Does Normal Pressure Hydrocephalus Have Mechanistic Causes?

Transcription

1 Does Normal Pressure Hydrocephalus Have Mechanistic Causes? Sub-Lieutenant Tonmoy Dutta-Roy, RAN B.Eng. (Mechanical), M.Eng. (Thesis) This thesis is presented for the Degree of Doctor of Philosophy of The University of Western Australia Intelligent Systems for Medicine Laboratory School of Mechanical Engineering Faculty of Engineering, Computing and Mathematics 2011

2 For my parents: Brigadier Prithwi Raj Dutta-Roy, ex. of the Indian Army and Lieutenant Colonel (Dr.) Nina Dutta-Roy, ex. of the Indian Army for encouraging, supporting and allowing me to indulge in my interests

3 ACKNOWLEDGEMENTS I must express my sense of gratitude for the following: Prof. Karol Miller, primary advisor for my thesis, for his research supervision, direction and encouragement when I was feeling frustrated with the research outcomes. Karol understands that a PhD is a development process of a philosophy for life in general and research in particular and not just about publishing papers, for which I am thankful. Also, he appreciates the importance of having a life away from PhD. He encouraged me to pursue my interests in Military History, Defence and Strategy by supporting my application for internship at the Australian Strategic Policy Institute. I have had some very wonderful conversations on World War II with him during my candidature. Dr. Adam Wittek, my co-advisor. Adam has an encyclopedic knowledge of Finite Element Analysis (FEA) applications especially in LS-DYNA. Because of him, I could make a smooth transition from vibration engineering into non linear FEA. He patiently sat through many hours of discussions with me, explaining, reasoning and correcting my ideas as well as pointing me to the right direction. Also, he has a great sense of humour which keeps the research group amused. Ashley Horton, my fellow PhD student, for many hours of discussion on his belief in religion and the reasoning behind it, under constant questioning. Grand Joldes, for inviting me to visit him in Romania. Brent Fillery, for sharing his knowledge in

4 ABAQUS and providing the right advice in times of need. Finally, Irving Aye, Nitin Repalle, Mahdi Memarpour, Ashkan Rafiee, Vinay Domal, Santiram and Divya Chatterjee and colleagues in the postgraduate room for their friendship and many hours spent eating at various places in Perth. IT support, especially Angus Stewart and the help of the administrative staff at Mechanical Engineering is much appreciated. indulgence. And most importantly, my grandparents and parents for their patient support and Financial support of the William and Marlene Schrader Trust (William and Marlene Schrader Post-Graduate Scholarship), Graduate School Travel Award and UWA Convocation travel award is gratefully acknowledged.

5 List of Figures Figure 2.1 Cerebrospinal Fluid (CSF) circulation pathways 8 Figure 3.1: Brain geometry, pressure loading and applied boundary conditions 26 for the brain Figure 3.2: Transmantle pressure difference v/s ventricular volume for single 44 phase and biphasic brain model Figure 3.3: Void ratio distribution for biphasic model 46 [loading pressure: 1 mm of Hg ( Pa)] Figure 3.4: Total fluid volume ratio distribution for biphasic model 47 [loading pressure: 1 mm of Hg ( Pa)] Figure 4.1: Brain surface after removal of Pia 52 Figure 4.2: Assembled experimental setup 53 Figure 4.3a: Top view of the cylindrical die 54 Figure 4.3b: Bottom view of the cylindrical die 55 Figure 4.4: Cylindrical sample inserted into the die 56 Figure 5.1: Methodology used for implementing time varying shear modulus 68 Figure 5.2: Sinusoidal time varying shear modulus 71 Figure A.1 Swine brain placed in a saline solution filled container 106 Figure A.2a Needle insertion: experimental set-up 107 i

