A stochastic modelling framework for cancer patient trajectories: combining tumour growth, metastasis, and survival

Abstract

Cancer is a major burden of disease around the globe and one of the leading causes of premature death. The key to improve patient outcomes in modern clinical cancer research is to gain insights into dynamics underlying cancer evolution in order to facilitate the search for effective therapies. However, most cancer data analysis tools are designed for controlled trials and cannot leverage routine clinical data, which are available in far greater quantities. In addition, many cancer models focus on single disease processes in isolation, disregarding interaction. This work proposes a unified stochastic modelling framework for cancer progression that combines (stochastic) processes for tumour growth, metastatic seeding, and patient survival to provide a comprehensive understanding of cancer progression. In addition, our models aim to use non-equidistantly sampled data collected in clinical routine to analyse the whole patient trajectory over the course of the disease. The model formulation features closed-form expressions of the likelihood functions for parameter inference from clinical data. The efficacy of our model approach is demonstrated through a simulation study involving four exemplary models, which utilise both analytic and numerical likelihoods. The results of the simulation studies demonstrate the accuracy and computational efficiency of the analytic likelihood formulations. We found that estimation can retrieve the correct model parameters and reveal the underlying data dynamics, and that this modelling framework is flexible in choosing the precise parameterisation. This work can serve as a foundation for the development of combined stochastic models for guiding personalized therapies in oncology.

Keywords: Cancer modelling; Mathematical oncology; Stochastic modelling.

Fig. 1

Simulations of tumour growth curves. a Depicts example curves for exponential growth (2) and Gompertz growth (3). The initial tumour size is set to $S_0=0.065{\text{mm}}^3$ and saturation for the Gompertz growth to $K=150{\text{cm}}^3$. For the corresponding growth parameters we chose $\beta \in \{0.48,0.49,0.5,0.51,0.52\}$ for exponential growth and $\beta \in \{0.14,0.17,0.2,0.24,0.3\}$ for the Gompertz growth indicated by the shade of the lines. b An example curve for the Gyllenberg–Webb model (4), where we choose $P(0)=1,Q(0)=0,De(0)=0$ and example parameter values $[b=1,\mu =0.05,d=0.01,r=1,k=2,a=1,m=2]$

Fig. 2

Simulations of the metastasis process. Visualization of the mean metastasis number (solid line) with $95\%$ pointwise credibility intervals (shaded regions) and 10 realisations (stepped lines) from 500 patients using the volume based metastatic spread (5) with exponential tumour growth (2) in a and using the Gompertz growth model (3) in b. The rates $m_{\text{basal}},m_{\text{size}}$ of the metastasis intensity are the same in both cases

Fig. 3

Illustration of the proposed model of cancer patient trajectories. a Outline of the three components of the model, the tumour size S with growth parameter $\beta$, the metastasis process N with rate ${\lambda }_N(S)$ and the death process D with rate ${\lambda }_D(S,n)$ and their interconnections. b Simulation results for a population of 500 cancer patients using parameters reported in Supplementary Table S2: (left) Survival curve and (right) tumour growth and metastasis development for an exemplary patient. Created in BioRender. Hasenauer, AG. (2024) BioRender.com/k04j643.

Fig. 4

Assessment of survival times distribution. a Provides a visualization of the estimated survival curve of model (M4) combining Gompertz growth with exponential proportional metastasis intensity (blue) against the estimated survival curve of 33 patients from the ACT2-trial (green). Additionally, the survival curve of a Weibull distribution with parameters $k=2.7,\lambda =26.7$ (red) is plotted. b Provides the parameter values used to simulate the data from the model

Fig. 5

Assessment of proposed model structure. a Visual outline of model assessment performed using the dataset by Engel et al. (2003) on overall survival and survival after metastasisation of breast cancer patients across different tumour sizes at time of diagnosis. b Comparison of observed (data) and simulated (model) survival curves. The simulation results are indicated using the mean survival curve (line) and the 95-percentile interval (shaded area) from 200 simulations of our model

Fig. 6

Comparison of likelihood formulations. We randomly sampled 100 parameter vectors and evaluated the analytical negative log-likelihood as well as the numerical negative log-likelihood on this vectors. a Visualizes the resulting values for the exponential proportional model and b for the cell division model. The black-dashed lines correspond to the ${45}^{\circ }$ lines

Fig. 7

Evaluation of computational efficiency. a, b show the mean computation times of evaluating the negative log-likelihood function on 100 randomly sampled parameter vectors for the two exponential growth based models (M1) and (M2). c, d, and e depict the trace of the current best negative log-likelihood value over time for 100 single optimisation runs of the exponential proportional model using different likelihood formulations and optimization algorithms

Fig. 8

Parameter inference results for model (M1) Maximum likelihood estimates (MLE) and sampling-based credibility intervals for the model parameters of the exponential proportional model

Fig. 9

Parameter inference results for model (M2) Maximum likelihood estimates (MLE) and sampling-based credibility intervals for the model parameters of the cell division model

Fig. 10

Parameter inference results for model (M3) Maximum likelihood estimates (MLE) and sampling-based credibility intervals for the model parameters of the Gompertz model

Fig. 11

Parameter inference for model (M4). Maximum likelihood estimates (MLE) and sampling-based credibility intervals for the model parameters of the Gyllenberg–Webb model

Fig. 12

Datasets for model selection. Visual comparison of the two datasets used for the model selection task

Fig. 13

Parameter inference for model (M5) Maximum likelihood estimates (MLE) and sampling-based credibility intervals for the model parameters of the treatment effect model