relationship linking progression-free-survival (PFS) and TVDT was also demonstrated (15).

PFS is defined as the timeframe between study randomization and objective tumor progression or death, whichever occurs first (16). It is inherently linked to tumor growth dynamics and, accordingly, tumor growth dynamics were found to be relevant predictors of PFS (17, 18). PFS is increasingly used as primary study endpoint in lieu of overall survival (OS), especially in the early clinical development stages for accelerate drug approvals. In these studies, the investigated treatments are shown or disproved to provide therapeutic benefits in terms of extended PFS typically compared to standard-of-care (SOC) therapies. Indeed, a placebo-controlled or untreated comparison is rarely available for obvious ethical reasons.

Given the clinical relevance of TVDT, multiple studies have tried to characterize the TVDT distributions in untreated cancer patients starting from measurements of tumor mass volumes. Based on these studies, a repository of TVDT values for several cancer types can be derived from the literature. Alternatively, a statistical approach has been recently proposed to predict TVDT distributions from PFS curves (15). Exploiting the quantitative relationship linking PFS and TVDT, the typical value and the associated inter-patient variability of TVDT in a given population was predicted for a panel of cancer types, obtaining an agreement with the literature information.

In this paper, a translational modeling framework to predict the TVDT distributions in untreated patients from tumor growth data in patient-derived xenograft (PDX) mice was presented and discussed. Eleven different solid cancers, classified by primary site and/or histological type, were considered. TVDT predictions were successfully validated against clinically observed TVDT ranges in untreated patients. Then, exploiting the intrinsic relationship between tumor growth dynamics and PFS, the expected PFS curves in absence of any anticancer treatment were generated.

Materials and Methods

Cancer Types

Eleven different solid cancers classified by primary site and/or histological type were considered: breast cancer (BR), cholangiocarcinoma (CC), colorectal cancer (CR), head and neck cancer (HN), gastric cancer (GA), liver cancer (LI), melanoma (ME), pancreatic cancer (PA), sarcoma (SA) and two subtypes of non-small-cell lung cancer (LU), i.e., adenocarcinoma (ADC) and squamous cell carcinoma (SCC). They were selected based on the concomitant availability of tumor growth studies in PDX mouse models and adequate characterization of TVDT in untreated cancer patients.

In Vivo Tumor Growth Studies in PDX Mice

A panel of 265 PDX mouse models (25 for each cancer type except for CC (n = 24) and SA (n = 16)), originally derived from cancers of 265 patients, were considered. For each PDX mouse model, a single study, involving several mice, was selected and data obtained from HuBase database (Crownbio Bioscience Inc., https://www.crownbio.com/). All the in vivo experiments were performed at Crownbio as follows: tumor fragments were inoculated subcutaneously in several immunodeficient mice and tumor dimensions were measured by caliper. Tumor volume was derived as (length ∙ width2)/2 and converted to weight assuming unit density. For each study, average tumor weight was made available. The complete list of the analyzed experiments is reported in Table S1.1 of Supplementary Material S1.

Collection and Curation of TVDT and PFS Data in Cancer Patients

TVDTs

For each cancer type, studies measuring the TVDT value of primary tumors in untreated cancer patients were retrieved from the literature (14, 19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43). Only studies reporting information about the median, minimum and maximum TVDT in the observed population as well as the population size, or alternatively individual TVDT values, were selected. Overall, twenty-six studies were included.

Clinical data showed that the distributions of TVDT, computed by the Schwartz’s formula (Eq. 1), is positively skewed, with some very high TVDTs (5, 44). Based on this observation, for each literature study, a range excluding the extreme values was computed from the reported minimum and maximum values. For cancer types for which multiple studies were available, medians and ranges of TVDT were gathered weighting each study by the corresponding number of patients. The detailed procedure is reported in the Supplementary Material S1 (Section Curation of TVDT data in untreated cancer patients).

The observed TVDTs stratified by study and cancer type are reported in Table I and Table S1.2 of Supplementary Material S1.

