UNIVERSITY COLLEGE LONDON EXAMINATION FOR INTERNAL STUDENTS MODULE CODE : ENGF0003 MODULE NAME : Mathematical Modelling and Analysis I LEVEL: : Undergraduate DATE : 26-April-2021 EXAM START TIME : 10:00 EXAM PAPER LENGTH : 24 hours This paper is suitable for candidates who attended classes for this module in the following academic year(s): Year 2020/21 Answer papers must be submitted to the Turnitin Assignment on the ENGF0004 Moodle site by the end time of the exam. Late submissions will receive a mark of zero. TURN OVER THIS EXAM PAPER CONSISTS OF FOUR QUESTIONS ANSWER ALL FOUR QUESTIONS Model 1: Approximating a Function (25 Marks) A certain physical quantity is time-dependent and can only be measured experimentally. It is known that can be approximated by a rational function given by: () ≈ () = + + 2 Where is time in hours and , and are parameters of this function. The experiment is performed at three points in time: 1 = 0, 2 = 0.5 and 3 = 1 h such that 1 = (0) = 0.5, 2 = (0.5) = 1 and 3 = (1) = 1.75 a) Write down a system of equations that describes the relationship between the unknown parameters , and and the known experimental values 1, 2, 3, 1, 2 and 3. [3] b) Express this system of equations in matrix form. [3] c) Use analytical matrix inversion to calculate , and and find an approximation of (). You need to show the step-by-step calculations of the inverse matrix and justify them. [9] d) Assume that = (, , , ) and find the partial derivative of with respect to , , and . Use these partial derivatives to determine to which of the parameters , , or the approximation is most sensitive to at points 1 and 3. [10] Hint: The derivative of a function with respect to a variable/parameter is a measure of the sensitivity of the function to that variable/parameter. Also, the function might be more sensitive to different parameters at different times. CONTINUED Model 2: A Simple Electrical Circuit (25 Marks) The complex impedance of the linear LCR circuit shown in Figure 1 is given by: () = + − where , and are physical constants for the resistance, inductance and capacitance of the system, = √−1 is the imaginary unit and is the angular frequency at which the circuit is being excited. Figure 1. Simple LCR circuit. a) Obtain an expression for the impedance of the circuit i.e. |()|, the magnitude of (). [6] b) Express () in polar form and represent it on the Gauss plane. In order to do this, you will need to identify the real and imaginary components of () and calculate its magnitude and argument. [6] c) Use differential calculus to find and characterise (maximum, minimum, or inflection) any real-valued stationary points of |()| as a function of . Assume that , and are strictly positive real numbers. [10] Hint: Look out for terms that will vanish at the critical point when considering the second derivative. For ease of notation, represent |()| as a function (). CONTINUED d) The resonance frequency of this system is given by = 1 √ . Using exclusively your mathematics knowledge, discuss the meaning of resonance in this problem. [3] NB: In this question we will not accept any disciplinary knowledge about the physics of electrical circuits. All statements in your answers should be supported by mathematical modelling or information provided in the question. CONTINUED Model 3: Hydrodynamic Oscillation (30 Marks) A barrel floats in still water that is at a temperature of 8 ˚C, as illustrated in Figure 2. The immersed depth of the barrel is represented by [m] and its circular faces both have area [m2]. The barrel is partially full and most of its total mass [kg] is concentrated near the bottom, so that it floats upright and does not flip over. Figure 2. Schematic of an oscillating floating barrel. The equation of motion for this barrel is given by 2 2 = − , where = 9.81 [m s-2] is the gravitational acceleration and [kg m-3] is the density of water at 8˚C. A high-speed camera was used to film the motion of the barrel floating in the water. This allowed for numerical data on the barrel’s position () and acceleration 2 2 () to be extracted from the images at different times. This is summarised in Table 1. Table 1. Measurements of the barrel’s position [m] and acceleration [m s-2] in time [s]. () 2 2 () 0 0.198882 -6.1606 0.4 0.056226 4.661359 0.8 0.17896 -4.34659 1.2 0.086789 2.572237 1.6 0.12908 -0.96643 2 0.139227 -1.3718 CONTINUED a) Use the equation of motion of the barrel and the sample data displayed in Table 1 to show that ≈ 120 [kg m-2]. [9] In order to appropriately reconstruct the oscillations of the barrel as a function of time, the camera needs to capture at least 50 images per full oscillation of the barrel. b) Show that the angular frequency of oscillation of the barrel is given by √ and suggest the ideal number of frames per second at which this barrel should be filmed. [9] Hint: Assume a solution of the form , where is a constant (real or complex). The complimentary function of the differential equation of motion of this barrel will then be a sum of complex exponentials. Analyse the complimentary function and look out for any mathematics that describes oscillatory phenomena. c) If the barrel has an initial velocity of 0 = 0 [m s -1], find the solution to the equation of motion of the barrel. [10] d) Plot the position of the barrel over three periods of oscillation. Discuss the stability of these oscillations. Propose a hypothesis (mathematical or physical) as to why this is the case. [2] CONTINUED Model 4: Pharmacokinetics (20 Marks) Reference: El-Kareh AW, Secomb TW. A mathematical model for comparison of bolus injection, continuous infusion, and liposomal delivery of doxorubicin to tumor cells. Neoplasia. 