6 Figure A.2b Needle tip geometry 107 Figure A.3 Typical needle insertion force time history recorded in this study 108 Figure A.4a General view of the swine brain mesh 111 Figure A.4b Refined mesh graft in the needle insertion area 111 Figure A.5 Loading through prescribed nodal velocity (5 mm/sec) 116 Figure A.6 Boundary conditions for the brain model when modelling the needle 117 insertion Figure A.7a Computed Force displacement relationship validated against 120 needle insertion experiment [Swine Brain Number 4 (7 insertions)] Figure A.7b Computed Force displacement relationship validated 120 against needle insertion experiments [3 Swine Brains (18 insertions)] Figure A.8 Needle insertion simulation: model cross-section at 8 mm insertion depth 121 ii

7 List of Tables Table 3.1: Brain and ventricular volumes for normal and NPH affected brain 25 Table 3.2: Material constants for the constitutive model of the brain tissue 27 Table 3.3: Material properties for incompressible, nearly incompressible 31 and compressible brain Table 3.4: Loading applied to single and biphasic model of brain parenchyma 32 Table 3.5: Ventricular volume when transmantle pressure difference (P trans1 ) 38 equivalent to 1 mm of Hg ( Pa) is applied to single phase and bi-phase brain parenchyma. Table 3.6: Ventricular volume produced when transmantle pressure difference 39 (Load Case 2 (P trans2 ) = Pa= 1.69 mm of Hg) is applied to incompressible/nearly incompressible single and biphasic model Table 3.7: Transmantle pressure difference (Load Case 3-P trans3 ) required to 40 produce NPH in single phase brain parenchyma model Table 4.1: Height of artificial CSF column applied on the brain sample 57 Table 4.2: Total volume flow of CSF solution through the cylindrical brain 60 parenchyma sample Table 5.1: Time varying shear modulus function and computed ventricular 70 volumes under 1 mm of Hg transmantle pressure difference Table 5.2: Computed ventricular volumes for time varying shear modulus 73 iii

8 Table A.1 Weight of Swine brain used for needle insertion experiments 105 Table A.2: Material constants for the constitutive model of the swine 114 brain tissue iv

9 ABSTRACT This thesis studies the biomechanics of Normal Pressure Hydrocephalus (NPH) growth using fully non-linear Finite Element procedures. A generic 3-D brain mesh of a healthy human brain was created. The brain parenchyma was modelled as single phase and biphasic continuum. In the computational model, hyperelastic constitutive law and finite deformation theory described deformations within the brain parenchyma. A value of Pa for the shear modulus (μ ) of the brain parenchyma was used. Additionally, contact boundary definitions constrained the brain outer surface inside the skull. A transmantle pressure difference was used to load the model. Fully non-linear, implicit finite element procedures in the time domain were used to obtain the deformations of the ventricles and the brain. Clinicians generally accept that at most 1 mm of Hg transmantle pressure difference ( Pa) is associated with the condition of NPH. The computations showed that transmantle pressure difference of 1 mm of Hg ( Pa) did not produce NPH for either single phase or biphasic model of the brain parenchyma. A minimum transmantle pressure difference of mm of Hg ( Pa) was required to produce the clinical condition of NPH. This suggested that the hypothesis of a purely mechanical basis for NPH growth needs to be revised. The computational results also showed that under equal transmantle pressure difference load, there were no significant differences between the computed ventricular volumes for biphasic and incompressible/nearly incompressible single phase model of the brain parenchyma. As a result, there was no advantage gained by using a biphasic model for the brain parenchyma. It is proposed that for modelling NPH, nearly incompressible single v

10 phase model of the brain parenchyma was adequate. Single phase treatment of the brain parenchyma simplified the mathematical description of the NPH model and resulted in significant reduction of computational time. Further, simple experiments using samples of lamb brain tissue were conducted to verify as to what extent the brain parenchyma responses are consistent with the models using biphasic (soil consolidation) continuum theory. The lamb brain (byproduct of a commercial slaughter process) was obtained from the butcher. The age of the animals at slaughter was approximately 6 months. The Pia mater was carefully teased out from the Sulci features and torn from the brain surface. An approximately cylindrical sample (diameter ~ 30mm and height ~ 20mm) was cut out of the region of the brain from which Pia mater was removed, using a sharp cylindrical punch and scalpel. The cylindrical brain sample was inserted into a die and three separate heights of Cerebrospinal Fluid (CSF) solution column were applied to it. Each height of the CSF solution column was applied for a period of 120 mins. A check was made after 120 minutes to observe any leakage of CSF solution through the brain tissue. No CSF solution leakage through the brain tissue was observed. The results indicated that the brain parenchyma behaviour is incompatible with biphasic (soil consolidation) continuum theory. Therefore, if there is a need to model fluid flow within the brain parenchyma, for applications such as drug delivery or nutrient transfer, using principles of transport phenomena should be considered. Subsequently, the idea that the pulsations observed in Intracranial Pressure (ICP) recordings of Normal Pressure Hydrocephalus (NPH) patients can explain the mechanics of NPH growth was investigated. A generic 3-D brain mesh of a healthy human brain was vi

