Midterm Exam

Please upload your answers by 8:30 am March 17 (Tue). You can either type or scan the solutions to the problems.

For problems which require data plotting, you can use ‘PrtSc’ to copy and paste the graphs. Alternatively you can submit the R output. I strongly encourage you to type your answers, as this will be a good practice on how to use a LaTex. 

The lowest scored problem won't be counted in the grade.

If you need a clarification on the problems, please drop me an email.

Problem 1. A child with acute lymphocytic leukemia (ALL) has approximately 10¹² leukemic cells when the disease is clinically apparent.

(a) If a cell is about 8 μm in diameter, estimate the total mass of leukemic cells.

(b) Cure requires killing every single cell. The doubling time for the cells is about 5 days. If all cells were killed except for one, how long would it take for the disease to become apparent again?

(c) Suppose that chemotherapy reduces the number of cells to 10⁹ and there are no changes of ALL cell properties (no mutations). How long a remission would you expect? What if the number were reduced to 10⁶?

Problem 2. Suppose that the experimental values y(x_j ) are exactly equal to the calculated values plus random noise for each data point: y(x_j ) = ycalc(x_j ) + n_j . What is Q?

Problem 3. This problem is designed to show you what happens when the number of parameters exceeds the number of data points. Suppose that you have two data points:

x y

0 1

1 4

Find the best fits for one parameter (the mean) and two parameters (y = ax + b). Then try to fit the data with three parameters (a quadratic). What happens when you try to solve the equations?

Problem 4. Consider a tissue with a specific heat of 3.6 J kg⁻¹ K⁻¹, a density of 1000 kg m⁻³, and a thermal conductivity of 0.5 Wm⁻¹ K⁻¹. Assume the specific heat of blood is the same, and that the tissue perfusion is 4.17 × 10⁻⁶ m³ kg⁻¹ s⁻¹. Find the thermal diffusivity, the time for the heat to flow 1 cm, and the thermal penetration depth.

Problem 5. Calculate the value of M²_x+ M²_y+ M²_z for relaxation Eqs. 18.16 when T₁ = T₂.

Problem 6. What does LAA and HAA abbreviations mean? How are LAA and HAA defined? Why do we need to measure these values? Please use the Mid_Term_Problem6_Paper.pdf (Files folder) for the answer.

Problem 7. Please use R to compute the auto-correlation and correlation between the times series. Why is it important to find whether two voxels of the time series are correlated or autocorrelated? 

# Download the 4D fMRI hyper-volume (3D space * 1D time)

library(brainR)  # for NIFTI file inpurt

fMRIURL <- "http://socr.umich.edu/HTML5/BrainViewer/data/fMRI_FilteredData_4D.nii.gz"

fMRIFile <- file.path(tempdir(), "fMRI_FilteredData_4D.nii.gz")

download.file(fMRIURL, dest=fMRIFile, quiet=TRUE)

(fMRIVolume <- readNIfTI(fMRIFile, reorient=FALSE))

# Extract the 1D time course of a specific 3D voxel (say the one at x=30, y=30, z=10):

plot(fMRIVolume[30, 30, 10,], type='l', main="Time Series of 3D Voxel \n (x=30, y=30, z=10)", col="blue")

# Smooth the noisy fMRI time-series

x1 <- c(1:180)

y1 <- loess(fMRIVolume[30, 30, 10,]~ x1, family = "gaussian")

lines(x1, smooth(fMRIVolume[30, 30, 10,]), col = "red", lwd = 2)

lines(ksmooth(x1, fMRIVolume[30, 30, 10,], kernel = "normal", bandwidth = 5), col = "green", lwd = 3)

legend("bottomright", legend=c("(raw) fMRI", "smooth(fMRI)", "ksmooth(fMRI"),

       col=c("blue", "red", "green"), lty=1, cex=0.8,

       y.intersp=0.8)

# Define a pair of fMRI timeseries at 2 separate voxel locations

y_30_30_10 <- fMRIVolume[30, 30, 10,]

y_20_30_15 <- fMRIVolume[20, 30, 15,]

# Examine the top-6 of the pair of intensities 

head(cbind(y_30_30_10, y_20_30_15))

# Plot y_30_30_10 (x-axis) against y_20_30_15 (y-axis)

plot(y_30_30_10, y_20_30_15)

# Compute their correlation

cor(y_30_30_10, y_20_30_15)

# Compute the Auto-correlation Function for the first time-series (y_30_30_10)

acf(y_30_30_10, lag.max = 20, plot = T)

# fit as linear model

lm1 <- lm(y_30_30_10 ~ y_20_30_15); summary(lm1)