A Mathematical Model for Pancreatic Cancer Growth and Response to Treatment
Allison Cruikshank
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03/28/2021
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Description
Pancreatic cancer is one of the leading causes of death due to cancer in the United
States. Analyzing the effects of radiation is extremely valuable in determining when
a patient is allowed surgical resection, which is, presently, the only potentially curative treatment for pancreatic cancer. This study examines pancreatic tumor growth
and shrinkage to predict tumor response and change of resectability for pancreatic
cancer patients undergoing radiation therapy. This is done using ordinary differential
equations as a mathematical model. Mathematical models have increasingly been
applied to various biological systems/processes to analyze the principles involved. In
our project, a population dynamical model is used along with suitable assumptions
to study the tumor growth, and the model parameters are carefully calculated from
references. To model the tumor response under radiotherapy treatment, a linear-
quadratic (LQ) model is incorporated with the tumor growth model. The coupled
model is used to observe the mechanisms involved in pancreatic cancer growth and
radioresistance. Numerical analysis of the model takes place using modeling soft-
ware (MATLAB). We found that the implementation of higher doses of radiation in
a smaller amount of fractions more effectively decreases the cancer cell size with and
without a radioresistance factor.
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- [00:00:01.840]Hi, my name is Allison Cruikshank,
- [00:00:04.210]and my UCare project was on A Mathematical Model
- [00:00:07.720]of Pancreatic Cancer Growth and Response to Treatment.
- [00:00:11.730]So the motivation behind this project
- [00:00:13.720]is that pancreatic cancer
- [00:00:15.180]is one of the leading causes of death due to cancer
- [00:00:17.750]in the United States.
- [00:00:19.310]And this is due to surgical resection
- [00:00:21.700]being the only potential curative treatment.
- [00:00:24.290]However, treatments exist like chemotherapy and radiotherapy
- [00:00:27.800]to decrease the tumor size.
- [00:00:29.630]Therefore, we wanted to create a model
- [00:00:31.800]to predict response of pancreatic cancer cells
- [00:00:34.970]to treatments of radiotherapy.
- [00:00:38.400]So in pancreatic cancer,
- [00:00:40.160]there are three cancer cell types:
- [00:00:42.540]cancer stem cells,
- [00:00:43.830]progenitor cells,
- [00:00:44.887]and differentiated cancer cells.
- [00:00:47.270]Cancer stem cells make up a small subpopulation in the tumor
- [00:00:51.220]and are more radioresistant than other cell types.
- [00:00:55.320]And cancer stem cells and progenitor cells can divide.
- [00:00:58.810]However, differentiated cancer cells do not.
- [00:01:02.470]And when cancer stem cells and progenitor cells divide,
- [00:01:06.370]they can either go symmetric,
- [00:01:08.456]or undergo symmetric division or asymmetric division.
- [00:01:13.940]In the bottom right,
- [00:01:14.960]you can see there models of asymmetric division
- [00:01:18.820]and symmetric division.
- [00:01:20.510]We model (indistinct) division through symmetric division,
- [00:01:25.270]the model on the right.
- [00:01:27.170]However, it has been shown in literature
- [00:01:30.560]that modeling through asymmetric division
- [00:01:33.940]is proven to result in equivalent conclusions
- [00:01:36.980]when modeling growth.
- [00:01:40.870]So for radiotherapy,
- [00:01:43.540]we will be comparing SBRT and CFRT.
- [00:01:48.720]SBRT implements higher doses in smaller fractions,
- [00:01:52.850]however, CFRT implements lower doses
- [00:01:56.050]in a larger amount of fractions,
- [00:01:57.760]and this takes longer.
- [00:01:59.940]Also, SBRT implements five to 25 gray,
- [00:02:05.300]which is a unit of radiation,
- [00:02:07.500]and CFRT implements one to three gray.
- [00:02:13.440]So the methods we use in this UCare project,
- [00:02:16.500]we're looking at three-compartmental cancer growth model,
- [00:02:21.450]this was using ordinary differential equations,
- [00:02:24.720]then we used a linear-quadratic model
- [00:02:27.170]for response to radiotherapy,
- [00:02:29.630]and then we combined these two to make a coupled model
- [00:02:33.860]which we can use to compare radiotherapy treatments.
- [00:02:37.430]So in the next slides, we will talk
- [00:02:39.050]about the three-compartmental cancer growth model,
- [00:02:42.750]and the LQ model.
- [00:02:46.000]The cancer growth model uses cancer stem cells,
- [00:02:49.510]progenitor cells and differentiated cancer cells
- [00:02:51.870]as discussed before.
- [00:02:53.630]Cancer stem cells are represented as C0,
- [00:02:56.570]progenitor cells, C1,
- [00:02:58.520]differentiated cancer cells C2.
- [00:03:01.380]Cancer stem cells can self-renew
- [00:03:03.810]with a probability p0,
- [00:03:06.150]or upon division,
- [00:03:08.890]they can create progenitor cells
- [00:03:10.770]with the probability one minus p0.
- [00:03:14.600]Similarly, the probability that progenitor cells
- [00:03:17.350]can self-renew is p1.
- [00:03:20.130]The other three parameters v0, v1 and d2
- [00:03:23.910]are growth and death rates.
- [00:03:26.040]So cancer stem cells progenitor cells
- [00:03:28.710]have a growth rates of v0 and v1 respectively,
- [00:03:33.630]and differentiated cancer cells have a death rate d2
- [00:03:38.640]as they do not divide.
- [00:03:43.150]So this is our system of ordinary differential equations
- [00:03:47.170]which represents the rate or change of each cell type.
- [00:03:50.270]So the first one represents the rate of change
- [00:03:53.750]of cancer stem cell population.