11 created. The brain parenchyma was modelled as single phase. Hyperelastic constitutive law and finite deformation theory described deformations within the brain parenchyma. The relaxed shear modulus (μ ) was varied according to the normalised cardiac cycle so that the brain parenchyma was stiffer during systolic part of the cardiac cycle when compared to the diastolic part. The results of the simulations showed that 1 mm of Hg ( Pa) did not produce the clinical condition on NPH, even after including ICP pulsatile effects. It is concluded from the results that it is highly unlikely for a phenomenon (such as pulsations in cerebral blood flow), which take place many times a second to be able to influence an occurrence such as NPH which presents itself over a time scale of hours to days. The main conclusion of this thesis is that the hypothesis of a purely mechanical basis for NPH growth needs to be revised. vii

12 CHAPTER 1 INTRODUCTION 1.1 Background The clinical condition of Normal Pressure Hydrocephalus (NPH) has been known for almost five decades now, yet it remains an unsolved problem for diagnosticians. Difficulties are encountered in effectively recognising, differentiating, diagnosing and treating the condition. Clinicians have only in the recent past been able to agree on a common classification system for NPH. Hakim (1965) and Adams et al. (1965) described NPH as a condition of ventricular enlargement with normal CSF pressure (< 18 mm of Hg) accompanied with a triad of symptoms, namely, ataxia of gait, urinary incontinence and dementia. However, not all cases of NPH show the three classical symptoms given above; the most common symptoms are ataxia of gait and dementia. Therefore even a commonly accepted clinical definition of NPH is difficult to reach and contentious. The triad of clinical symptoms and outcomes of diagnostic tests (MRI and CT scans) are quite similar to other medical conditions (e.g. Alzheimer s disease) and could be misrecognised as such. Hence, diagnosis of NPH remains one of the most challenging problems in clinical practice. Though a lot of work done in this field is from a clinical point of view (identification, diagnosis and treatment of NPH), the reasons behind the growth of NPH is not well understood.

13 Chapter 1 Introduction It was postulated by successive generations of researchers that NPH has mechanistic causes (Hakim et al., 1976; Hakim, 1977; Nagashima et al., 1987; Tada et al., 1990; Kaczmarek et al., 1997; Stastna et al., 1999; Stastna et al., 1999; Tenti et al., 1999; Pena et al., 1999; Pena et al., 2002, Taylor and Miller, 2004; Drapaca, 2005; Sivaloganathan, 2005; Sivaloganathan, 2005; Smillie et al, 2005; Drapaca, 2006; Sobey and Wirth, 2006; Wirth and Sobey, 2006 and Momjian and Bichsel, 2008), yet they cannot reconcile their findings with clinical observations. The major drawbacks of these works are the use of small strain theory (Smillie et al, 2005; Sobey and Wirth, 2006 and Wirth and Sobey, 2006) and elastic constitutive law (Kaczmarek et al., 1997; Taylor and Miller, 2004; Smillie et al, 2005; Sobey and Wirth, 2006 and Wirth and Sobey, 2006) for calculating the deformations of the brain parenchyma as well as simplified brain geometry (Hakim et al., 1976; Hakim, 1977; Kaczmarek et al., 1997; Smillie et al, 2005; Sobey and Wirth, 2006 and Wirth and Sobey, 2006 etc.). Small strain theory is incapable of capturing the geometric changes and elastic constitutive law is unable in to handle material behaviour at large deformations (Bathe, 1994) occurring during hydrocephalus. In this thesis, an attempt is made to rectify the above deficiencies by using realistic 3-D brain geometry, fully non linear (geometric, constitutive and contact) Finite Element model of NPH. To the best of the knowledge of the author, this is the first, realistic 3-D brain geometry, fully non-linear (geometric, constitutive and contact) model investigating NPH growth. 2