Table I Observed and Predicted TVDT in Untreated Cancer Patients
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PFS Curves

For each cancer type, literature was searched to retrieve appropriate PFS data to validate PFS predictions. Because placebo arms are rarely available in clinical trials, published PFS data from different clinical trials including patient cohorts receiving placebo, supportive care or active treatments exerting little to no effect on tumor growth (i.e., overall response rate, ORR, ≤ 10%) were selected (45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69). For BR cancer, ADC and SCC of lung, works reporting suitable PFS data were not found. The selected references (n = 25) are listed in Table S1.3 in Supplementary Material S1 together with details about the clinical studies. Kaplan–Meier (KM) plots from these studies were digitalized, their 95% CIs were reconstructed and the median PFS values, i.e., time at which the PFS curve passes cross the 50% point, extracted (see Collection and curation of PFS data in Supplementary Material S1 for details).

Mathematical Modeling Framework

A multistep procedure was developed to predict TVDTs and PFS curves in untreated cancer patients from tumor growth data from PDX mice studies (Fig. 1). For each cancer type, a population model describing the tumor growth in PDX mice was built and used to characterize the distribution of the exponential tumor growth rate in mice. An allometric scaling approach was adopted to translate the tumor growth rates from PDX mice to humans, assuming an exponential growth of the untreated tumor mass in human. Then, the obtained tumor growth rates were used to predict TVDTs and PFS curves in patients. TVDT and PFS predictions were finally validated against the literature data.

Fig. 1
figure 1

Flow chart representing the translational modeling approach developed to predict TVDTs and PFS in untreated cancer patients from tumor growth data in PDX mice

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Mathematical Modeling of Tumor Growth in PDX Mice

For each cancer type, the unperturbed tumor growth model developed by Simeoni et al. (70) was used to describe the time-course of the average (stratified by study) weight of tumor mass. It assumes that the unperturbed tumor growth consists of an initial exponential growth phase, characterized by the rate λ0 [1/day], followed by a linear one (with slope λ1 [cm3/day]). A population (non-linear-mixed effect) approach was adopted to simultaneously analysis all the PDX studies involving the same cancer type. Individual model parameters were supposed to be log-normally distributed; model fitting was performed assuming a proportional residual error, i.e., y = f + b ∙ f ∙ε, where y was the measurement, b a coefficient representing the coefficient of variation, f the model prediction and ε a standardized random variable normally distributed.

Therefore, for each cancer type, a different set of model parameters was identified and, in particular, a log-normal distribution for the exponential growth rate, \({{\varvec{\lambda}}}_{0}\), was estimated. It was parametrized as \({{\varvec{\lambda}}}_{0,{\varvec{i}}}\)=\({{\varvec{\lambda}}}_{0,{\varvec{p}}{\varvec{o}}{\varvec{p}}}\) exp(\({{\varvec{\upeta}}}_{{{\varvec{\lambda}}}_{0},{\varvec{i}}}\)) where \({{\varvec{\lambda}}}_{0,{\varvec{p}}{\varvec{o}}{\varvec{p}}}\) was the typical population parameter (median) of a specific cancer type and \({{\varvec{\upeta}}}_{{{\varvec{\lambda}}}_{0}}\) a normally distributed variable with mean zero and standard deviation \({{\varvec{\omega}}}_{{{\varvec{\lambda}}}_{0}}\), accounting for the associated inter-PDX model (i.e., inter-patient) variability. In addition, the individual \({{\varvec{\lambda}}}_{0}\) values were obtained for all the 265 analyzed PDX mouse models.

Scaling the Exponential Tumor Growth Rate from Mice to Humans

Based on clinical evidence (2), human cancers were assumed to grow exponentially with a rate \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\), at least during the clinically observable period. The exponential tumor growth rates in PDX mice (parameter \({{\varvec{\lambda}}}_{0}\) of the Simeoni model) were scaled up to human. An allometric scaling approach was applied (71,72,73):

$${{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}[1/\mathbf{m}\mathbf{o}\mathbf{n}\mathbf{t}\mathbf{h}]=30\;\mathbf{*}\;{{\varvec{\uplambda}}}_{0}{\left(\frac{{\mathbf{B}\mathbf{W}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}}{{\mathbf{B}\mathbf{W}}_{{\varvec{M}}{\varvec{i}}{\varvec{c}}{\varvec{e}}}}\right)}^{-{\varvec{\upalpha}}}$$
(2)

where \({\mathbf{B}\mathbf{W}}_{{\varvec{h}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\) and \({\mathbf{B}\mathbf{W}}_{{\varvec{m}}{\varvec{i}}{\varvec{c}}{\varvec{e}}}\) represented the typical body weight of human (70 kg) and mice (0.025 kg), respectively, 30 was the unit conversion factor from day to month and \({\varvec{\upalpha}}\) the allometric exponent.

