Solving a non-linear partial differential equation for the simulation of tumour oxygenation
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1 Solving a non-linear partial differential equation for the simulation of tumour oxygenation Julian Köllermeier, Lisa Kusch, Thorsten Lajewski MathCCES, RWTH Aachen Talk at Karolinska Institute, J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
2 About Us CES- Computational Engineering Science mathematics computer science mechanical engineering project thesis mandatory in 6th semester J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
3 Task Theoretical modelling based on Dasu et al. Theoretical simulation of tumour oxygenation and results from acute and chronic hypoxia, PMB 48, 2003 design specialised software focusing on reduced memory consumption possible inclusion of more detailed models extension to third dimension J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
4 Outline 1 Introduction 2 Theoretical Modelling Tissue Generation Mathematical Modelling Discretisation Numerics 3 Software Presentation 4 Results Validation Acute and Chronic Hypoxia Memory Requirements Application Example 5 Conclusion J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
5 Introduction Introduction J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
6 Introduction Tumour Oxygenation tumour micro-structure non-regular vascular structure poor blood supply hypoxia chronic hypoxia (diffusion limited) acute hypoxia (perfusion limited) importance for radiotherapy radiosensitivity depends on oxygenation J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
7 Introduction Tumour Oxygenation tumour micro-structure non-regular vascular structure poor blood supply hypoxia chronic hypoxia (diffusion limited) acute hypoxia (perfusion limited) importance for radiotherapy radiosensitivity depends on oxygenation J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
8 Introduction Tumour Oxygenation tumour micro-structure non-regular vascular structure poor blood supply hypoxia chronic hypoxia (diffusion limited) acute hypoxia (perfusion limited) importance for radiotherapy radiosensitivity depends on oxygenation J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
9 Introduction Tumour Oxygenation tumour micro-structure non-regular vascular structure poor blood supply hypoxia chronic hypoxia (diffusion limited) acute hypoxia (perfusion limited) importance for radiotherapy radiosensitivity depends on oxygenation J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
10 Introduction Challenges measurement methods poor resolution small area influence oxygenation theoretical modelling commercial software not specialised high memory consumption only small domains design specialised software J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
11 Introduction Challenges measurement methods poor resolution small area influence oxygenation theoretical modelling commercial software not specialised high memory consumption only small domains design specialised software J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
12 Introduction Challenges measurement methods poor resolution small area influence oxygenation theoretical modelling commercial software not specialised high memory consumption only small domains design specialised software J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
13 Introduction Challenges measurement methods poor resolution small area influence oxygenation theoretical modelling commercial software not specialised high memory consumption only small domains design specialised software J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
14 Theoretical Modelling Theoretical Modelling J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
15 Theoretical Modelling Tissue Generation Domain Description properties two-dimensional, rectangular domain discoid blood vessels J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
16 Theoretical Modelling Tissue Generation Distribution Properties vessel distribution normal distribution of intervascular distances characterised by mean µ and variance σ 2 Example 1 scattered grid 2 dart throwing J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
17 Theoretical Modelling Tissue Generation Scattered Grid place vessels on equidistant grid for each vessel do choose normally distributed distance choose uniformly distributed angle move vessel according to distance and angle end for J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
18 Theoretical Modelling Tissue Generation Dart Throwing while vessel fits in domain do choose normally distributed distance choose random coordinates place vessel at coordinates check distance end while J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
19 Theoretical Modelling Tissue Generation Vessel Data Format few vessels easy to manipulate data structure 1 x-coordinate 2 y-coordinate 3 radius 4 pressure J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
20 Theoretical Modelling Mathematical Modelling The Problem Equation variable D p + q(p) = 0 Laplacian: p = 2 p x + 2 p 2 y 2 p Consumption: q(p) = q max p + k partial oxygen pressure p,[p] = mmhg parameters diffusion coefficient D = µm 2 s maximum consumption rate q max = 15 mmhg s consumption at half pressure k = 2.5mmHg J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
21 Theoretical Modelling Mathematical Modelling The Problem Equation variable D p + q(p) = 0 Laplacian: p = 2 p x + 2 p 2 y 2 p Consumption: q(p) = q max p + k partial oxygen pressure p,[p] = mmhg parameters diffusion coefficient D = µm 2 s maximum consumption rate q max = 15 mmhg s consumption at half pressure k = 2.5mmHg J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
22 Theoretical Modelling Mathematical Modelling The Problem Equation variable D p + q(p) = 0 Laplacian: p = 2 p x + 2 p 2 y 2 p Consumption: q(p) = q max p + k partial oxygen pressure p,[p] = mmhg parameters diffusion coefficient D = µm 2 s maximum consumption rate q max = 15 mmhg s consumption at half pressure k = 2.5mmHg J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
23 Theoretical Modelling Mathematical Modelling The Problem Equation variable D p + q(p) = 0 Laplacian: p = 2 p x + 2 p 2 y 2 p Consumption: q(p) = q max p + k partial oxygen pressure p,[p] = mmhg parameters diffusion coefficient D = µm 2 s maximum consumption rate q max = 15 mmhg s consumption at half pressure k = 2.5mmHg J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
24 Theoretical Modelling Mathematical Modelling Discretisation of Laplacian p i,j p i 1,j p i,j 1 + 4p i,j p i,j+1 p i+1,j h 2 = h p i,j commercial software central finite differences five-point stencil for Laplacian second order accuracy J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
