1. Volume 148, Issue 1, Pages 362-375 (January 2012)
Computational Modeling of Pancreatic Cancer Reveals Kinetics of Metastasis Suggesting Optimum Treatment Strategies 
Hiroshi Haeno, Mithat Gonen, Meghan B. Davis, Joseph M. Herman, Christine A. Iacobuzio-Donahue, Franziska Michor 
Cell 
Volume 148, Issue 1, Pages (January 2012)
DOI: /j.cell
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2. Cell  , DOI: ( /j.cell )
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3. Figure 1
A Mathematical Framework of Pancreatic Cancer Progression Allows the Prediction of Growth and Dissemination Kinetics
(A) Computed tomography (axial view) of one representative patient at initial diagnosis, one intermediate time point five months later, and then again at 7 months after diagnosis, which was also one week before death. In each image the primary pancreatic cancer is indicated in dashed yellow outlines and the liver metastases by dashed red outlines.
(B) The mathematical framework. The model considers three cell types: type-0 cells, which have not yet evolved the ability to metastasize, reside in the primary tumor where they proliferate and die at rates r and d. They give rise to type-1 cells at rate u per cell division; these cells have evolved the ability to metastasize but still reside in the primary tumor, where they proliferate and die at rates a1 and b1, respectively, and disseminate to a new metastatic site at rate q per time unit. Once disseminated, cells are called type-2 cells and proliferate and die at rates a2 and b2, respectively. This mathematical framework can be used to determine quantities such as the risk of metastatic disease at diagnosis and the expected number of metastasized cells at death.
(C and D) Estimated mutation and dissemination rates allow the prediction of the probability of metastasis at diagnosis. The color represents the deviations between the data and the results of the mathematical model; we used patient data on the number of metastatic sites and metastatic cells for the estimation, and then calculated the geometric mean of the two values for each point. Darker colors represent the region of fit between theory and data. Panel (D) provides a more detailed analysis of the data shown in panel (C).
(E) The panel shows the probability of metastasis at diagnosis (red curve) and the probability of the existence of cells in the primary tumor that have evolved the potential to metastasize (blue curve). Parameters are u = 6.31 · 10-5, q = 6.31 · 10-7, r = a1 = 0.16, a2 = 0.58, d = b1 = 0.01r, and b2 = 0.01a2.
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4. Figure 2
The Predictions of the Mathematical Framework Are Validated Using Patient Data
(A and B) The panels show the distribution of survival times of patients who were diagnosed with primary tumors with a diameter of  cm (A) and of 3.5 – 4.4 cm (B).
(C and D) The panels show the distribution of the number of metastatic cells at autopsy of patients who were diagnosed with primary tumors with a diameter of  cm (C) and of  cm (D).
(E and F) The panels show the distribution of the number of primary tumor cells at autopsy of patients who were diagnosed with primary tumors with a diameter of 2.5–3.4 cm (E) and of 3.5–4.4 cm (F).
In all panels, the red curves represent the prediction of the mathematical framework and the black lines represent the data. We observed no significant difference between the predictions and the data; the p values are (A) 0.26, (B) 0.63, (C) 0.54, (D) 0.47, (E) 0.13, and (F) Parameters are u = 6.31 · 10-5, q = 6.31 · 10-7, d = b1 = 0.01r, b2 = 0.01a2 and γ = 0.7. Tumor size at autopsy was obtained from the normal distribution with mean 11.2 and variance 0.46 in a base 10 logarithmic scale for each calculation. The growth rate of primary tumor cells and metastatic tumor cells are obtained from the normal distribution with mean 0.16 and variance 0.14, and mean 0.58 and variance 2.72, respectively.
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5. Figure 3
Validation of Our Framework Using an Independent Patient Cohort
(A) The distribution of the primary growth rate from the original dataset including 101 patients is shown in blue and that from the additional data in black; for the latter, only 10 patients had sufficient follow-up measurements (size at diagnosis, intermediate, and death) such that the growth rate could be determined.
(B) The panel shows the distribution of survival times of patients after resection of the primary tumor with 2 (1.5–2.4) cm diameter after diagnosis. The red curve represents the prediction of the mathematical framework and the black line represents the data.
(C) The panel shows the distribution of survival times of patients after resection of the primary tumor with 3 (2.5–3.4) cm diameter after diagnosis. The red curve represents the prediction of the mathematical framework and the black line represents the data.
(D) The panel shows the distribution of survival times of patients after resection of the primary tumor with 4 (3.5–4.4) cm diameter after diagnosis. The red curve represents the prediction of the mathematical framework and the black line represents the data. We observed no significant difference between the predictions and the data; the p values are (A) 0.45, (B) 0.44, (C) 0.40, and (D) Parameters used are u = 6.31 · 10-5, q = 6.31 · 10-7, d = b1 = 0.01r, b2 = 0.01a2 and γ = 0.7. Tumor size was obtained from a normal distribution with mean 11.2 and variance 0.46 in a base 10 logarithmic scale for each calculation. The growth rate of primary tumor cells and metastatic tumor cells were obtained from a normal distribution with mean 0.16 and variance 0.14, and mean 0.58 and variance 2.72, respectively.