For each cancer type, a log-normal distribution of \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\) was predicted propagating the \({{\varvec{\uplambda}}}_{0}\) variability. Distribution of \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\) was parametrized as \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}},{\varvec{i}}}\)=\({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}},{\varvec{p}}{\varvec{o}}{\varvec{p}}}\) exp(\({{\varvec{\upeta}}}_{{{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}},{\varvec{i}}}\)), where \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}},{\varvec{p}}{\varvec{o}}{\varvec{p}}}={30\mathbf{*}{\varvec{\uplambda}}}_{0,\mathbf{p}\mathbf{o}\mathbf{p}}{({\mathbf{B}\mathbf{W}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}/{\mathbf{B}\mathbf{W}}_{{\varvec{M}}{\varvec{i}}{\varvec{c}}{\varvec{e}}})}^{-{\varvec{\upalpha}}}\) was the typical value (median) and \({{\varvec{\upeta}}}_{{{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}}\) a normally distributed variable with mean zero and standard deviation \({{\varvec{\omega}}}_{{{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}}={{\varvec{\omega}}}_{{{\varvec{\lambda}}}_{0}}\). The 265 individual \({{\varvec{\uplambda}}}_{0}\) values, computed for all the PDX mouse models, were translated to the corresponding patient values through Eq. 2, too.

Predicting TVDT in Untreated Cancer Patients

TVDTs in untreated cancer patients can be directly computed from \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\) through the following formula:

$${{\varvec{T}}{\varvec{V}}{\varvec{D}}{\varvec{T}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}=\frac{{\varvec{l}}{\varvec{n}}(2)}{{{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}}.$$
(3)

Therefore, for each cancer type, \({{\varvec{T}}{\varvec{V}}{\varvec{D}}{\varvec{T}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\) resulted log-normally distributed: \({{\varvec{T}}{\varvec{V}}{\varvec{D}}{\varvec{T}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}},{\varvec{i}}}={{\varvec{T}}{\varvec{V}}{\varvec{D}}{\varvec{T}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}},{\varvec{p}}{\varvec{o}}{\varvec{p}}}\) exp(\({{\varvec{\upeta}}}_{\mathbf{T}\mathbf{V}\mathbf{D}\mathbf{T},\mathbf{H}\mathbf{u}\mathbf{m}\mathbf{a}\mathbf{n},\mathbf{i}}\)) where \({{\varvec{T}}{\varvec{V}}{\varvec{D}}{\varvec{T}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}},{\varvec{p}}{\varvec{o}}{\varvec{p}}}={\varvec{l}}{\varvec{n}}(2)/{{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}},{\varvec{p}}{\varvec{o}}{\varvec{p}}}\) was the typical value (median) and \({{\varvec{\upeta}}}_{\mathbf{T}\mathbf{V}\mathbf{D}\mathbf{T},\mathbf{H}\mathbf{u}\mathbf{m}\mathbf{a}\mathbf{n}}=-{{\varvec{\upeta}}}_{{{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}}\) a normally distributed variable. Individual TVDT values were also computed from individual \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\) estimates.

The predicted TVDTs (individual values and distributions) were finally compared with the clinical TVDTs collected from the literature (Table S1.2) to assess the translational approach.

