publication . Master thesis . 2016

Estimation of Metabolic Oxygen Consumption From Optical Measurements in Cortex

Sætra, Marte Julie;
Open Access English
  • Published: 01 Jan 2016
Abstract
We seek a standardized method for estimating cerebral metabolic rate of oxygen (CMRO2) from optical measurements of partial pressure of oxygen (pO2). This parameter is critical for understanding how the brain responds to changes in metabolism and oxygen delivery. Such changes are associated with clinical conditions like stroke and Alzheimer’s disease. The oxygen consumption rate is further important for the interpretation of functional magnetic resonance imaging. We approach two different methods for estimating CMRO2: The Krogh method and the Laplace method. They are based on Fick's law of diffusion and carried out using different assumptions. When oxygenated bl...

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.

Abstract
We seek a standardized method for estimating cerebral metabolic rate of oxygen (CMRO2) from optical measurements of partial pressure of oxygen (pO2). This parameter is critical for understanding how the brain responds to changes in metabolism and oxygen delivery. Such changes are associated with clinical conditions like stroke and Alzheimer’s disease. The oxygen consumption rate is further important for the interpretation of functional magnetic resonance imaging. We approach two different methods for estimating CMRO2: The Krogh method and the Laplace method. They are based on Fick's law of diffusion and carried out using different assumptions. When oxygenated bl...

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.

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