A cancerous tumor is spherical in shape with a radius of 10 mm. After initiating chemotherapy treatment, the tumor volume decreases at a rate of 25π mm³ / day. At approximately what speed does its radius change, when the radius is 0.3 mm?
Select one:
1.13 mm / day
-20.83 mm / day
-69.44 mm / day
-0.0625 mm / day
Answers
Step-by-step explanation:
Bulletin of mathematical biology
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Estimating tumor growth rates in vivo
Anne Talkington and Rick Durrett
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Abstract
In this paper we develop methods for inferring tumor growth rates from the observation of tumor volumes at two time points. We fit power law, exponential, Gompertz, and Spratt’s generalized logistic model to five data sets. Though the data sets are small and there are biases due to the way the samples were ascertained, there is a clear sign of exponential growth for the breast and liver cancers, and a 2/3’s power law (surface growth) for the two neurological cancers.
1 Introduction
Finding formulas to predict the growth of tumors has been of interest since the early days of cancer research. Many models have been proposed, but there is still no consensus about the growth patterns that solid tumors exhibit [7]. This is an important problem because an accurate model of tumor growth is needed for evaluating screening strategies [18], optimizing radiation treatment protocols [27, 2], and making decisions about patient treatment [5, 6].
Recently, Sarapata and de Pillis [29] have examined the effectiveness of a half-dozen different models in fitting the growth rates of in vitro tumor growth in ten different types of cancer. While the survey in [29] is impressive for its scope, the behavior of cells grown in a laboratory setting where they always have an ample supply of nutrients is not the same as that of tumors in a human body.
One cannot have a very long time series of observations of tumor size in human patients because, in most cases, soon after the tumor is detected it will be treated, and that will change the dynamics. However, we have found five studies where tumor sizes of different types of cancers were measured two times before treatment and the measurements were given in the paper, [11], [13], [28], [21], and [22]. We describe the data in more detail in Section 4. Another data set gives the time until death of 250 untreated cases observed from 1805 to 1933, see [1]. That data is not useful for us because there is no information on tumor sizes.
In the next section, we review the models that we will consider. Each model has a growth rate r. Given the volumes V1 and V2 at two time points t1 and t2, there is a unique value of r that makes the tumor grow from volume V1 to V2 in time t2 − t1. We use the average of the growth rates that we compute in this way as an estimate for the growth rate. Chingola and Foroni [3] used this approach to fit the Gompertz model to data on the growth of multicellular tumor spheroids. Here, we extend their method to other commonly used growth models.
A new feature of our analysis is that in order to find the best model we plot the estimated values of r versus the initial tumor volume V1 and look at trends in the sizes of the rates. To explain our method, we begin by noting that all of our models have the form