Simulating PFS Curves in Untreated Cancer Patients

PFS predictions in untreated cancer patients were obtained following a strategy based on the work of Kay et al. (15). First, only progressive disease (PD) events due to target lesion progression were considered. Simplifying the response evaluation criteria in solid tumor (RECIST) (74), they were defined as a 20% increase of the sum of longest diameters (SLDs) of target tumor lesions. Second, tumors were supposed to be composed by a single lesion with a spherical shape (i.e., same growth dynamics along the three spatial dimensions), that could be considered representative of the overall tumor burden due to target lesions. Under these assumptions, the tumor volume change associated to a 20% increase in diameter was found to be 1.728-fold (see Supplementary Materials S2). Finally, assuming that tumor volume grows exponentially with a rate \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\), the time of tumor progression, \({{\varvec{t}}}_{{\varvec{e}}{\varvec{v}}{\varvec{e}}{\varvec{n}}{\varvec{t}}},\) was computed as:

$${{\varvec{t}}}_{{\varvec{e}}{\varvec{v}}{\varvec{e}}{\varvec{n}}{\varvec{t}}}=\frac{{\varvec{l}}{\varvec{n}}(1.728)}{{{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}}.$$
(4)

For each cancer type, 200 values were sampled from the log-normal distribution of \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\), the corresponding times of progression were computed through Eq. 4 and used to construct the KM curve for PFS. This process was repeated 1000 times propagating uncertainty of parameter estimates, i.e., 1000 studies, each composed by 200 patients, were simulated. The 1000 simulated KM curves were used to construct the KM-visual predict check (VPC) plots, where simulations were overlaid with the published PFS plots (when available). The predicted and observed median PFS (i.e., time at which the PFS curve passes cross the 50% point) were also compared.

Software

The population tumor growth Simeoni model was implemented in Monolix (Version 2021R1 http://lixoft.com/products/monolix/), while diagnostic plots were produced in R (Version 3.6.0) after post-processing of the results.

Human TVDT predictions as well as simulated PFS curves were computed in R (Version 3.6.0). Functions included in the “mvlognCorrEst” (75) and “fitdistrplus” packages were used to sample from log-normal distributions and to compute the PFS curves, respectively.

Results

Mathematical Modeling of Tumor Growth in PDX Mice

For each cancer type, the log-normal distribution of \({{\varvec{\lambda}}}_{0}\) and the individual \({{\varvec{\lambda}}}_{0,{\varvec{i}}}\) values of each PDX mouse model were successfully derived by fitting data with a population Simeoni growth model (see Fig. S3.1 in Supplementary Material S3 for diagnostic plots). The model parameter estimates are reported in Table S3.1 and S3.2 together with their uncertainty. In Fig. S3.2 the densities of \({{\varvec{\lambda}}}_{0}\) distributions are plotted propagating uncertainty of \({{\varvec{\lambda}}}_{0,{\varvec{p}}{\varvec{o}}{\varvec{p}}}\) and \({{\varvec{\omega}}}_{{{\varvec{\lambda}}}_{0}}\) estimates.

Scaling the Exponential Tumor Growth Rate from Mice to Human

Obtained \({{\varvec{\uplambda}}}_{0}\) values were scaled up from mice to human through Eq. 2, where the allometric exponent \({\varvec{\upalpha}}\) was fixed to 1/3, embracing the 2/3-power scaling rule (71). For each cancer type a log-normal distribution of \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\) was obtained. The uncertainty affecting mice-related parameter estimates was propagated to the \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\) distribution parameters, which are reported in Table S3.3. Density plots are displayed in Fig. S3.2 of Supplementary Material S3. Individual \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}},{\varvec{i}}}\) values were computed and are reported in Table S3.3.

Predictions of TVDTs in Untreated Cancer Patients

For each cancer type, the log-normal distribution of TVDT in untreated cancer patients was derived from \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\), by Eq. 3 (Table S3.4 of Supplementary Material S3), also considering uncertainty affecting \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\) distribution parameters. Median values and 80% PIs of TVDTs were computed (See Supplementary Material S3, Section “Predicting the TVDT in untreated cancer patients”) and shown in Table I together with the typical values and the ranges observed in cancer patients as reported in the literature. A very good agreement was found between the predicted and observed TVDTs, with all cancer types showing some degree of overlap.

Predicted TVDT median values are very close to the typical values observed in clinics, with 91% of the predictions within 1.5-fold of the observations (additional fold-error metrics are reported in Table S3.5 of Supplementary Material S3) and a root mean squared error (RMSE) of 0.9244 months (see Fig. 2, panel A). Considering the individual TVDTs, directly derived from the 265 analysed PDX mouse models (Table S3.4 of Supplementary Material S3), 83.59% of the predictions falls within the reported clinical 80%CI ranges. Among the considered cancer types, sarcoma is the worst predicted case study, with predicted TVDTs generally higher than the observed.

