Fig 1.
NLME model of early cancer growth.
A: Shows the population model, p(ψ|θ), for θ = (10, 1, 2, 0.5) in shades of blue and the parameters of a randomly chosen individual in red. The shades of blue indicate the bulk 20%, 40%, 60% and 80% probability of the distribution. B: Shows the distribution of tumour volumes across individuals in the population, p(y|θ, t), in blue and the tumour volume of a randomly chosen individual over time in red. The blue lines indicate the 5th and 95th percentile of the tumour volume distribution at each time point. The quantities are shown in arbitrary units. L denotes length dimensions and T denotes time dimensions.
Table 1.
Outline of an example tumour volume dataset.
The dataset contains (fictitious) time series measurements of tumour volumes across patients. Patients are labelled with unique IDs. The time and tumour volume are presented in arbitrary units. T indicates the time dimension and L the length dimension.
Fig 2.
The figure shows a Gaussian filter, a lognormal filter and a Gaussian KDE filter of the early cancer growth model for S = 100 simulated individuals at time t = 1. Each filter is illustrated by a randomly chosen realisation, illustrated in red, and the 5th to 95th percentile of the filter distribution for different sets of simulated individuals. As a reference, the exact population measurement distribution is illustrated in black.
Fig 3.
Filter inference versus traditional NLME inference.
A: Shows 90 snapshot measurements in arbitrary units, generated from the early cancer growth model, and the fitted NLME models obtained using filter inference with a Gaussian filter (blue) and traditional Bayesian NLME inference (red). The measurements are illustrated with jitter on the time axis. The fitted models are illustrated by the medians and the 5th to 95th percentile range of the inferred measurement distributions, . The filter is constructed using S = 100 simulated individuals. B: Shows the inferred posterior distributions obtained using filter inference (blue) and NLME inference (red). The data-generating parameters (solid lines) as well as the prior distributions used for the inference (dashed lines) are also shown.
Fig 4.
Quality of IIV estimates for varying dataset sizes.
A: Shows inferred population distributions from snapshot measurements of varying numbers of individuals (IDs) using filter inference with a Gaussian filter and S = 100 simulated individuals in blue. The different shades of blue indicate the bulk 20%, 40%, 60% and 80% probability regions. The distribution inferred from measurements of 90 individuals using NLME inference from Fig 3 is illustrated by red dashed lines. The data-generating distribution is illustrated in black. B: Shows the KL divergences between the data-generating population distribution and the inferred distributions from A.
Fig 5.
Inference results of EGF pathway model I.
A: Shows snapshot measurements of active and inactive EGFR concentrations across cells. The cells are exposed to one of two EGF concentrations: data (low) with cl = 2ng/mL (black scatter points); and data (high) with cl = 10ng/mL (grey scatter points). The shaded areas illustrate the 5th to 95th percentile of the inferred measurement distributions using filter inference with Gaussian filters and S = 100 simulated cells. B: Shows the inferred posterior distributions of the model parameters illustrated by blue density plots, together with the data-generating parameter values illustrated by black solid lines. The density of the prior distribution is illustrated by dashed lines for each parameter. C: Shows the inferred population distribution of the production rate and the activation rate in blue. The different shades of blue indicate the bulk 20%, 40%, 60% and 80% probability regions. The probability regions of the data-generating distribution are illustrated by black contours.
Fig 6.
Inference results of EGF pathway model II.
A: Shows the joint posterior distribution of the mean activation rate, , and the deactivation rate,
from Fig 5B. B: Shows the inferred posterior distribution and the inferred population distribution from a separate inference run, where we fixed the deactivation rate to its data-generating value. All other inference settings, including data and priors, remain unchanged from the inference approach used to generate Fig 5.
Fig 7.
Computational costs of filter inference and traditional NLME inference I—Evaluation time of log-posterior.
The figure shows the evaluation time of the traditional NLME log-posterior and its gradient, defined in Eq 7, (blue lines) and the filter inference posterior and its gradient, defined in Eq 10, with a Gaussian filter and S = 50, S = 100 and S = 150 simulated individuals (red lines). The left and right panel illustrate the results for the early cancer growth model and the EGF pathway model, respectively.
Fig 8.
Computational costs of filter inference and traditional NLME inference II—Number of log-posterior evaluations.
The figure shows the number of log-posterior evaluations of traditional NLME inference (blue lines) and filter inference, with a Gaussian filter and S = 100 simulated individuals, (red lines) for the early cancer growth model and the EGF pathway model using varying sizes of snapshot datasets. Each log-posterior evaluation includes the evaluation of its gradient. The posterior distributions are inferred using 1000 MCMC iterations of NUTS after calibrating the algorithm for 500 iterations.
Fig 9.
Sampling efficiency of filter inference variants.
The figure shows the minimum ESS across dimensions as a function of log-posterior evaluations for different posterior distributions inferred with: 1. NUTS and the deterministic filter posterior (blue); and 2. MH and the stochastic filter posterior (orange). For NUTS, the number of evaluations include evaluations of the log-posterior gradient. For MH, the log-posterior gradient is not evaluated. Panels 1, 2, 3 and 4 show the minimum ESS of the cancer growth model posteriors from Fig 4 and panel 5 shows the EGF pathway model posteriors from Fig 6B.
Fig 10.
ABC interpretation of filter inference.
The figure shows the accepted means and variances of the Gaussian filter at t = 0.6 from the inference results in Fig 4, where filter inference is performed on datasets with 90, 270, 810 and 2430 snapshot measurements of the cancer growth model. The accepted summary statistics are illustrated as black scatter points with the corresponding KDE plots shown in blue. The summary statistics of each dataset is illustrated by a red scatter point. The sample variation of the dataset summary statistics is represented by the 5th to 95th percentile of the summary statistic distribution (red bars), estimated from 1000 realisations of each dataset. The exact mean and variance of the data-generating distribution is close to the intersection of the bars.
Fig 11.
Information loss I—Number of simulated individuals.
The figure shows inferred population distributions from 2430 snapshot measurements of the early cancer growth model, using filter inference with a Gaussian filter and S = 3, S = 10, S = 100 and S = 500 simulated individuals. The inferred distributions are visualised in blue. The different shades of blue indicate the bulk 20%, 40%, 60% and 80% probability regions. The data-generating distribution is illustrated by black contours.
Fig 12.
Information loss II—Choice of filter.
The figure shows inferred population distributions from 120 (top row) and 3000 (bottom row) snapshot measurements, using filter inference with different filter choices and S = 100 simulated individuals. The inferred distributions, , are visualised in blue. The different shades of blue indicate the bulk 20%, 40%, 60% and 80% probability regions. The data-generating distribution is illustrated by black solid lines. The posterior inferred with NLME inference is illustrated by black dashed lines.
Fig 13.
Accepted simulated measurements during filter inference.
The figure shows the histograms over all accepted simulated measurements at t = 0.6 (blue) during inference with a Gaussian filter (left panel) and a Gaussian KDE filter (right panel) from the bottom panel in Fig 12. The data-generating measurement distribution is illustrated in black. A typical realisation of each filter is illustrated in red. The inset figure in the right panel shows a typical kernel (blue) placed on a simulated measurement (black cross) during the construction of the Gaussian KDE filter. The simulated measurement is placed at a tumour volume of 350 for illustration purposes. The scale of the kernel is taken from the filter realisation.