Table 1.
Patient-derived sample information.
Fig 1.
Overview of mathematical models.
A: The exponential model assumes that all cells in the organoid divide at the same rate a, while the power law model assumes that cell divisions are restricted to a subset of cells, for example at the surface of the organoid. The Gompertz and logistic models each assume that initial growth is exponential at rate a and that the growth slows down over time. This can either be due to the cell division rate decreasing uniformly across the organoid, or due to a decreasing subset of actively dividing cells over time. The von Bertalanffy model assumes that cell proliferation follows the power law model, but in addition that cells die uniformly across the organoid at rate b. B: The exponential and power law models are models of unconstrained growth, while the Gompertz, logistic and von Bertalanffy models all assume that the growth eventually stops, with the organoid reaching a so-called “carrying capacity” K.
Fig 2.
Overview of organoid mathematical modeling.
A: Colorectal cancer patient organoids were imaged at multi-time points and analyzed using an AI-based method. The exported results were analyzed using our mathematical models. The diagram was created using LucidChart and BioRender software. B: NN training with H2B-GFP US organoid data. Pre-processing with extended focal imaging was done with an automated macro. Live/dead organoid training labels were generated from untreated and 1uM ST-treated organoids. Training was performed with 750,000 iterations and validated with auto-selected images. C: NN processing with brightfield images and tracking. NN processing and data export were automated with macro for batch processing.
Fig 3.
Organoid tracking and morphology measurement.
A: Each patient organoid segmented with NN. Representative images show the changes of tracked organoids over time. Blue: live organoids, Red: dead organoids, Green: tracks. B: Area measurements of tracked individual UK, UP and US organoids at Day 0, 3 and 5. Thick dotted lines = Median, Thin dotted lines = Quartile. C: Distribution of sphericity at Day 5. Sphericity is approximately the squared quotient of width and length. D: Distribution of shape factor at Day 5. The shape factor is an area relative to the area of a circle with an equal perimeter. E: Distribution of convexity at Day 5. The convexity is an area relative to the area of object’s convex hull. UK = 303 organoids, UP = 496 organoids, US = 136 organoids.
Table 2.
Model comparison results using BIC.
Table 3.
Model comparison results using normalized fitting error.
Fig 4.
Intrapatient heterogeneity in tumor organoid growth.
A: Distribution of log10(a) within the UK/UP/US datasets, where log10(a) denotes the logarithm of a with base 10. For each organoid in each dataset, the initial exponential growth rate a of the organoid was estimated using the Gompertz model (Sections “Gompertz model” and “Model fitting”). Each panel shows how the estimated values of a are distributed across individual organoids within the indicated dataset, under a logarithmic transformation of a. The logarithmic transformation is applied since the estimated values of a vary across an order of magnitude within each dataset. For each dataset, the distribution of log10(a) is consistent with a normal distribution, meaning that we fail to reject the null hypothesis of normality under a Kolmogorov-Smirnov statistical test at the 5% significance level. B: Distribution of the Gompertz parameters (a, b) within the UK/UP/US datasets, shown on a logarithmic scale. The different datasets for each patient have been combined. For each organoid in each dataset, the initial exponential growth rate a and the rate of growth decay b were estimated using the Gompertz model (Sections “Gompertz model” and “Model fitting”). Each dot in each panel represents a single organoid, where the horizontal position of the dot indicates the value of a for that organoid and the vertical position indicates the value of b. When fitting the Gompertz model to individual organoids, we set b = 10−4 as the smallest possible value for b and treat it as effectively zero. Organoids with b = 10−4 are referred to as “exponential organoids”, while the remaining organoids are referred to as “nonexponential”. The exponential organoids are all situated on the horizontal axis and their position on the axis represents their rate of exponential growth. The slanted lines indicate carrying capacities of K = 10, 102, 103, 104, where the carrying capacity is the predicted final size of the organoid under the Gompertz model (Fig 1). All organoids falling on the same line share the indicated carrying capacity. Organoids falling below the lowest line are predicted to have a final size above 104 cells.
Fig 5.
Interpatient heterogeneity in tumor organoid growth.
A: Comparison of the distributions of log10(a) between the different patient samples, where the datasets for each patient have been combined. B: Comparison of the distributions of carrying capacities between the different patient samples. Only nonexponential organoids are considered (b > 10−4). For each organoid in each dataset, the carrying capacity K = ea/b of the organoid was estimated using the Gompertz model (Sections “Gompertz model” and “Model fitting”). The columns indicate for each patient the proportion of organoids falling within each category K < 10, 10 ≤ K < 102, 102 ≤ K < 103, 103 ≤ K < 104 and K ≥ 104. C: Comparison of the distributions of log10(n1) between the different patient organoids, where the datasets for each patient have been combined, and n1 is the observed size of the organoid on Day 0. Here, log10(n1) denotes the logarithm of n1 with base 10. For each patient sample, the graph of the cumulative distribution function (CDF) of log10(n1) is shown, which gives for each value of x the proportion of organoids satisfying log10(n1) ≤ x. The CDF has been estimated using the ksdensity function in MATLAB. The fact that the UK organoid graph lies farthest to the left means that UK organoids are the smallest on average on Day 0, while the UP organoids are the largest on average on Day 0. D: Comparison of the distributions of log10(ae−bτ) between the different patient organoids, where ae−bτ is the growth rate of the organoid on Day 0 according to the Gompertz model. The UK organoids have the largest growth rates on average on Day 0, while the US organoids have the smallest growth rates on average. E: Comparison of the distributions of log10(n3) between the different patient organoids, where n3 denotes the observed size of the organoid on Day 5. The UP organoids are slightly larger than the UK organoids overall on Day 5, and both the UP and UK organoids are significantly larger than the US organoids overall. F: Comparison of the distributions of log10(ae−b(τ+5)) between the different patient organoids, where ae−b(τ+5) is the growth rate of the organoid on Day 5 according to the Gompertz model. The UK organoids have the largest growth rates on average on Day 5. G: Median organoid size projected to Day 20 for each patient sample. To generate the curves, we sampled 100,000 sets of Gompertz parameters (a, b) from the observed parameter distributions for each patient, and computed the medians of the sampled curves. H: UK organoid size projected to Day 20 with error bars. The limits of the error bars represent the 5th and 95th percentile, respectively, of the 100,000 sampled curves from part G.