Fig. 2
figure 2

a Predicted versus observed TVDTs stratified by cancer type. Dots mark the typical values, vertical and horizontal bars the 80%-intervals of the observed and predicted TVDTs, respectively. Identity line and 1.5-fold areas are represented by solid and dashed black line, respectively. b Box plots (medians and 80%-intervals) of the observed (lightblue) and predicted (grey) TVDTs in untreated cancer patients stratified by cancer type. Dots mark the individual TVDTs predicted from the 265 analysed PDX mouse models. c Box plots (medians and 80%-intervals) of the observed (colored) and predicted (grey) TVDTs in untreated cancer patients stratified by cancer type and literature study. In all the panels, the 80%-intervals represent the 80%CIs for observed TVDTs and 80%PI for the predicted TVDTs

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Regarding the TVDT inter-patient variability, values vary significantly among studies reported in the literature, even if they are related to the same cancer type (panel C of Fig. 2). The 80%PIs of TVDTs are quite consistent with the clinically observed 80%CIs for almost all the cancer types; however, the comparison of the observed and predicted standard deviation of TVDTs (i.e., \({{\varvec{\omega}}}_{{\varvec{T}}{\varvec{V}}{\varvec{D}}{\varvec{T}}}\) in Table S1.2 and S3.4) reveals an underestimation of the clinically observed inter-patient variability for some cancer types.

Predictions of PFS Curve in Untreated Cancer Patients

PFS curves were generated from the distributions of the scaled exponential growth rates, \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}},\) accounting for the uncertainty of parameter estimates. Figure 3 shows the KM-VPC plots generated from 1000 simulated clinical studies (each composed by 200 individuals) superimposed over the observed PFS curves for cancer types for which literature data were available. KM-VPC plots for the remaining cancer types are reported in Fig. S3.5 in Supplementary Material S3. Simulated KM curves were overlaid to the literature PFS curves throughout the time course, demonstrating that the translational modeling approach is able to adequately predict PFS in untreated cancer patients. Predicted median PFS (i.e., time at which the PFS curve passes cross the 50% point) together with its 95%PIs are reported in Table II and S3.6 in comparison with observations and plotted in Fig. 4. Predictions and observations are in close agreement with 95%-ranges overlapping in most of the cases.

Fig. 3
figure 3

KM-VPC plots (grey areas) generated from 1000 simulated clinical studies composed of 200 individuals each superimposed to PFS curves taken from literature studies (colored areas), stratified by cancer types. Median and 95% CI are reported

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Table II Observed and Predicted Median PFS [month] in Untreated Cancer Patients
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Fig. 4
figure 4

Box plot of the observed (colored) and predicted (grey) median PFS in untreated cancer patients stratified by cancer type and references. For observations, patient cohorts receiving placebo or active treatments were marked in blue and green, respectively. Medians and 95%PI are reported. For predictions, 1000 simulated clinical studies each composed by 200 individuals were generated

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Discussion

In this work, a translational modeling approach to predict TVDT distributions and PFS curves in population of untreated cancer patients from tumor growth data in PDX mice was developed and successfully applied on eleven types of solid cancers. It is a multistep procedure that essentially relied on the assumption of an exponential growth of human solid tumors. First, for each cancer type, a set of tumor growth studies in PDX mice was analyzed through a mathematical modeling approach, allowing to derive the exponential tumor growth rate characterizing the initial tumor growth phase in mice. To this aim, the Simeoni tumor growth model was applied and combined with a population (non-linear mixed effect) approach. A log-normal distribution for the exponential tumor growth rate (\({{\varvec{\lambda}}}_{0}\) parameter) was identified for each cancer type, accounting for the variability between different PDX mouse models (called inter-PDX variability). Then, the exponential growth rates were scaled up from mice to humans. According to the allometric scaling rules, the values of \({{\varvec{\lambda}}}_{0}\) were adjusted based on body weight (Eq. 2). The inter-PDX variability was propagated, and a log-normal distribution was derived for the exponential tumor growth rates in humans, \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}},\) for each cancer types. The individual rates of all the 265 analyzed PDX mouse models were translated in human, too. The obtained \({{\varvec{\lambda}}}_{{\varvec{H}}{\varvec{u}}{\varvec{m}}{\varvec{a}}{\varvec{n}}}\) (individual values and distributions) were used to predict the TVDT distributions and PFS in untreated cancer patients.