25 Theoretical Modelling Discretisation Vessel Discretisation translate continuous into discrete information equidistant grid constant pressure inside vessels (Dirichlet boundaries) determine vessel points use efficient data structure 3 different discretisation methods J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
26 Theoretical Modelling Discretisation Vessel Discretisation translate continuous into discrete information equidistant grid constant pressure inside vessels (Dirichlet boundaries) determine vessel points use efficient data structure 3 different discretisation methods J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
27 Theoretical Modelling Discretisation Vessel Discretisation translate continuous into discrete information equidistant grid constant pressure inside vessels (Dirichlet boundaries) determine vessel points use efficient data structure 3 different discretisation methods J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
28 Theoretical Modelling Discretisation Discretisation Methods method 1 only points inside the vessel method 2 points inside and adjacent points method 3 points inside and close to the vessel boundary J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
29 Theoretical Modelling Discretisation Vessel Points Data Format requirements many vessel points efficient data format needed consecutive data access for calculations = store information rowwise data structure 1 leftmost vessel point 2 number of vessel 3 rightmost vessel point J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
30 Theoretical Modelling Discretisation Vessel Points Data Format requirements many vessel points efficient data format needed consecutive data access for calculations = store information rowwise data structure 1 leftmost vessel point 2 number of vessel 3 rightmost vessel point J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
31 Theoretical Modelling Discretisation Vessel Points Data Format memory consumption increases linearly with resolution while number of vessel points increases quadratically J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
32 Theoretical Modelling Discretisation Vessel Points Data Format memory consumption increases linearly with resolution while number of vessel points increases quadratically J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
33 Theoretical Modelling Discretisation Structure of the matrix p i,j p i 1,j p i,j 1 + 4p i,j p i,j+1 p i+1,j h 2 = h p i,j results in a sparse symmetric positive definite matrix typical Laplacian matrix except for vessel boundary points J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
34 Theoretical Modelling Numerics Non-linear System it is q(p) = 0 for all vessel points f(p) := D Ap + q(p) D b = 0. = use Newton s algorithm (quadratic order of convergence) application of Newton s algorithm leads to a linear system of equations f (p k )y k+1 = f(p k ) with f (p) = D A + q (p) for each iteration k J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
35 Theoretical Modelling Numerics Non-linear System it is q(p) = 0 for all vessel points f(p) := D Ap + q(p) D b = 0. = use Newton s algorithm (quadratic order of convergence) application of Newton s algorithm leads to a linear system of equations f (p k )y k+1 = f(p k ) with f (p) = D A + q (p) for each iteration k J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
36 Theoretical Modelling Numerics Linear System requirements exploit matrix structure direct evaluation of matrix operations = use iterative methods like Conjugate Gradient or Conjugate Residual implementation details solvers exchangeable J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
37 Theoretical Modelling Numerics Linear System requirements exploit matrix structure direct evaluation of matrix operations = use iterative methods like Conjugate Gradient or Conjugate Residual implementation details solvers exchangeable J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
38 Software Presentation Software Presentation... and now a short example... J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
39 Results Results J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
40 Results Validation Exact Solution of Simplified Problem assumptions no consumption (linear problem) only one vessel in the center of quadratic domain = compare different discretisation methods J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
41 Results Validation Comparison Between Discretisation Methods results error occurs near vessel boundary method 3 gives best results J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
42 Results Validation Comparison Between Discretisation Methods results error occurs near vessel boundary method 3 gives best results J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
43 Results Validation Comparison Between Discretisation Methods (2) results method 3 gives best results linear order of convergence J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
44 Results Validation Comparison Between Discretisation Methods (2) results method 3 gives best results linear order of convergence J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
45 Results Acute and Chronic Hypoxia Modelling Chronic Hypoxia D = 2000 mmhg D = 1000 mmhg J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
46 Results Acute and Chronic Hypoxia Modelling Acute Hypoxia 25 % deleted 40 % deleted J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
47 Results Acute and Chronic Hypoxia Modelling Acute Hypoxia (2) smaller diameter smaller partial pressure J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
48 Results Memory Requirements Memory Measurements results discretisation memory increases linearly solver memory increases quadratically = almost no dependency on number of vessels J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
49 Results Application Example Large Domain Simulation J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
50 Conclusion Conclusion J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
51 Conclusion achievements new software developed resolved memory consumption problem simulation of real tumours possible outlook extension to third dimension parallelisation of code implementation of other numerical methods J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
52 Conclusion achievements new software developed resolved memory consumption problem simulation of real tumours possible outlook extension to third dimension parallelisation of code implementation of other numerical methods J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
53 Conclusion Thank you for your attention! J. Köllermeier, L. Kusch, T. Lajewski (RWTH) Virtual Tumour / 35
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