Langmuir model without transport
In this worksheet I am working on the identifiability of the parameters of a model of antibody binding.
create usefull sets
In this worksheet we shall obtain results and look at the case where the inlet concentration, is known
or alternately unknown
set up equations
B equations
Notably I present this equation in terms of the output y_ij
set up y_ij equations.
We label the equation for the nth derivative of
as
(2.1)
(2.2)
obtain y and derivatives multinomial
eliminate B1
take a derivative
substitute in the definition of
Substitute an equation for
obtained by rearanging y1
Put it in a form that is easy to read
(3.1)
look at coefficients
We know that if two multinomials on the same variables
are equal for
all vlaues of these variables, then their coefficents will also be equal.
Now we consider only the coefficients
(4.1)
(4.2)
(4.3)
Evaluate for the two stages of the model
We also know that for two models too give the same outputs, the coefficients of the multinomials
when C is 0
(4.1.1.1)
(4.1.1.2)
(4.1.1.3)
(4.1.1.3)
when C is C_T
(4.1.2.1)
(4.1.2.2)
(4.1.2.3)
Create equation lists and solve
For known C_T
(5.1.1)
For unknown C_T
(5.2.1)