TVDT predictions were compared to the literature values in untreated cancer patients, showing a very good agreement. Predicted and observed TVDT medians are very close (RMSE = 0.9244 months), with predictions within 1.5-fold of observations for all the cancer types except for sarcoma that results the worst predicted case. 83.59% of the 265 individual TVDTs falls within the reported clinical ranges (80%CI). Regarding the inter-patient variability, the 80%PI of predicted TVDTs were aligned with the 80%CI of the clinical values However, the comparison of the observed and predicted standard deviation of TVDTs (i.e., \({{\varvec{\omega}}}_{{\varvec{T}}{\varvec{V}}{\varvec{D}}{\varvec{T}}}\) in Table S1.2 and S3.4) highlighted an underestimation of the inter-patient variability for some cancer type. This is at least partially due to the limited number of considered PDX mouse models (n = 25) for each cancer type versus the wider cohorts of patients on which the clinical estimates of TVDT were derived. Indeed, the estimated inter-PDX variability increased by increasing the number of PDX mouse models (intermediate results not shown), supporting previous hypothesis. However, other causes, such as the intrinsic limitation of PDX mouse models in completely grasping the inter-patient heterogeneity or non-exponential growth of tumors in humans, could not be excluded. TVDT predictions were further compared with the values derived from PFS curves by the Kay et al. (15) for cancer types included in both the studies (i.e., BR, CR, GA, LI, LU, PA cancers and ME). TVDTs predicted by Kay and co-authors are systematically faster than the observed values and, also, than ours that, in turn, are closer to the clinical observations (RMSE between predicted and observed median TVDTs is 3.24 and 1.12 months for Kay’s and our approach, respectively). For what regards TVDT heterogeneity, ranges derived from PFS curves are much broader than the ones derived from PDX mice, and even wider than the ones observed in clinics.

For what concerns PFS curves, KM-VPC plots were constructed comparing 1000 simulated clinical trials, each composed by 200 patients, with PFS plots taken from published clinical studies. No useful PFS data were found in the literature for BR cancer, ADC and SCC of lung. In addition, studies including patient cohorts receiving placebo were available only for some of the cancer types (i.e., CC, CR and LI cancers and SA). For the others, PFS predictions were compared with those reported in studies involving patient cohorts treated with supportive care or therapies exerted limited anticancer effect on tumor growth (ORR ≤ 10%). Even if the comparison should be considered only from a qualitative point of view, it is interesting to note that in most of the cases the observed and simulated curves are close, suggesting that the translational modeling approach is suitable to predict PFS in absence of drug anticancer effects. Results are further confirmed by the agreement between the observed and predicted median PFS.

Further considerations about the assumptions and hypothesis made in this work are necessary. First, the approach relies on the assumption that natural growth of human solid tumors can be approximated by an exponential growth, at least during the clinical observational time period. Indeed, because anticancer therapies are generally started soon after diagnosis, measurements of untreated tumors are usually available only for short time intervals, during which tumor growth appears to be exponential. Even if a tumor growth slowdown is rarely observed in clinics, some studies showed that exponential tumor growth in humans may decelerate as the tumor burden on the host increases (43, 76). These observations support the hypothesis of an S-shaped growth pattern in which an initial rapid growth is followed by a deceleration, and eventually a plateau, when tumor size results in nutrient restrictions (1). Under this more plausible hypothesis, the exponential model would well grasp only the initial tumor growth phase. Unfortunately, as tumor growth dynamics depend on a multitude of factors, such as tumor-host energetic interactions and oxygen/nutrient restrictions, it is not possible to identify a range of applicability, for example in terms of tumor volume, where the exponential growth assumption holds. The large TVDT inter-patient variability observed in clinics might partially resulted as an artefact of the non-exponential tumor growth kinetics (4). However, the limited availability of data in the clinical context prevents the development of an adequate non-exponential growth model, making the exponential model the only viable solution. Literature modeling efforts to describe clinical tumor growth data were aligned with this choice. Indeed, among the models (linear, exponential, logistic, Gompertz) commonly used to characterize the natural growth of human tumors (77, 78), the exponential one remained the most widely adopted (79,80,81,82,83,84,85). In addition, due to the exponential growth assumption, the computation of time to double and to tumor progression is independent from the tumor dimension, allowing to predict TVDT and PFS in absence of any information about initial patient tumor mass dimensions.