14 Chapter 1 Introduction 1.2 Motivation of Research A vast plethora of literature is available on the recognition, diagnosis and treatment of NPH. However, there is the need for deeper understanding of the causes for NPH growth. The present work attempts to fulfil this void by using computational techniques. The growth of NPH is investigated using a 3-D geometry of the brain and fully non-linear (geometry, material and contact) Finite Element procedures. A better understanding of NPH growth biomechanics could lead to improved protocols for diagnosis and enhanced treatment methodologies. 1.3 Thesis Outline A review of relevant literature is presented in Chapter 2. This sets up the mathematical and computational basis to model NPH presented later in the thesis. Motivation for conducting the work is also given in this chapter. A biomechanical model of NPH is presented in Chapter 3. Realistic 3-D brain geometry and fully non-linear Finite Element procedures are utilised to understand NPH growth. The suggestion that 1 mm of Hg ( ) transmantle pressure difference is associated with the clinical condition of NPH is investigated. Also, it is investigated if soil consolidation (biphasic) based models of the brain parenchyma gave any major advantages over single phase models of the brain parenchyma for modeling NPH. 3

15 Chapter 1 Introduction For modelling NPH, the brain tissue is treated as a biphasic continuum (soil consolidation theory). To verify as to what extent the parenchyma responses are consistent with the models using soil consolidation (biphasic) theory, simple experiments using samples of lamb brain tissue were conducted. The experimental procedure and results are discussed in Chapter 4. Clinical techniques used to enhance the diagnosis of NPH indicated that pulsations observed in Intracranial Pressure (ICP) recordings of NPH patients could explain the mechanics of NPH growth. This idea was investigated in Chapter 5 by varying the relaxed shear modulus (μ ) so that the brain parenchyma was stiffer during systolic part of the cardiac cycle when compared to the diastolic part. Again realistic 3-D brain geometry and fully non-linear Finite Element procedures are utilised for this purpose. Chapter 6 details the conclusions drawn from the present research work and also provides suggestions for future research into NPH. The use of computational mechanics for modelling other complicated brain tissue related phenomena such as needle insertion into brain tissues is illustrated in Appendix A. 4

16 CHAPTER 2 LITERATURE REVIEW 2.1 Modelling the Brain for Neurosurgery Computational Mechanics has proven to be a powerful and an effective tool for understanding complex physical phenomena. It finds huge application in engineering and is widely used in many disciplines (e.g. sub-sea engineering, vehicle crash simulation, drilling and reservoir modelling etc.). Recently, computational mechanics has been used to understand and model complex biological and medical phenomena (e.g. surgical simulation, development of diseases, rehabilitation, cellular and subcellular mechanics, orthodontics etc.). One of the most promising areas for use of computational mechanics is Neurosurgery (Roberts et al., 1998; Warfield et al., 2002; Bucholz, et al., 2004; Nakaji et al., 2004; Warfield et al., 2005). Neurosurgery calls for advanced surgical skills and complex after-surgery care for patients. As a result, the costs associated with neurosurgery are increasing as more neurosurgical procedures are being performed. The use of robots in conjunction with a neurosurgeon can possibly reduce costs as well as minimise the side effects of the surgery. Reduction in side effects would reduce the costs associated with neurosurgery as well as benefit the patient. Four important inputs are needed to apply computational methods for neurosurgery; subject/patient specific brain mesh, appropriate mathematical formulations for computing the deformations of the brain parenchyma, suitable material model for the brain tissue and correct boundary conditions. Finite deformation