2000 Jul-Aug;2(4):325-38. doi: 10.1038/sj.neo.7900096. PMID: 11005567; PMCID: PMC1550297. Doxorubicin is an anti-cancer drug that slows or stops the development of tumours by blocking a particular enzyme that is required by cancer cells in order for them to divide and grow. Doxorubicin is administered into the bloodstream, which then transports the drug to all cells in the body which are in the vicinity of any blood vessels. Doxorubicin flows out of the vasculature into the extracellular space, where it can then be absorbed by the cells and perform its therapeutic effect. Figure 3. Schematics of drug transport to tumours. A simple model of doxorubicin concentration as a function of time is represented in terms of the three main compartments where it is present: the vascular blood, the extracellular space and the inside of the tumour cells. With reference to Figure 3, • () is the concentration of doxorubicin in vascular blood [g/ml] • () is the concentration of doxorubicin in the extracellular space [g/ml] • () is the concentration of doxorubicin inside tumour cells [ng/10 5 cells] CONTINUED Part 1: The dynamics of drug uptake and exchange between these three compartments is given by a system of ordinary differential equations: = [ () () + − () () + ] = [() − ()] − [ () () + − () () + ] In this system of equations , and are real-valued and strictly positive, [min] represents time, [ng/105 cells/min] is a kinetic constant for transport across the cell membrane, [g/ml] and [ng/10 5 cells] are Michaelis constants for cellular transmembrane transport, = 0.4 [non-dimensional] is the volume fraction of the extracellular space, [min-1] is a measure of the permeability of vascular walls and [108 cells/ml] is the tumour cell density. All parameters in this system are strictly positive. a) Check the dimensional homogeneity of this system of differential equations. [5] CONTINUED Figure 4 shows the numerical solution of this system via the Euler method for an initial administration of 100 mg of doxorubicin through injection with initial conditions ( = 0) = 0 [ng/10 5 cells] and ( = 0) = 0 [g/ml]. Figure 4. Concentrations of doxorubicin after bolus injection of 100 mg. The red curve shows vascular concentrations, the blue curve shows extracellular concentrations, and the black curve shows intracellular concentrations. The red circle over the black curve shows the peak intracellular concentration of doxorubicin. b) Prove mathematically that the intracellular concentration of doxorubicin reaches a maximum when the vascular concentration becomes smaller than the extracellular concentration of doxorubicin. Support your results by carefully analysing the numerical solution of this system of ODEs plotted in Figure 4. [5] Hint: You will need to demonstrate that () () + − () () + = 0 is a condition that must be met. You will also need to show that, at the critical point, reduces to: = [() − ()]. CONTINUED Part 2: The administration of doxorubicin can be done via injection, where the total amount of drug is administered all at once, or continuous infusion, where the drug is administered over a period of time . For doxorubicin infusion over a total time , the vascular concentration at ≤ is ( ≤ ) = (1 − −), and after the infusion is stopped ( > ) the vascular concentration is described by ( > ) = ( − 1)− . For injection, the vascular concentration of doxorubicin is given all at once, and decays exponentially as given by () = − . In these equations, [min] is time, [mg] is the amount of doxorubicin administered, [litre- 1] is the inverse volume of drug distribution in blood plasma, [min-1] is a measure of the half-life of doxorubicin in blood plasma and [min] is the infusion time. The area under the curve (AUC) of each of these concentration functions is a measure of the therapeutic dose given to the patient. The AUC helps us understand whether one mode of administration exposes the patient to a higher therapeutic dose of doxorubicin than the other. c) Prove that for a same amount of doxorubicin, the AUC for both modes of administration (infusion or injection) is . In other words, show that [10] ∫ () ∞ 0 = ∫ () ∞ 0 = Hint: Remember that () has two definitions within the integration interval. Also, to evaluate an antiderivative at +∞, you can use limits or alternatively replace +∞ by a very large number in MATLAB and check the results. NB: We will not accept any disciplinary knowledge about the pharmacokinetics of chemotherapeutics. All statements in your answers should be supported by mathematical modelling or information provided in the question. END OF PAPER
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