Second, a model-based approach was applied to characterize the exponential tumor growth in PDX mice. From a theoretical point of view, \(\boldsymbol{ }{{\varvec{\lambda}}}_{0}\) could be computed directly from tumor volume measurements through a simple linear regression of the log-transformed tumor volume time profiles. However, because a slowdown of tumor growth was always observed in the PDX studies, only tumor measurements collected during the exponential growth phase should be considered to derive the exponential rate directly from the experimental data. This requires identifying the end of the exponential growth phase in advance that could be a non-trivial task. The use of a mathematical model describing the entire tumor growth dynamics, and not only the exponential growth phase, allows us to provide more robust estimates of \({{\varvec{\lambda}}}_{0}\) based on all the available data. In this work, the well-known exponential + linear growth model proposed by Simeoni et al. twenty years ago was applied. The choice was based on its widely assessed ability to describe tumor growth in cell- and patient-derived xenograft mice (86, 87) and to support preclinical-to-clinical translation (88, 89). Alternative tumor growth models could also be applied. For example, we developed a tumor-in-host growth model, based on the Dynamic Energy Budget (DEB) theory, (90,91,92,93) that describes a sigmoidal tumor growth with an initial exponential phase (94) that seems to be scalable from xenograft mice to humans.

Third, the exponential tumor growth rates were allometrically scaled from mice to humans accordingly to body weight raised to the power of –\({\varvec{\upalpha}}\), as typically done for metabolic rates. Indeed, it has been shown that there is a strictly relationship between tumor growth and host metabolism (90, 94, 95). Even if the allometric scaling approach to tumor growth has already been applied in other translational works (96,97,98), this is the first study where tumor growth dynamics were simultaneously scaled in a multitude of cancer types using the same allometric mouse-human conversion factor. The value of the allometric exponent \({\varvec{\upalpha}}\) has been the objective of a fervent debate for decades and a multitude of theories, based on different garments and degree of formality, has been proposed to justify the occurrence of a particular value of \({\varvec{\upalpha}}\) (72, 99,100,101,102,103). It could be either estimated on experimental data (especially when more than two species were available) or a priori fixed to reference values, typically \(1/4\) or \(1/3\) (71). In this work \({\varvec{\upalpha}}\) was first fixed to \(1/3\), and, then, estimated minimizing (least squares algorithm, “nls” function of “stats” R-package) the differences between the observed and predicted median TVDTs. The value simultaneously estimated on all the eleven cancer types (\({\varvec{\upalpha}}\) = 0.3444, SE = 0.00684) was very close to the fixed allometric factor, supporting the initial choice. \({\varvec{\upalpha}}\) values separately estimated for each cancer type were strongly consistent (\({\varvec{\upalpha}}\) ranged in [0.3068, 0.3753] for all cancer types, except to SA for which \({\varvec{\upalpha}}\) = 0.1070 – Table S3.7 in Supplementary material S3) and always close to 1/3, giving further empirical support to the adopted allometric scaling rule.

