A beam of particles is directed at a 0.016-kg tumor. There are 2.1 x 1010 particles per second reaching the tumor, and the energy of each particle is 5.4 MeV. The RBE for the radiation is 14. Find the biologically equivalent dose given to the tumor in 16 s.
To find the biologically equivalent dose given to the tumor, we can follow these steps:
Step 1: Calculate the total energy delivered per second to the tumor.
First, let's calculate the energy delivered by each particle:
Energy per particle = 5.4 MeV = 5.4 x 10^6 electron volts
Now, let's calculate the total energy delivered per second to the tumor:
Energy per second = Energy per particle x Number of particles per second
Energy per second = (5.4 x 10^6 eV) x (2.1 x 10^10 particles/s)
Step 2: Convert the energy to joules.
To convert electron volts (eV) to joules (J), we use the conversion factor:
1 eV = 1.602 x 10^-19 J
Using this conversion factor, let's convert the energy from electron volts to joules:
Total energy per second (in joules) = Energy per second (in eV) x (1.602 x 10^-19 J/eV)
Step 3: Calculate the dose per second.
The dose is a measure of the energy absorbed per unit of mass. In this case, the dose will be calculated in joules per kilogram (J/kg).
Since the tumor mass is given as 0.016 kg, we can calculate the dose per second:
Dose per second = Total energy per second (in joules) / Tumor mass
Step 4: Calculate the biologically equivalent dose.
The biologically equivalent dose (BED) takes into account the relative biological effectiveness (RBE) of the radiation. In this case, the RBE is given as 14.
Biologically equivalent dose = Dose per second x RBE
Step 5: Calculate the biologically equivalent dose for the given time.
To calculate the biologically equivalent dose for 16 seconds, we need to multiply the dose per second by the number of seconds:
Biologically equivalent dose for 16 seconds = Biologically equivalent dose (per second) x 16
By following these steps, we can find the biologically equivalent dose given to the tumor in 16 seconds.