Consider the following problem in Radiation Therapy Radiatio

Consider the following problem in Radiation Therapy. Radiation therapy involves using an external beam treatment machine to pass ionizing radiation through the patients body, damaging both cancerous and healthy tissues. Our goal is to have enough radiation dosage throughout tumoir region while keep the damage to health tissues within a certain thresholod We consider one special case. The table below shows the dosage amount delivered to different areas by two beams. Bcam 1 Bcam 2 Arca Hcalthy anatomy Critical tissues Tumor region Center of tumor 31 32 Suppose we are required to limit dosage reaching to critical tissues by bi, and we want to send exactly b2 dosage amount to tumor region and at least bz dosage amount to center of tumor. (a) Formulatc a lincar program to minimizc thc total dosc to the cntirc hcalthy anatomy. (b) If Beam 1 delivers ci dose to the healthy anatomy, is previous mathematical programming model still a linear program\'?

Solution

Area

Beam 1

Beam 2

Healthy Anatomy

c1

c2

Critical Tissues

d11

d12

Tumor Region

d21

d22

Center Of Tumor

d31

d32

a) Decision Variables:

Let x1 and x2 represent the dosage at the entry point for beam 1 and the beam 2, respectively.

Objective function:

Minimize Z= c1*x1+c2*x2

s.t.:

x1, x2>=0

non-negativity

d11*x1+d12*x2<=b1

limited dosage to critical tissues

d21*x1+d22*x2=b2

exact dosage to tumor region

d31*x1+d32*x2>=b3

minimum dosage to center of tumor

b) Yes, if Beam 1 Delivers c1^2 dose to the healthy anatomy, the model still remains linear model as c1 here is a mathematical value (i.e. it is a constant and not a variable in the model) that needs to be known before solving the model, and hence the value of c1^2 will also be mathematical. If any of the variables in the constraints were in squared form, then the model wouldn\'t have been linear.

Area	Beam 1	Beam 2
Healthy Anatomy	c1	c2
Critical Tissues	d11	d12
Tumor Region	d21	d22
Center Of Tumor	d31	d32
a) Decision Variables:
Let x1 and x2 represent the dosage at the entry point for beam 1 and the beam 2, respectively.
Objective function:
Minimize Z= c1*x1+c2*x2
s.t.:
x1, x2>=0	non-negativity
d11*x1+d12*x2<=b1	limited dosage to critical tissues
d21*x1+d22*x2=b2	exact dosage to tumor region
d31*x1+d32*x2>=b3	minimum dosage to center of tumor
b) Yes, if Beam 1 Delivers c1^2 dose to the healthy anatomy, the model still remains linear model as c1 here is a mathematical value (i.e. it is a constant and not a variable in the model) that needs to be known before solving the model, and hence the value of c1^2 will also be mathematical. If any of the variables in the constraints were in squared form, then the model wouldn\'t have been linear.