17 Chapter 2 Literature Review formulations (Miller, 2002) and non-linear constitutive law (Miller and Chinzei, 1997, Miller and Chinzei, 2002 and Miller, 2002) are suitable for use in neurosurgery applications. The issue of automatic generation of appropriate subject/patient specific brain mesh is still unresolved (Owen, 1998; Owen, 2001; Couteau et al., 2000; Castellano-Smith et al., 2001; Viceconti et al., 2003; Luboz et al., 2005; Wittek et al., 2008 etc.) and correct boundary conditions remain partially resolved (Wittek et al., 2003; Wittek et al, 2008, Ji et al., 2009 etc.). A subject specific brain mesh, finite deformation formulations, non-linear constitutive law for the brain tissue and appropriate boundary conditions are used. are used later in the thesis (Chapters 3 and 5) to model the growth of Normal Pressure Hydrocephalus. 2.2 Normal Pressure Hydrocephalus Cerebrospinal Fluid Circulation The brain is a part of the Central Nervous System (CNS) along with the spinal cord. It is contained within the cranial cavity and the spinal cord within the spinal cavity. Both the brain and the spinal cord are surrounded by meninges. Two fluids flow through the CNS: blood and the Cerebrospinal Fluid (CSF). Both the brain and the spinal cord are bathed in CSF. The CSF covers the outer surface of the brain. It is commonly accepted that CSF is formed within the cerebral ventricles (Figure 2.1) with the majority (around 80%) being formed in the choroid plexus. After formation in the lateral 6

18 Chapter 2 Literature Review ventricles, the CSF passes through the foramina of Monro to reach the third ventricles (Figure 2.9). It then flows along the aqueduct of Sylvius into the fourth ventricles. From the fourth ventricles, CSF exits through the two lateral foramina of Luschka and the midline foramen of Magendie into the cisterna magna. Thereafter CSF flows through the subarchanoid space into the superior sagittal sinus, being absorbed by the archnoid villi Normal Pressure Hydrocephalus: Background The clinical syndrome of Hydrocephalus has been known to mankind for great many years. Vesalius (1543) was the first to give a clear description of internal hydrocephalus. He confirmed that the exact location of accumulation of fluid in hydrocephalus was in the ventricles. Prior to this observation, anatomists and pathologists were not clear about the exact location of fluid accumulation during hydrocephalus. They believed that fluid accumulated beneath the scalp. Hydrocephalus is the condition in which too much Cerebrospinal Fluid (CSF) is present in the ventricles. More precisely it is a disorder of CSF hydrodynamics. Though there are many other disorders of CSF hydrodynamics, the distinguishing feature between hydrocephalus and the other disorders of CSF hydrodynamics is the visible increase in CSF volume occurring during hydrocephalus. In a healthy individual there is a situation of normal production, flow and absorption of CSF within the CNS and equilibrium exists between production and absorption of CSF. Hydrocephalus occurs 7

19 Chapter 2 Literature Review Superior Sagittal Sinus Dura Mater Arachnoid Villi Fourth Ventricle Aqueduct of Sylvius Pons Cerebellum Superior Sagittal Sinus Dura Mater Brain Convexity Cerebellum Pia Mater Figure 2.1 Cerebrospinal Fluid (CSF) circulation pathways [adapted from Milhorat (1972) and Hakim (1985)] 8

20 Chapter 2 Literature Review when there are problems in production, flow or absorption of CSF in the nervous system leading to unbalance between production and absorption of CSF. As a result CSF is mainly accumulated in the ventricles and the ventricular space is dilated. Hydrocephalus is generally classified as non-communicating hydrocephalus or communicating hydrocephalus. In non-communicating hydrocephalus, there is obstruction to CSF flow within the ventricular system. When the obstruction to CSF flow occurs within the subarchnoid space, it is termed as communicating hydrocephalus. Normal Pressure Hydrocephalus (NPH) is of the communicating kind. It can be further divided into idiopathic and secondary NPH. Idiopathic NPH has no known etiology (causes) and mostly occurs in patients above the age of 60 years. It is accepted that conditions such as hypertension, diabetes, ischemic heart disease (Casmiro et al., 1989) and cardiac as well as vascular conditions (Krauss et al., 1996) are related to idiopathic NPH. Secondary NPH results from other underlying causes such as head injury, intracranial tumour, intracranial surgery, intracranial haemorrhage etc. The condition of Normal Pressure Hydrocephalus (NPH) was first recognised by Hakim (1965) and Adams et al. (1965). The authors described a condition of dilation in ventricular size with normal CSF pressure (< 18 mm of Hg) as recorded by performing a lumbar puncture. Diagnostic tools (MRI and CT) indicated dilated lateral and/or third and fourth ventricles. The patients also suffered from a triad of clinical symptoms: a) Ataxia of gait b) Incontinent Urine c) Dementia 9

