Advanced Modelling Case Studies Case Study 2 Assignment • This assignment has 4 questions worth a total of 50 marks. • You answers must be written up in a single word (or similar) document with the full answers to the questions. Any supporting material - for example matlab files - can also be submitted and any code used given in the appendix of the document. • Answers to the questions and any relevant Matlab (or other) code should be submitted via the www.maths.shu.ac.uk server on Friday 14th July 2017 by midnight. • Your attention is drawn to the University’s regulations on plagiarism. Answers to this assignment must be individual work; any evidence suggesting the possibility of collusion will be investigated. • You should answer all questions. • Any questions should be directed to Angharad Ugonna. Room: Norfolk 607; e-mail: [email protected]. Part 1 1.1. Consider an enzymatic reaction in which an enzyme can be activated or inactivated [8] by a chemical X as follows: E + X k1 −* )− k −1 E1; E1 + X k2 −* )− k −2 E2; E1 + S −! k3 P + Q + E: (1) Suppose further that X is supplied at a constant rate, and removed at a rate proportional to its concentration. (a) Write down differential equations for the evolution of E; E1; E2; X and S. (b) Write down a set of meaningful initial conditions. Justify your answer. (c) Which elements of this system are conserved? Explain your reasoning. 1.2. We are now going to consider the dynamics of this reaction system. Using Matlab [12] (or otherwise), solve the system of equations. Let all rates of reaction and non-zero initial conditions be 1. 1AMCS Case Study 2 2016/2017 Figure 1: Plot of enzyme dynamics for given equation 1 (a) Plot E; E1; E2; X; S and P. Ensure that the plots are fully labelled and that they are presented in such a way as to meaningfully demonstrate the dynamics of the system. (b) Investigate the steady state of the system. What does this tell us about the dynamics? Relate your answer to the rates of reaction and initial conditions. (c) Explain what happens as we increase the initial reactant and enzyme concentrations. Why do we see these particular dynamics? (d) Under what conditions will we see an increase in product formation? Under what conditions will we see more E2 produced than P? (e) Look at figure 1. Estimate the parameter values for the plot given. Explain your reasoning. Part 2 You have been introduced to the concept of physiologically based pharmacokinetic (PBPK) modelling. In this part of the assignment you will be asked to investigate the pros and cons of such models, to build a simple model of your own and to critique a model for yourself. 2.1. One of the earlier PBPK models was developed by Bischoff et al. in their [12] 1971 paper. In this paper a multi compartment representation to simulate the movement of an immunosuppressive drug through the body is given. The full Page 2 of 4AMCS Case Study 2 2016/2017 list of equations are: VP dCP dt = Mg(t) + Q RL LCL + Q RK K CK + Q RM M CM − (QL + QK + QM)CP VM dCM dt = QM CP − R CM M  VK dCK dt = QK CP − R CK K  − kK R CK K VL dCL dt = (QL − QG) CP − R CL L + QG  R CG G − R CL L − r τ dr1 dt = r − r1 τ dr2 dt = r1 − r2 τ dr3 dt = r2 − r3 VG dCG dt = QG CP − R CG G + 1 4 P4 i=1( Kk G G+ CC i 1 + bCi) dCGL dt = 14 P4 i=1 dC dti VGL 4 dC1 dt = r3 − kFVGLC1 − 1 4  K k G G+ CC 1 1 + bC1 VGL 4 dCi dt = kFVGL(Ci−1 − Ci) − 1 4  K k G G+ CC i i + bCi The full nomenclature and diagram are given in the paper Bischoff et al.. How do we obtain the value for r where r = kL(CL=RL) KL+(CL=RL)? What do we mean by saturation level? (b) Run the model (Bischoff Matlab Model). Modify the dose applied between 3mcg/kg and 300mcg/kg. How does this effect the maximum values of CM, CK, CG and CGL. Ensure that any plots given are clearly labelled. (c) The parameters given are for mice. Modify these parameters for humans. What are the key differences that we observe? Give evidence of these observations and state the parameters used. (d) Given your observations of this model what do you see as the main advantages and disadvantages of this model? How do you believe that it could be improved? 2.2. We are now going to build our own PBPK model for exposure to chloroform. Think [18] about how the compound enters the body, how it is metabolised and what parameter values are available. (a) i. Give a schematic outlining your PBPK model. This schematic should clearly indicate absorption, distribution, metabolism and excretion. ii. Discuss how you decided which compartments to use when building your PBPK model. In particular, state which compartments are lumped together and how this can be justified. (b) i. Write down a set of possible equations to describe the interactions between these compartments. Explain your choices - stating clearly what assumptions you have made. ii. How do you go about finding parameters for these equations? Give examples of potential parameters and cite your sources. Page 3 of 4AMCS Case Study 2 2016/2017 (c) Solve your equations using Matlab, or otherwise. Where you do not have specific parameter values use guestimates with full reasoning. What do your results tell you about the metabolism of ethanol in the body? Use plots to help answer this. (d) Discuss the strengths and weaknesses of your model? How could we improve on this model. Give specific examples with details of how this can be achieved. Page 4 of 4