Creatinine is a waste product of creatine, a compound used to supply energy to muscles in the human body. Creatinine is cleared from the body entirely through the kidneys and the urinary tract. We can model the production of creatinine in the muscle as an input to a two-compartment model, one piece containing muscle mass and the other containing blood plasma. A diagram of the system is provided at right. Each compartment (m for muscle, p for plasma) has a fixed and constant volume, and at any point contains a number of moles of creatinine n. After a workout, the muscles produce a creatinine at a constant rate n, which has units [mol/time]. nin k1 Vm Vp nm Пр k₂ K3 The transfer of creatinine between compartments is governed by a first-order rate law based on transfer coefficients k. The mass balances on each compartment, treated as well-mixed control volumes, are: dnp(t) = kmnm(t) (k₁ + k3)np(t) dt dnm(t) dt = k₁np(t) - k₂nm(t) + nin(t) 2.1. Assuming the initial conditions for all states and inputs are zero and that the concentration of creatinine in a given control volume i can be calculated as c¿(t) = ni(t), derive the following first-order transfer functions G₁(s), G₂(s) and G3(s): vi G₁(s) Cp(s) Cm(s) Cm(s) G₂(s) G3(s) Cp(s) Cm(s) Nin(s) 1 Note: G₁(s) is not the same as G₂(s) 2.2. Draw a block diagram of this system, labeling the blocks as G₁(s), G₂(s) and G3(s) 2.3. For a similar model (but different to avoid error propagation) with constant parameters and normalized units, assume that the transfer function relating the muscle concentration Cm(s) to an input Nin(s) is: Cm(s) Nin(s) s+2 s² + 5s +6 Using the table of Laplace Transforms and/or partial fraction expansion, determine the time-domain expression for the concentration of creatinine in the muscle volume: cm (t) responding to a step change in nin(t).
Creatinine is a waste product of creatine, a compound used to supply energy to muscles in the human body. Creatinine is cleared from the body entirely through the kidneys and the urinary tract. We can model the production of creatinine in the muscle as an input to a two-compartment model, one piece containing muscle mass and the other containing blood plasma. A diagram of the system is provided at right. Each compartment (m for muscle, p for plasma) has a fixed and constant volume, and at any point contains a number of moles of creatinine n. After a workout, the muscles produce a creatinine at a constant rate n, which has units [mol/time]. nin k1 Vm Vp nm Пр k₂ K3 The transfer of creatinine between compartments is governed by a first-order rate law based on transfer coefficients k. The mass balances on each compartment, treated as well-mixed control volumes, are: dnp(t) = kmnm(t) (k₁ + k3)np(t) dt dnm(t) dt = k₁np(t) - k₂nm(t) + nin(t) 2.1. Assuming the initial conditions for all states and inputs are zero and that the concentration of creatinine in a given control volume i can be calculated as c¿(t) = ni(t), derive the following first-order transfer functions G₁(s), G₂(s) and G3(s): vi G₁(s) Cp(s) Cm(s) Cm(s) G₂(s) G3(s) Cp(s) Cm(s) Nin(s) 1 Note: G₁(s) is not the same as G₂(s) 2.2. Draw a block diagram of this system, labeling the blocks as G₁(s), G₂(s) and G3(s) 2.3. For a similar model (but different to avoid error propagation) with constant parameters and normalized units, assume that the transfer function relating the muscle concentration Cm(s) to an input Nin(s) is: Cm(s) Nin(s) s+2 s² + 5s +6 Using the table of Laplace Transforms and/or partial fraction expansion, determine the time-domain expression for the concentration of creatinine in the muscle volume: cm (t) responding to a step change in nin(t).
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![Creatinine is a waste product of creatine, a compound used to
supply energy to muscles in the human body. Creatinine is cleared
from the body entirely through the kidneys and the urinary tract.