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6. Figure 4
The Mathematical Framework Predicts Optimum Treatment Strategies for Pancreatic Cancer Patients
The panels show the predictions of different quantities for a tumor size of 1 cm diameter at diagnosis (left column) and 3 cm at diagnosis (right column). The tumor size at autopsy in a 10 base logarithmic scale was obtained from a normal distribution with mean 11.2 and variance 0.46 for each calculation. The growth rates of primary tumor cells and metastatic tumor cells were obtained from a normal distribution with mean 0.16 and variance 0.14; and mean 0.58 and variance 2.72, respectively. The black curve represents mathematical predictions of the survival time without treatment or resection, the blue curve with resection (removal of 99.99% of the primary tumor by surgery), the red and green curves with treatment (90% [red] and 50% [green] reduction of the growth rate), and purple curve with resection and treatment (removal of 99.99% of the primary tumor by surgery and 90% reduction of the growth rate). Parameters are u = 6.31 · 10-5, q = 6.31 · 10-7, d = b1 = 0.01r, b2 = 0.01a2, ε = , and γ = 0.9 (red and purple curve) and γ = 0.5 (green curve). (A and B) Survival time. (C and D) The number of metastatic sites at autopsy. (E and F) The number of primary tumor cells. (G and H) The number of metastatic tumor cells. (I and J) The number of metastatic tumor cells per site. See also Figure S2, Figure S3, Figure S4, Figure S5, Figure S6, and Figure S7.
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7. Figure 5
A Delay in the Initiation of Therapy Significantly Increases Tumor Volume and Shortens Survival
The panels show the prognosis after surgery with different theoretical treatment options and treatment delays. Panel (A) shows the median of the number of tumor cells in 100 trials over time. Panel (B) shows the fraction of surviving patients in 100 trials at each time point. Panel (C) shows the numbers of tumor cells and the fraction of surviving patients. The tumor size at autopsy was obtained from a normal distribution with mean 11.2 and variance 0.46 in a 10 base logarithmic scale. The growth rates of primary tumor cells and metastatic tumor cells were obtained from a normal distribution with mean 0.16 and variance 0.14; and mean 0.58 and variance 2.72, respectively. The black curve represents the case with no treatment after surgery, the red curve with starting treatment immediately after surgery, and the green, blue, and yellow curves with starting treatment 2, 4, and 8 weeks after surgery, respectively. Parameters are u = 6.31 · 10-5, q = 6.31 · 10-7, d = b1 = 0.01r, b2 = 0.01a2, ε = , and γ = 0.7.
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8. Figure S1
Statistical Analysis of the Pancreatic Cancer Patient Dataset Containing 101 Patients
(A) Correlations between growth rates of primary and metastatic tumors and log-survival time. Primary tumors are shown in black while metastatic tumors are shown in white.
(B) Distribution of the standardized residuals from the multiplicative robust regression model of survival times and the factors listed in Table S2C. There is no evidence of poor fit based on this residual plot.
(C) Q-Q plot for the residuals. The horizontal axis represents the quantile from the normal distribution and the vertical axis the quantile of the standardized residual. With most points on or near the line of equality, this figure indicates no serious departures from the presumed regression model.
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9. Figure S2 Survival after Treatment
The figure shows the effect of several treatment options on the survival of patients, in dependence of different parameter values. Panels (A), (B), (C), (D), and (E) show the case in which the second mutation rate is relatively small (v2=v1/2) and panels (F), (G), (H), (I), and (J) show the case in which the second mutation rate is large (v2=0.1). The circles connected by lines indicate the results of the direct computer simulations. Black, red dotted, blue, red, and purple curves respectively represent no treatment, 50% reduction of growth rates by drug, 99% of primary tumor removed by resection, 90% reduction of growth rates by drug and both 90% reduction of growth rates and 90% primary tumor reduction after diagnosis. The parameter region where the net growth rate of primary tumor becomes negative is not shown (panels [B] and [G]). Parameter values are M1=250000, M2= , r=s1=s2=0.11, s3=0.21, d=d1=d2=d3=0.01, v1=10−3, and q=10−4.
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10. Figure S3
The Dependence of Metastatic Quantities on the Grow Rate of Primary Tumor Cells
The figure shows the dependence of the three quantities, probability of metastasis, expected number of metastatic sites, and expected number of metastatic cells, on the growth rate of primary tumor cells. The circles connected by lines indicate the results of the direct computer simulations. Panels (A), (B), and (C) show the case in which the second mutation rate is relatively small (v2=v1/2), panels (D), (E), and (F) show the case in which the second mutation rate is large (v2=0.1), and panels (G), (H), and (I) show the case in which the product of the mutation rates in the new model is fixed as the mutation rate in the original model. Black and red curves in panels g, h, and i, indicate the results of the original model in the main text and the new model where two (epi)genetic mutations confer metastatic ability to tumor cells, respectively. Parameter values are M1=250000, M2= , a2=s3=0.21, d=d1=d2=d3=b1=b2=0.01,, v1=10−3 (a-f), v1=10−2 (g-i), v2=10−2 (g-i), u=10−4 (g-i), q=10−4, ε=0, and γ=0.
