Maths of Treatment Planning

Question 1

What is digital signal processing?

Answer 1

A process that converts signals from sensory data in the real world to digital form

Question 2

What is the definition of a signal?

Answer 2

A description of how one parameter relates to another, whether it is discrete or analogue, 1D, 2D or multi-D

Question 3

What is the definition of a system?

Answer 3

Any process that produces an output signal in response to an input signal

Question 4

What is the process of superposition?

Answer 4

A complex signal is broken down into simple components, decompostion, which are processed and the results reunited, synthesis

Question 5

What are the prerequisites of superposition being used on a signal?

Answer 5

Needs to be linear:
Homogeneous: a change in the input signal’s amplitude results in a corresponding change in the output signal’s amplitude
Additive: signals added at input results in added signals at output

Question 6

What are the methods for decomposing signal in the TPS?

Answer 6

Impulse decomposition

Fourier decomposition

Question 7

How are dose calculations decomposed?

Answer 7

Split by fluence map (input) which models the primary beam photons and scattered photons from the head, and scatter kernels (output) which models scatter and dose deposition in the patient

Question 8

What is the process of performing a dose calculation for a patient?

Answer 8

Decompose beam into primary and scatter components
Adjust each component based on beam shape, intensity, surface topography, internal tissue density
Sum the contribution from all other scattering elements throughout the patient

Question 9

How does treating a scatter kernel as a response function allow for the correction of lateral disequilibrium of electrons in narrow photon beams?

Answer 9

Apply an impulse-response analysis to the primary electrons launched by photons

Question 10

What does the fluence map show?

Answer 10

How the beam exits the linac, how it diverges and how it is attenuatd as it travels
Can be modified in a heterogeneous medium to account for the change in density

Question 11

What is a scatter kernel?

Answer 11

It describes how dose is deposited around an interaction site by electrons released at the site and by photons scattered at the site that cause electrons distant from the original site

Question 12

What are the alternative names for a scatter kernel?

Answer 12

PSF, impulse response, convolution kernel

Question 13

How are scatter kernels made?

Answer 13

Monte Carlo simulations

Question 14

How is superposition achieved and what is the output?

Answer 14

Convolution - produces a third signal- the impulse response

Question 15

What is the impulse response?

Answer 15

Impulse - the input fluence of primary beam and scattered photons
Response - how the system responds to the impulse - spread of dose

Question 16

What is the equation and process of convolution?

Answer 16

Phi x K = D

Reverse impulse response, translate across input, multiply and summate the 2 to give the output signal

Question 17

What is an example equation of numerical convolution?

Answer 17

y(i) = sum(h(j)-x(i-j))

Question 18

What kind of filter is used in 2D convolution?

Answer 18

Window filter

Question 19

How is 3D convolution sped up?

Answer 19

Use fourier transforms: Phi x K = FT(phi).FT(k)

Question 20

What is a Fourier series?

Answer 20

Any function that periodically repeats itself can be expressed as a sum of sines and cosines with different coefficients

Question 21

What is the process of discrete fourier tansforms?

Answer 21

Stimulus written as superposition of infinitely many sinusoids which are each analysed separately and their response computed due to superposition
The response is a sinusoid of the same frequency as the stimulus but different amplitude and phase - sinusoidal fidelity
Response to original stimulus is sum of all individual sinusoidal responses

Question 22

How is Fourier decomposition achieved?

Answer 22

Correlation method- find a known waveform in a signal, multiply the 2 and summate, resulting in a single numerical result which is a measure of similarity

Question 23

What are the equations for discrete Fourier transforms?

Answer 23

R(X(k)) = sum(x(i).cos(2.pi.k.i/N))
Im(X(k)) = -sum(x(i).sin(2.pi.k.i/N))
Frequency domain signals ar N/2 + 1 points long, and run from 0 to N/2