21 Chapter 2 Literature Review The triad of clinical symptoms and outcomes of diagnostic tests (MRI and CT scans) are quite similar to other medical conditions (e.g. Alzheimer s disease) and could be misrecognised as such. This is a major cause for misdiagnosis of NPH patients. Hence, diagnosis of NPH remains one of the most challenging problems in clinical practice. The term NPH has lately run into controversy. NPH is also used to describe patients presenting the classical triad of symptoms and dilated ventricles but without the measurement of ventricular or lumbar CSF pressure. Similarly, not all cases of NPH show the three classical symptoms given above; the most common symptoms are ataxia of gait and dementia. Also other symptoms may be present (Relkin et al., 2005). Therefore a commonly accepted clinical definition of NPH is difficult to reach (Raimondi, 1994). The commonly accepted treatment of NPH is surgical placement of a ventriculoperitoneal shunt (Owler, 2004). The practice of shunting in medicine refers to a hole or a passage which allows movement of fluid from one part of the body to another. The ventriculoperitoneal shunt drains excess CSF from the ventricles to the abdomen where CSF is absorbed. Mechanically, the ventriculoperitoneal shunt is a one way valve and can be of the fixed pressure valve or the adjustable valve type. The ventriculoperitoneal shunt is placed behind the ears in the space between the skin and the skull. Some clinicians are of the opinion that NPH can only be recognised and diagnosed if the patients respond positively to shunting (Conn, 2007). 10

22 Chapter 2 Literature Review As noted above, diagnosis of NPH remains one of the most challenging problems in clinical practice. This is due to the fact that clinical symptoms and diagnostic findings of NPH can overlap with other known medical condition such as neurodegenerative disorders (Parkinson s and Alzheimer s disease), vascular dementia, infectious diseases (HIV, Syphilis) etc. To diagnose NPH, clinicians usually rely on clinical symptoms, diagnostic tools (MRI and CT scans) and monitoring CSF pressure. After the patient presents the classical triad of symptoms or at-least the most common symptoms (ataxia of gait and dementia), the clinicians will further use MRI/CT scans as well as lumbar or intraventricular monitoring of CSF pressure (infusion tests) to confirm their diagnosis. The MRI scan of NPH patients show diluted ventricular space and prominent Sulci features. This is common with MRI findings on other neurodegenerative diseases (e.g. Alzheimer s) and further complicates diagnosis of NPH. Thereafter, in some hospitals (e.g. Addenbrooke s Hospital, Cambridge, UK) the patients are subjected to computerised infusion tests (Avezaat and Eijndhoven, 1984; Borgesen et al., 1978; Borgesen et al., 1992; Czosnyka et al., 1990; Czosnyka et al., 1996; Czosnyka et al., 1999; Czosnyka et al., 2000; Czosnyka et al., 2001; Czosnyka et al., 2002; Czosnyka et al., 2004; Czosnyka et al., 2004; Ekstedt, 1977; Gjerris et al., 1987; Katzman et al., 1997; Lundberg, 1960; Marmarou et al., 1978; Marmarou et al., 1996; Shulman and Marmarou, 1968 etc.). During these tests, fluid infusion (saline solution) is made into accessible CSF space [either into the lumbar compartment or intraventricular through a pre-implanted ventricular access device (e.g. Ommaya reservoir)]. For the infusion test, irrespective of the site for infusion, two needles are used. One needle is connected to a pressure transducer while the second needle is connected to an infusion pump. The pressure transducer records the pressure in the CSF spaces and the pressure data is 11