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11. Figure S4 The Dependence of Metastatic Quantities on the Death Rate
The figure shows the dependence of the three quantities, probability of metastasis, expected number of metastatic sites, and expected number of metastatic cells, on the death rate. The circles connected by lines indicate the results of the direct computer simulations. Panels (A), (B), and (C) show the case in which the second mutation rate is relatively small (v2=v1/2), panels (D), (E), and (F) show the case in which the second mutation rate is large (v2=0.1), and panels (G), (H), and (I) show the case in which the product of the mutation rates in the new model is fixed as the mutation rate in the original model. Black and red curves in panels (G), (H), and (I), indicate the results of the original model in the main text and the new model where two (epi)genetic mutations confer metastatic ability to tumor cells, respectively. Parameter values are M1=250000, M2= , r=a1=s1=s2=0.11, a2=s3=0.21, v1=10−3 (a-f), v1=10−2 (g-i), v2=10−2 (g-i), u=10−4 (g-i), q=10−4, ε=0, and γ=0.
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12. Figure S5
The Dependence of Metastatic Quantities on the Number of Tumor Cells at Autopsy
The figure shows the dependence of the three quantities, probability of metastasis, expected number of metastatic sites, and expected number of metastatic cells, on the number of tumor cells at autopsy. The circles connected by lines indicate the results of the direct computer simulations. Panels (A), (B), and (C) show the case in which the second mutation rate is relatively small (v2=v1/2), panels (D), (E), and (F) show the case in which the second mutation rate is large (v2=0.1), and panels (G), (H), and (I) show the case in which the product of the mutation rates in the new model is fixed as the mutation rate in the original model. Black and red curves in panels (G), (H), and (I), indicate the results of the original model in the main text and the new model where two (epi)genetic mutations confer metastatic ability to tumor cells, respectively. Parameter values are M1=250000, r=a1=s1=s2=0.11, a2=s3=0.21, d=d1=d2=d3=b1=b2=0.01, v1=10−3 (a-f), v1=10−2 (g-i), v2=10−2 (g-i), u=10−4 (g-i), q=10−4, ε=0, and γ=0.
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13. Figure S6
The Dependence of Metastatic Quantities on the Mutation and Metastatic Rates
The figure shows the dependence of the three quantities, probability of metastasis, expected number of metastatic sites, and expected number of metastatic cells, on the mutation and metastatic rates. The circles connected by lines indicate the results of the direct computer simulations. Panels (A), (B), and (C) show the dependence on the mutation rate, while panels (D), (E), and (F) show the dependence on the metastatic rate. Blue and red curves indicate the case in which the second mutation rate is relatively small (v2=v1/2), and in which the second mutation rate is large (v2=0.1). Black curve indicates the results of the original model in the main text, while green and yellow curves indicate the results of the new model where two (epi)genetic mutations confer metastatic ability to tumor cells. In panels a, b, and c, green curves examine different values of the first mutation rate in the new model (from v1=10−3 to v1=10−0.5) with the second mutation rate fixed as v2=10−2 and the horizontal axes in the panels are shown as the product of two mutation rates to compare the dependence on the mutation rate in the original model; yellow curves examine different values of the second mutation rate in the same way as green curves. Parameter values are M1=250000, M2= , r=a1=s1=s2=0.11, a2=s3=0.21, d=d1=d2=d3=b1=b2=0.01, q=10−4 (a-c), v1=10−3 (d-f, red and blue), v1=10−2 (d-f, green and yellow), v2=10−2 (d-f, green and yellow), u=10−4, ε=0, and γ=0.
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14. Figure S7 The Dependence of Metastatic Quantities on Treatment Options
The figure shows the dependence of the three quantities, probability of metastasis, expected number of metastatic sites, and expected number of metastatic cells, on the resection and reduction of growth rate by chemotherapy. The circles connected by lines indicate the results of the direct computer simulations. Panels (A), (B), and (C) show the dependence on the resection effect, while panels (D), (E), and (F) show the dependence on the reduction of growth rate. Blue and red curves indicate the case in which the second mutation rate is relatively small (v2=v1/2),and in which the second mutation rate is large (v2=0.1). Black curve indicates the results of the original model in the main text, while green curve indicates the results of the new model where two (epi)genetic mutations confer metastatic ability to tumor cells. Parameter values are M1=250000, M2= , r=a1=s1=s2=0.11, a2=s3=0.21, d=d1=d2=d3=b1=b2=0.01, v1=10−3 (red and blue), v1=10−2 (green), v2=10−2 (green), u=10−4, q=10−4, ε=0 (d-f), and γ=0 (a-c).
Cell  , DOI: ( /j.cell )
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