We can model the production of creatinine in the muscle as an
input to a two-compartment model, one piece containing muscle
mass and the other containing blood plasma. A diagram of the
system is provided at right.
Each compartment (m for muscle, p for plasma) has a fixed and
constant volume, and at any point contains a number of moles of
creatinine n. After a workout, the muscles produce a creatinine at a
constant rate n, which has units [mol/time].
nin
k1
Vm
Vp
nm
Пр
k₂
K3
The transfer of creatinine between compartments is governed by a first-order rate law based on transfer coefficients
k. The mass balances on each compartment, treated as well-mixed control volumes, are:
dnp(t)
=
kmnm(t) (k₁ + k3)np(t)
dt
dnm(t)
dt
=
k₁np(t) - k₂nm(t) + nin(t)
2.1. Assuming the initial conditions for all states and inputs are zero and that the concentration of creatinine in
a given control volume i can be calculated as c¿(t) = ni(t), derive the following first-order transfer functions
G₁(s), G₂(s) and G3(s):
vi
G₁(s)
Cp(s)
Cm(s)
Cm(s)
G₂(s)
G3(s)
Cp(s)
Cm(s)
Nin(s)
1
Note: G₁(s) is not the same as
G₂(s)
2.2. Draw a block diagram of this system, labeling the blocks as G₁(s), G₂(s) and G3(s)
2.3. For a similar model (but different to avoid error propagation) with constant parameters and normalized
units, assume that the transfer function relating the muscle concentration Cm(s) to an input Nin(s) is:
Cm(s)
Nin(s)
s+2
s² + 5s +6
Using the table of Laplace Transforms and/or partial fraction expansion, determine the time-domain
expression for the concentration of creatinine in the muscle volume: cm (t) responding to a step change in
nin(t).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca6b5de9-d666-4be4-bec5-372f49facd74%2Fbbb7f289-f17b-4c2c-aa42-8c87f8e0694d%2Fkgdn0gg_processed.png&w=3840&q=75)
Transcribed Image Text:Creatinine is a waste product of creatine, a compound used to
supply energy to muscles in the human body. Creatinine is cleared
from the body entirely through the kidneys and the urinary tract.
We can model the production of creatinine in the muscle as an
input to a two-compartment model, one piece containing muscle
mass and the other containing blood plasma. A diagram of the
system is provided at right.
Each compartment (m for muscle, p for plasma) has a fixed and
constant volume, and at any point contains a number of moles of
creatinine n. After a workout, the muscles produce a creatinine at a
constant rate n, which has units [mol/time].
nin
k1
Vm
Vp
nm
Пр
k₂
K3
The transfer of creatinine between compartments is governed by a first-order rate law based on transfer coefficients
k. The mass balances on each compartment, treated as well-mixed control volumes, are:
dnp(t)
=
kmnm(t) (k₁ + k3)np(t)
dt
dnm(t)
dt
=
k₁np(t) - k₂nm(t) + nin(t)
2.1. Assuming the initial conditions for all states and inputs are zero and that the concentration of creatinine in
a given control volume i can be calculated as c¿(t) = ni(t), derive the following first-order transfer functions
G₁(s), G₂(s) and G3(s):
vi
G₁(s)
Cp(s)
Cm(s)
Cm(s)
G₂(s)
G3(s)
Cp(s)
Cm(s)
Nin(s)
1
Note: G₁(s) is not the same as
G₂(s)
2.2. Draw a block diagram of this system, labeling the blocks as G₁(s), G₂(s) and G3(s)
2.3. For a similar model (but different to avoid error propagation) with constant parameters and normalized
units, assume that the transfer function relating the muscle concentration Cm(s) to an input Nin(s) is:
Cm(s)
Nin(s)
s+2
s² + 5s +6
Using the table of Laplace Transforms and/or partial fraction expansion, determine the time-domain
expression for the concentration of creatinine in the muscle volume: cm (t) responding to a step change in
nin(t).
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