23 Chapter 2 Literature Review processed by an IBM compatible personal computer running custom coded software (ICM+, Academic Neurosurgery Unit, Department of Clinical Neuroscience, School of Clinical Medicine, University of Cambridge, UK). Initially, baseline CSF pressure is monitored for 10 minutes. Thereafter, saline solution is infused at a constant rate (0.5 1 ml/min) until CSF pressure reaches a steady state plateau. During infusion tests, the IBM compatible personal computer calculates various factors on the basis of the recorded CSF pressure. The infusion tests determine CSF dynamics using factors such as resistance to CSF flow (R csf ), Baseline CSF pressure (P b ), Pressure Volume Index (PVI), Elastance (E) etc (Borgesen et al., 1979; Borgesen and Gjerris, 1982; Tans and Poortvliet, 1989; Albeck et al., 1991; Czosnyka and Pickard, 2004; Czosnyka et al., 2004; Kim et al., 2009 etc.). Armed with clinical symptoms, MRI/CT scans and results from the infusion tests, the clinicians maybe able to make the diagnosis of NPH (Boon et al., 1997; Boon et al., 1998; Boon et al., 1998; Boon et al., 1999; Boon et al., 2000; Petrella et al., 2008; Vanneste et al., 1993 etc.). However, other clinicians cannot corelate the findings of infusion tests with diagnosis of NPH (Brean and Eide, 2008; Eide, 2003; Eide et al., 2003; Eide, 2006; Sorteberg et al., 2004 etc.). If the diagnosis of NPH is confirmed, shunting is done to relieve the pressure and/or volume in the ventricular space. 2.3 Biomechanical Models of Normal Pressure Hydrocephalus Previous Generation Linear Models 12

24 Chapter 2 Literature Review After the condition of Normal Pressure Hydrocephalus (NPH) was recognised (Hakim; 1965 and Adams et al., 1965), it was postulated that NPH has mechanistic causes (Hakim, 1971). Hakim (1971) proposed an explanation based on pressure differential between the intra-ventricular CSF and brain parenchyma vascular system as a cause for development of NPH. He considered the brain parenchyma to be a sponge like viscoelastic structure. This approach of Hakim (1971) was conceptual in nature and lacked a mathematical framework. To give a mathematical basis to the conceptual understanding (Hakim, 1971), Hakim et al. (1976), Hakim (1977) and Hakim and Hakim (2001) used thick shell theory to analyse the biomechanics of NPH. The brain was considered as a hollow sphere and the brain parenchyma as elastic undergoing infinitesimal deformations (both these assumptions are intrinsic for thick shell theory). Akin to the model based on pressure differential between ventricular CSF and brain parenchyma vascular system (Hakim, 1971, Hakim et al., 1976 and Hakim, 1977), Berger et al. (1978) attempted to conceptually analyse NPH using force differential between intra-ventricular and extra-ventricular spaces. The first attempt to model NPH using Finite Element Method (FEM) was made by Nagashima et al. (1987). They used infinitesimal deformation theory, a 2-D model of the brain anatomy, linear elastic constitutive law and soil consolidation theory to compute deformations of the brain parenchyma. The results of the computation help in understanding periventricular edema during NPH. Tada et al. (1990) and Pena et al. (1999, 2002) modelled NPH using the principles set down earlier by Nagashima et al. (1987). Both the authors listed above used 2-D model of the brain anatomy, elastic constitutive law, infinitesimal deformations and soil consolidation theory to model NPH. Péna et al. (1999, 2002) showed that a pressure gradient needed to exist between the intra-ventricular and 13