- [00:03:55.960]The second one, the rate or change
- [00:03:57.520]of progenitor cell population.
- [00:04:01.700]And the third represents the rate of change
- [00:04:04.500]of differentiated cancer cell population.
- [00:04:09.990]So the linear-quadratic model we used for radiotherapy
- [00:04:13.890]is this equation right here,
- [00:04:15.820]this exponential decay equation.
- [00:04:19.050]SF is the surviving fraction of cells,
- [00:04:21.780]D is the cumulative dose in gray,
- [00:04:24.200]and alpha and beta are radiobiological parameters
- [00:04:27.240]of the cell.
- [00:04:28.840]So this model implements some mechanisms
- [00:04:31.870]of the damage that radiation causes,
- [00:04:36.370]which are double-stranded breaks in DNA
- [00:04:38.710]and random lethal lesions in DNA.
- [00:04:41.130]The alpha term in the equation
- [00:04:45.270]represents the unrepairable lethal lesions
- [00:04:48.720]and the beta term refers to double-stranded breaks
- [00:04:52.000]that can not be repaired.
- [00:04:54.350]Then we actually look at the alpha beta ratio
- [00:04:58.590]in our results section
- [00:05:01.360]just because this is what the parameters there are
- [00:05:04.230]that literature uses to compare the resistance
- [00:05:08.190]seen in cancer cells.
- [00:05:14.460]So this is our results.
- [00:05:19.420]When comparing SBRT and CFRT in our coupled model,
- [00:05:24.830]we can see,
- [00:05:25.770]on the left, we did simulations
- [00:05:28.750]that were done without a resistance factor,
- [00:05:31.370]so they had an equivalent alpha beta ratios
- [00:05:33.760]for all cell types.
- [00:05:35.280]And as you can see, for small ratios,
- [00:05:38.540]SBRT results in lower cancer cells
- [00:05:40.830]at the end of the treatment.
- [00:05:42.960]As the ratios increase,
- [00:05:44.750]the difference in say arriving fraction
- [00:05:46.690]is smaller between treatments.
- [00:05:48.740]In the middle panel,
- [00:05:49.730]we assume different radiation parameters
- [00:05:52.450]as cancer stem cells are known to be more resistant.
- [00:05:56.400]Therefore, we implemented a resistance factor of 0.5
- [00:06:00.250]for cancer stem cells.
- [00:06:02.200]This resulted in a lower alpha beta ratio for these cells.
- [00:06:06.560]With this resistance factor,
- [00:06:07.830]the surviving fraction of all cell types
- [00:06:09.980]is lower for SBRT than that of CFRT at low ratios.
- [00:06:15.440]This is same as seen in the previous simulation
- [00:06:17.720]with no resistance factor.
- [00:06:19.500]This similarity show that the added resistance
- [00:06:21.850]of cancer stem cells does not change the trend seen
- [00:06:24.320]when comparing both treatments.
- [00:06:27.150]Comparing treatments with and without
- [00:06:29.370]the resistance factor for cancer stem cells,
- [00:06:31.550]which is the very right,
- [00:06:34.450]we could see the difference at high ratios.
- [00:06:37.800]The amount of progenitor and differentiated cancer cells
- [00:06:40.570]do not change significantly
- [00:06:42.230]with and without resistance factor.
- [00:06:45.090]Therefore, CFRT is seen to be more sensitive to resistance
- [00:06:49.750]while SBRT is not as effected to this out of resistance.
- [00:06:54.840]Overall, in this figure,
- [00:06:56.610]comparing two treatments with and without resistance factor,
- [00:07:00.880]it is clear that SBRT can accommodate the resistance more.
- [00:07:05.980]Our next step was to implement some hypothetical treatment.
- [00:07:10.100]So we implemented a high dose per fraction,
- [00:07:14.170]with 13.3 gray in three fractions,
- [00:07:17.050]and a medium dose per fraction,
- [00:07:18.263]2.6 gray in 15 fractions,
- [00:07:22.400]and a low dose per fraction,
- [00:07:24.310]one gray in 40 fractions.
- [00:07:27.070]And as you can see here,
- [00:07:28.640]the high does per fraction shows the most promising results
- [00:07:32.350]with a low dose per fraction
- [00:07:33.730]has very similar results to CFRT.
- [00:07:37.750]However, although these results
- [00:07:40.500]show that a high dose per fraction like 13.3 gray
- [00:07:45.520]is the most effective,
- [00:07:46.650]surrounding tissue needs to be taken into account.
- [00:07:49.480]Radiotherapy needs to be a compromise
- [00:07:51.360]between sparing healthy cells while killing cancer cells.
- [00:07:57.470]Therefore, our conclusions on this part of the project
- [00:07:59.880]is that SBRT is the more effective treatment
- [00:08:04.140]as it has a lower sensitivity
- [00:08:06.080]to resistance factor of cancer stem cells
- [00:08:09.060]and hypothetical treatments
- [00:08:10.560]can look at a higher dose per fraction.
- [00:08:15.120]Future directions for our project
- [00:08:17.080]can be looking at the long-term effects of treatment.
- [00:08:20.070]Our simulations focused
- [00:08:21.350]on the short-term effects of radiotherapy.
- [00:08:23.300]Also, it can be important to incorporate
- [00:08:25.810]the different dynamics seen in cancer stem cells.
- [00:08:28.400]So radiation can induce progenitor cells
- [00:08:30.650]to become cancer stem cells
- [00:08:32.337]and fractioning radiation
- [00:08:33.820]causes more cancer stem cells to developed.
- [00:08:38.570]That is all for my presentation.
- [00:08:41.070]Thank you so much for your attention,
- [00:08:44.690]and bye.
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