Difference between revisions of "Compartment model"

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(Two compartment model)
(Two compartment model)
 
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in each compartment and that the transport between the compartments
 
in each compartment and that the transport between the compartments
 
is driven by concentration differences. We further assume that a drug
 
is driven by concentration differences. We further assume that a drug
with concentration <math>c_0</math> is injected in compartment 1 at a volume flow
+
with concentration <amsmath>c_0</amsmath> is injected in compartment 1 at a volume flow
rate of <math>u</math> and that the concentration in compartment 2 is the output.
+
rate of <amsmath>u</amsmath> and that the concentration in compartment 2 is the output.
Let <math>c_1</math> and <math>c_2</math> be the concentrations of the drug in the
+
Let <amsmath>c_1</amsmath> and <amsmath>c_2</amsmath> be the concentrations of the drug in the
compartments and let <math>V_1</math> and <math>V_2</math> be the volumes of the compartments.
+
compartments and let <amsmath>V_1</amsmath> and <amsmath>V_2</amsmath> be the volumes of the compartments.
  
 
The dynamics of the system can be obtained by keeping track of the flow rates into and out of each compartment.
 
The dynamics of the system can be obtained by keeping track of the flow rates into and out of each compartment.
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represents the flow rate out of compartment 1 that is not going to  
 
represents the flow rate out of compartment 1 that is not going to  
 
compartment 2.
 
compartment 2.
Introducing the variables <math>k_0=q_0/V_1</math>, <math>k_1=q/V_1</math>, <math>k_2=q/V_2</math> and <math>b_0=c_0/V_1</math> and using matrix
+
Introducing the variables <amsmath>k_0=q_0/V_1</amsmath>, <amsmath>k_1=q/V_1</amsmath>, <amsmath>k_2=q/V_2</amsmath> and <amsmath>b_0=c_0/V_1</amsmath> and using matrix
 
notation, the model can be written as
 
notation, the model can be written as
 
<center><amsmath>
 
<center><amsmath>
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   \end{aligned}
 
   \end{aligned}
 
</amsmath></center>
 
</amsmath></center>
A representative set of parameter values are <math>k_0=0.1</math>, <math>k_1=0.1</math>, <math>k_2=0.5</math> and <math>b_0=1.5</math>.
+
A representative set of parameter values are <amsmath>k_0=0.1</amsmath>, <amsmath>k_1=0.1</amsmath>, <amsmath>k_2=0.5</amsmath> and <amsmath>b_0=1.5</amsmath>.

Latest revision as of 02:31, 9 October 2012

Teorell-model.png

Compartment models can be used to model the transport processes between interconnected volumes, such as the flow of drugs and hormones in the human body (right). Compartment models assume that there is perfect mixing so that the drug concentration is constant in each compartment. The complex transport processes are approximated by assuming that the flow rates between the compartments are proportional to the concentration differences in the compartments. One of the early uses of compartment modes was by Widmark in the 1920s, who modeled the propagation of alcohol in the body. Compartment models are now important for the screening of all drugs used by humans.

Two compartment model

Compartment-model.png

A simple two-compartment model is shown to the right. We assume that there is perfect mixing in each compartment and that the transport between the compartments is driven by concentration differences. We further assume that a drug with concentration math is injected in compartment 1 at a volume flow rate of math and that the concentration in compartment 2 is the output. Let math and math be the concentrations of the drug in the compartments and let math and math be the volumes of the compartments.

The dynamics of the system can be obtained by keeping track of the flow rates into and out of each compartment. The mass balances for the compartments are

math

where math represents flow rate between the compartments and math represents the flow rate out of compartment 1 that is not going to compartment 2. Introducing the variables math, math, math and math and using matrix notation, the model can be written as

math

A representative set of parameter values are math, math, math and math.