BIOE 340 Fall, 2012 Modeling Physiological Systems and Laboratory Fischell Department of Bioengineering University of Maryland College Park, MD Lab 4: Numerical Integration Laboratory Assignment 4 Introduction Albumin is the most abundant protein in the human blood plasma, and plays a major role in the maintenance of osmotic pressure. In order to model the development of osmotic pressure, we can consider what happens when a solute particle is excluded from a single pore in a microporous membrane. According to Fick’s First Law of Diffusion, there is an expected net diffusion of solute particles into the pore. However, in this case, the particles are too large and hit the rim of the pore, reflecting back into the bulk solution. During this reflection, the solute particles experience a force, and the magnitude of the force can be obtained from a modified Fick’s equation. 𝜕𝐶 𝑥
𝐷
𝑗! = −𝐷
+
𝐹𝐶(𝑥) 𝜕𝑥
𝑅𝑇
Since js =0 for an impermeable membrane, 𝜕𝐶(𝑥)
𝐹𝐶 𝑥 = 𝑅𝑇
𝜕𝑥
We will consider the forces acting on an element of fluid (see figure above) with an area A from a point x (within the bulk fluid) to a point x+Δx (within the pore near its surface). It can be assumed that the volume element is in mechanical equilibrium. The forces on this volume are the forces acting on the BIOE 340 Fall, 2012 Modeling Physiological Systems and Laboratory Fischell Department of Bioengineering University of Maryland College Park, MD Lab 4: Numerical Integration Laboratory Assignment 4 solute particles and the forces acting on the surfaces in contact with the adjacent fluid. The sum of the forces on the element of fluid must be zero in order to maintain mechanical equilibrium. These forces can be described as “body” forces FB and “contact” forces FC. The total body force is the body force per unit volume integrated over the volume: !!∆!
𝐹! =
𝐹𝐶 𝑥 𝑑𝑉 !
We can insert 𝑑𝑉 = 𝐴𝑑𝑥 and 𝐹𝐶 𝑥 = 𝑅𝑇
!"(!)
!"
and obtain: !!∆!
𝐹! = 𝐴𝑅𝑇
!
𝜕𝐶(𝑥)
𝑑𝑥 𝜕𝑥
Part A The data in the table below contains the change in albumin concentration over the length of the fluid element (
!"(!)
!"
). In order to compute the total albumin concentration reflected from the pore, the data must be numerically integrated. Use the data to write an M-­‐file that will compute the body force (FB) acting on the total albumin concentration reflected by the pore using both the trapezoidal rule and Simpson’s 1/3 rule. The trapezoidal rule can be computed using the MATLAB function trapz. The Simpson’s rule must be manually coded using the equations provided in lecture. Be sure to pay attention to units. 𝝏𝑪(𝒙)
Length (µm) 𝝏𝒙
(μM/µm) 0 3.90 0.6 1.95 1.2 1.21 1.8 2.02 2.4 0.48 3.0 0.66 3.6 4.71 4.2 4.78 4.8 2.87 5.4 0.29 6.0 3.94 BIOE 340 Fall, 2012 Modeling Physiological Systems and Laboratory Fischell Department of Bioengineering University of Maryland College Park, MD Lab 4: Numerical Integration Laboratory Assignment 4 Part B: In a brief summary (no more than one page), compare the two numerical integration methods. Be sure to mention advantages and disadvantages of each method. Suppose you could estimate the integral as a difference (i.e. !!∆! !"(!)
𝑑𝑥
!
!"
= 𝐶 𝑥 + ∆𝑥 − 𝐶(𝑥)). Develop the equations for FB and FC using this assumption and solve for osmotic pressure (𝑃 𝑥 − 𝑃(𝑥 + ∆𝑥)). Remember that !
𝑃 = . Your solution should be in variable form only (do not plug in the values from the table !
above). Based on your solution, hypothesize what is behind the development of osmotic pressure across a microporous membrane. Email your completed m-­‐file to [email protected] by 3:30 PM on Thursday, October 11th. Your code must produce the required figures when executed. Turn in a hard copy of your code along with the written portion of the homework (Part B) at the start of class on 10/11/12.