Some cancers are detectable with a simple blood test (if the cancer cells secrete a chemical that circulates in the blood). A spherical tumor 3 cm in diameter may result in a concentration in the blood of 20 nanograms/milliliter of the chemical to be detected. The most common type of blood test can only detect the presence of this chemical in the blood if the concentration is greater than about 0.1 ng/ml. Another way to check is with an MRI scan, which can detect a tumor down to about a diameter of 1 mm. Which test is more sensitive? (Hint: this is a scaling prolem. Determine the size of a tumor that would produce a blood concentration of just under 0.1 ng/ml, which is undetectable by a blood test. Would an MRI be able to detect it?). Answer: In scaling problems, we need a linear scale factor. 0.1 ng/ml is 200 times smaller than 20 ng/ml, and we want to know what diameter of tumor will produce that amount. Assuming the secretion is based on the volume of the tumor, then we have λ 3 = 200 → λ = 5.85, which is how much we scale down the diameter: 3 cm / 5.85 = 0.51 cm (about 5 mm). Thus, any tumor smaller than 5 mm can’t be detected by a blood test, but is detectable by an MRI. Using surface area instead of volume results in λ = 14.1 and a diameter of about 2 mm. Still detectable by MRI. I need an explnation of the calculations step by step please I also do not understand how they got 5.85
Some cancers are detectable with a simple blood test (if the cancer cells secrete a chemical that circulates in the blood). A spherical tumor 3 cm in diameter may result in a concentration in the blood of 20 nanograms/milliliter of the chemical to be detected. The most common type of blood test can only detect the presence of this chemical in the blood if the concentration is greater than about 0.1 ng/ml. Another way to check is with an MRI scan, which can detect a tumor down to about a diameter of 1 mm. Which test is more sensitive? (Hint: this is a scaling prolem. Determine the size of a tumor that would produce a blood concentration of just under 0.1 ng/ml, which is undetectable by a blood test. Would an MRI be able to detect it?).
Answer: In scaling problems, we need a linear scale factor. 0.1 ng/ml is 200 times smaller than 20 ng/ml, and we want to know what diameter of tumor will produce that amount. Assuming the secretion is based on the volume of the tumor, then we have λ 3 = 200 → λ = 5.85, which is how much we scale down the diameter: 3 cm / 5.85 = 0.51 cm (about 5 mm). Thus, any tumor smaller than 5 mm can’t be detected by a blood test, but is detectable by an MRI. Using surface area instead of volume results in λ = 14.1 and a diameter of about 2 mm. Still detectable by MRI.
I need an explnation of the calculations step by step please
I also do not understand how they got 5.85
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