Estimation of Metabolic Oxygen Consumption From Optical Measurements in Cortex

1 Introduction 3 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Goal and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 My Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Background 7 2.1 Experimental Measurement of Oxygen in Brain Tissue . . . . . . 7 2.1.1 A Short Introduction to Two-Photon Imaging . . . . . . . 7 2.1.2 Interpreting Data . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.3 Extracting Data . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Physics of Oxygen Dynamics . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Fick's Laws of Di usion . . . . . . . . . . . . . . . . . . . 10 2.3 Estimation of the Oxygen Consumption Rate . . . . . . . . . . . 12 2.3.1 Estimation using the Krogh Method . . . . . . . . . . . . 12 2.3.2 Estimation using the Laplace Method . . . . . . . . . . . . 15 2.3.3 Estimating CMRO2 From the Parameter M . . . . . . . . 15

3 Mathematical Methods 19 3.1 Optimization Theory . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.1 Starting Point . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.2 The Chi Square Function . . . . . . . . . . . . . . . . . . . 20 3.1.3 Solution by use of Singular Value Decomposition . . . . . 20 3.1.4 Goodness-of- t Measure . . . . . . . . . . . . . . . . . . . 22 3.1.5 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Smoothing in Laplace Method Estimation . . . . . . . . . . . . . 23 4.1.3 Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.4 Evaluate the Physical Parameters . . . . . . . . . . . . . . 30 4.2 Testing the Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Results for the Krogh Method 33 5.1 Three Fitting Parameters . . . . . . . . . . . . . . . . . . . . . . 33 5.1.1 Model Form . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1.2 Evaluation of the Parameters . . . . . . . . . . . . . . . . 34 5.1.3 Evaluation of the Goodness-of- t and Error Estimates . . 34 5.2 Two Fitting Parameters . . . . . . . . . . . . . . . . . . . . . . . 37 5.2.1 Model Form . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2.2 Evaluation of the Parameters . . . . . . . . . . . . . . . . 38 5.2.3 In uence of Rt on the Relationship Between Pves and M . 39 5.2.4 In uence of Rves on the Relationship Between Pves and M 39

6 Exploration of the Laplace Method 43 6.1 Building a Data set . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.1.1 Spatial Grid and pO2 Values . . . . . . . . . . . . . . . . . 44 6.1.2 Choosing a Noise Function . . . . . . . . . . . . . . . . . . 44 6.1.3 Adding Noise . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.2 Exploration of the Method . . . . . . . . . . . . . . . . . . . . . . 49 6.2.1 Estimation of M From a Dataset Without Noise . . . . . . 49 6.2.2 Estimation of M from a Noisy Dataset . . . . . . . . . . . 52

7 Results for the Laplace Method 55 7.1 A Study of Datasets 3, 4 and 5 . . . . . . . . . . . . . . . . . . . 55 7.1.1 Overview of Dataset 5 . . . . . . . . . . . . . . . . . . . . 56 7.1.2 Parting the Grid . . . . . . . . . . . . . . . . . . . . . . . 56 7.1.3 Estimation of M for Di erent Regions Around the Vessel . 59 7.2 A Study of Datasets 8, 9 and 10 . . . . . . . . . . . . . . . . . . . 64 7.2.1 Parting the Grid . . . . . . . . . . . . . . . . . . . . . . . 65 7.2.2 Estimation of M for Di erent Regions Around the Vessel . 65

8 Discussion, Conclusion and Future Work 73 8.1 Summary and Discussion of the Krogh Method . . . . . . . . . . 73 8.2 Summary and Discussion of the Laplace Method . . . . . . . . . . 74 8.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 50 30 20 20

[16] R Sander. \Compilation of Henry's law Constants (Version 4.0) for Water as Solvent". In: Atmospheric Chemistry and Physics 15 (2015), p. 4399.

[17] Daniel Goldman. \Theoretical Models of Microvascular Oxygen Transport to Tissue". In: Microcirculation 15 (2008), p. 795.

[18] Doug Hull. MakeColorMap. May 2016. url: http://www.mathworks.com/ matlabcentral/ leexchange/17552-makecolormap.