Fourth, PFS curves take into account, by definition, both PD events and deaths (104). In RECIST 1.1 (74), PD events were defined as: i) target lesion progression (i.e., at least a 20% increase in SLD of target lesions with an absolute increase of at least 5 mm, taking the smallest sum as reference), ii) unequivocal progression of existing non-target lesions or iii) appearance of one or more new lesions. However, in this work, PFS curves were simulated considering only PD events triggered by a target lesion progression, that was simply defined as a 20% SLD increase. Of course, neglecting not-target progression and death events could led to a possible overprediction of PFS, inducing some bias if, for example, clinical trials involve patients with advance disease stage for which the life expectation is short (this could be a possible explanation for the extremely rapid drop observed in some of the PFS curves reported in the literature for CR cancer patients). Conversely, not imposing the 5 mm threshold on the absolute SLD increase of the target lesions could potentially result in an overprediction of PD events and, thus, in an underprediction of PFS, balancing the previous bias. Note that the target lesion progression is generally the main trigger of PD events and the risks of not-target lesion progression and death are strongly correlated with it (17), limiting the impact of neglecting death and not-target PD events. Furthermore, the presence of a single tumor mass representative of the overall tumor burden was the basic hypothesis on which time to progression was computed. From a methodological point of view, this is a strong simplification of the actual clinical settings, where PFS analysis is generally performed on metastatic cancer patients for which tumor progression is evaluated based on measurements of multiple lesions. To this regard, it is also important to add that primary tumors and metastasis could exhibit differences in the tumor growth rate. However, it is remarkable that, even in presence of such strong simplifying assumptions, the translational exercise and in particular the PFS prediction shows excellent performances on almost the totality of the considered solid tumors, empirically confirming the robustness and the potential usefulness of the here proposed approach.

Finally, in this work, eleven different solid cancer types were considered, clustered and classified by primary site and/or histological type. More specific sub-stratification based on molecular characteristics and/or gene expression was hampered by the lack of adequate panels of PDX mouse models and clinical data on which validating predictions. This led to an over-generalization of certain cancer types, such as BR cancer for which molecular classification on the expression status of hormone receptors has been shown to have an impact on TVDT (21). Incorporating covariates such as molecular characteristics and genetic alterations could significantly improve the proposed translational framework, also increasing the clinical relevance of the obtained predictions.

Overall, the translational framework proposed in this work is able to adequately predict TVDT distributions in untreated cancer patients starting only from tumor growth data in PDX mice. It represents a powerful tool to increase our understanding on TVDT in humans and to compensate the limited availability of data in the clinical setting. Increasing knowledge on typical TVDT can contribute to plan and evaluate optimal screening and follow-up programs, to draw up recommendations for treatment options and to estimate the impact of treatment delay (2, 10, 105). The proposed translational approach could be also potentially applied to derive TVDT values in individual cancer patients to optimize and personalize treatments, if the corresponding PDX mouse models were available. Even if in this work a complete validation of the individual TVDT predictions is missing due to the absence of actual correspondence between PDX mouse models and clinical subjects, the fact that individual TVDT predictions fall within observed ranges is a promising result for a potential application in the context of precision medicine. For example, co-clinical trials where PDX mouse models are directly derived from patients enrolled in clinical studies (106, 107), is an interesting field of investigation. Of course, current timelines (2–8 months) and success rate for the establishment of PDX mouse models represents a bottleneck.

Together with TVDT, the proposed approach allowed to predict the expected PFS curves in patient cohorts receiving placebo. Given the increasing use of PFS as clinical endpoint in lieu of OS, the possibility of predicting PFS from preclinical studies in PDX mice could be of relevant value. Here, PFS curves were predicted in absence of any anticancer treatment. However, the proposed translational framework could be easily extended to cancer treatments. Research efforts are currently focused on this direction. To date, PFS curves in populations of PA and LI cancer patients receiving SOC therapy were successfully predicted starting from panels of tumor growth inhibition studies in PDX mice (108). Once these preliminary results will be confirmed, the predicted PFS curves could be used to reduce/substitute control arm (either placebo or SOC) against which to compare the agent investigated in the clinical trial. Even the effect of investigated treatments on PFS could be early predicted from preclinical data in PDX mice, hence supporting early decision making in oncology.

Conclusions

We developed a model-based translational framework able to predict TVDT distributions and PFS in untreated cancer patients from tumor growth data in PDX mice. It represents a powerful tool to increase the knowledge on TVDT in humans, without the need of direct measurements of tumor dimensions. In addition, the predictions of the expected PFS curves in absence of any anticancer treatment could potentially be used as ‘virtual’ placebo-controlled arms against which to compare experimental agents during clinical trials. However, further validation and refinement are needed to fully assess their effectiveness in this regard.

Finally, the obtained results demonstrates that, when coupled with a smart use of mathematical modeling, PDX mouse models provide a wealth of information regarding tumor dynamics in humans.