BENG 221 Mathematical Methods in Bioengineering. Fall 2017 Midterm

Transcription

1 BENG Mathematical Methods in Bioengineering Fall 07 Midterm NAME: Open book, open notes. 80 minutes limit (end of class). No communication other than with instructor and TAs. No computers or internet, except for access to posted class materials.

2 Table : aplace and Fourier Transforms u(t) U(s) u(t) U(jω) t 0 δ(t) s e at s + a u(t t 0 ); t 0 0 t 0 du dt e st 0 U(s) s U(s) u(0) u(t 0 ) dt 0 s U(s) f(t 0 ) h(t t 0 ) dt 0 H(s) F (s) e at + δ(t) for t 0; 0 otherwise t u(t t 0 ) du dt jω jω + a e jωt 0 U(jω) jω U(jω) u(t 0 ) dt 0 jω U(jω) f(t 0 ) h(t t 0 ) dt 0 H(jω) F (jω) Table : Green s Functions for Diffusion in -D B.C. G(x, t; x 0, t 0 ) x = 0 x = t > t 0 N (x 0, D(t t 0 )) = 4πD(t t0 ) exp( (x x 0) 4D(t t 0 ) ) u(0, t) = 0 N (x 0, D(t t 0 )) N ( x 0, D(t t 0 )) x (0, t) = 0 N (x 0, D(t t 0 )) + N ( x 0, D(t t 0 )) u(0, t) = 0 u(, t) = 0 u(0, t) = 0 (, t) = 0 x k= sin((k + )π k=0 x (0, t) = 0 u(, t) = 0 (0, t) = 0 x k=0 cos((k + )π sin(kπ x 0) sin( kπ x) exp( (kπ ) D(t t 0 )) x 0 ) sin( (k + )π x 0 ) cos( (k + )π x) exp( ( (k + )π ) D(t t 0 )) x) exp( ( (k + )π ) D(t t 0 )) x (, t) = 0 + cos(kπ x 0) cos( kπ x) exp( (kπ ) D(t t 0 )) k=

3 Problem (0 points): Short answer problems. Provide brief explanations (no lengthy derivations!) for each problem.. (5 points): Among all the eigenvalue-eigenvector pairs obtained by singular value decomposition of a matrix containing multi-dimensional data samples, which one explains most of the variance in the data?. (5 points): Give an example that demonstrates why diffusion over a bounded interval is not space-invariant. 3. (5 points): To arrive at the diffusion transfer function H(k, s), does it matter in which order the Fourier and aplace transforms are applied to the diffusion equation, and why? 4. (5 points): Solve the following integral: + δ(x x 0 ) exp( jkx) dx =

4 Problem (30 points): Consider a two-segment lumped model of diffusion along a passive cable of length, with line resistivity r and line capacitance c, and with zero-voltage boundary conditions on both ends, as shown below. The length of each of the two segments is x = /. v 0 = 0 r x v r x v = 0 c x. (0 points): Write the ordinary differential equation governing the dynamics of the voltage v (t) at the center of the cable. Is this ODE homogeneous or inhomogenous?

5 . (0 points): Show that the solution to this ODE is given by a decaying exponential over time, v (t) = A exp( t/τ). Identify the amplitude constant A and the time constant τ in terms of the initial condition v (0), cable length, and diffusivity D. 3. (0 points): Compare the time constant τ that you obtained in part for this twosegment lumped approximation of the passive cable, with the time constant τ for the infinite-segment continuum limit, corresponding to the dominant (first) term in the eigenmode series of the homogeneous solution for the cable with same diffusivity D, same total length, and same zero-voltage boundary conditions on both ends.

6 Problem 3 (50 points): Here we will consider diffusion of estrogen through tissue into the bloodstream. The diffusivity D is uniform across the tissue spanning length from the skin at x = 0, to the vasculature at x =. The skin is completely impermeable to estrogen at x = 0. The vasculature completely absorbs any estrogen at x =. A subcutaneous patch implanted at x = x 0 continuously supplies a constant but infinitely concentrated source of estrogen into the tissue: f(x, t) = q 0 δ(x x 0 ) where q 0 is a constant, and δ( ) is the delta-dirac function. Initially the estrogen concentration u(x, t) is given by: q 0 D ( x 0) for 0 x x 0 u(x, 0) = g(x) = q 0 D ( x) for x 0 < x. (5 points): Write the partial differential equation governing the estrogen concentration u(x, t) in the tissue. Express initial and boundary conditions.. (5 points): Solve for a particular solution u p (x) for the estrogen concentration in the tissue at steady state.

7 3. (5 points): Write the homogeneous problem, with homogeneous partial differential equation and boundary conditions. 4. (0 points): Find the eigenmode decomposition for the general solution of the homogenous problem.

8 5. (5 points): Solve for the estrogen concentration in the tissue over time u(x, t) from the given initial conditions. Hint: KISS (Keep It Simple & Stupid). Don t get alarmed if your answer appears too simple. But do get alarmed if you re wielding through pages of integrals...