25 Chapter 2 Literature Review subarchanoid space for development of NPH. Moreover, the computational model Péna et al. (1999, 2002) also explained the phenomenon of Periventricular Lucency (Mori et al., 1980). Their findings of pressure gradient between the intra-ventricular and subarchanoid space (transmantle pressure difference) is not supported by clinical observation in NPH patients (Penn et al., 2005) State of the Art Non Linear Models Kaczmarek et al (1997) observed that due to the large ventricular dilation in NPH, the brain tissue deformed significantly. As a result, the assumption of infinitesimal deformation theory (Hakim et al., 1976; Hakim, 1977; Nagashima et al., 1987; Tada et al., 1990; Péna et al., 1999 and Péna et al., 2002) used to model NPH was invalid. Hence, Kaczmarek et al (1997) proposed the use of finite deformation theory to describe the deformations within the brain parenchyma. Though Kaczmarek et al. (1997) used simplified brain geometry (cylindrical) geometry, linear elastic constitutive law for the brain parenchyma (brain tissue exhibits strongly non-linear characteristics, for e.g. Miller and Chinzei, 1997 and Miller and Chinzei, 2002) and soil consolidation theory to calculate deformations in the brain parenchyma, their contribution was a major improvement over earlier work done on modelling NPH. Following Kaczmarek et al. (1997), Taylor and Miller (2004) used finite deformation formulations to model NPH. They used a realistic 2-D model of brain anatomy, soil consolidation theory and linear elastic constitutive law for the brain parenchyma in their work. Taylor and Miller (2004) showed that the brain parenchyma elastic modulus needed to be revised (584 Pa compared to Pa used in other works) to accurately model NPH. More 14

26 Chapter 2 Literature Review recently small strain formulations, non-linear soil consolidation theory, linear elastic constitutive law for brain parenchyma and simplified spherical model for the brain geometry (Smillie et al, 2005 and Sobey and Wirth, 2006) was used to model hydrocephalus. Wirth and Sobey (2006) further investigated hydrocephalus using simplified 3 D, axisymmetric model. The major drawbacks of these works are the use of small strain theory (Smillie et al, 2005; Sobey and Wirth, 2006 and Wirth and Sobey, 2006) and elastic constitutive law (Kaczmarek et al., 1997; Taylor and Miller, 2004; Smillie et al, 2005; Sobey and Wirth, 2006 and Wirth and Sobey, 2006) to calculate the deformations for the brain parenchyma as well as simplified brain geometry. Small strain theory is incapable of capturing the geometric changes and elastic constitutive law is unable to handle material behaviour at large deformations (Bathe, 1994) occurring during hydrocephalus. Furthermore, complicated mathematical basis for modelling hydrocephalus have been laid down by Sivaloganathan (1998), Stastna et al. (1999), Stastna et al. (1999), Tenti et al. (1999), Sivaloganathan (2005), Sivaloganathan (2005), Drapaca (2005) and Drapaca (2006) albeit on simplified brain geometry. Recently, a stress-relaxation type ( poroplastic ) computational model of NPH was presented (Momjian and Bichsel, 2008) for a realistic 2-D model of brain anatomy. The authors claim that their model correctly predicted the magnitude and shape of the ventricular space for acute and chronic hydrocephalus and the same methodology might be able to explain the biomechanics behind NPH. Soil consolidation theory, small strain formulation and linear elastic constitutive law for simplified (spherical) model of the brain parenchyma was also used to predict average intracranial pressure increase and CSF pressure pulsation amplitude rise (Wirth and Sobey, 2009) during infusion tests (Avezaat and Eijndhoven, 1984; Borgesen et al., 1978; Borgesen et al., 1992; Czosnyka 15

27 Chapter 2 Literature Review et al., 1990; Czosnyka et al., 1996; Czosnyka et al., 1999; Czosnyka et al., 2000; Czosnyka et al., 2001; Czosnyka et al., 2002; Czosnyka et al., 2004; Czosnyka et al., 2004; Gjerris et al., 1987; Katzman et al., 1997; Lundberg, 1960; Marmarou et al., 1978; Marmarou et al., 1996; Shulman and Marmarou, 1968 etc.) on NPH patients. Innovative attempts using mass transport theory (Garcia and Smith, 2009 and Smith and Garcia, 2009) were recently made to model constant pressure and constant flow infusion tests on NPH patients. Finite deformation formulations and hyperelastic constitutive law (Garcia and Smith, 2009 and Smith and Garcia, 2009) were used to calculate the deformations of the brain parenchyma but the brain geometry was approximated as spherical. The literature review in the above Sections (Section 2.2.2) showed that most of the work done in the field of Normal Pressure Hydrocephalus (NPH) was carried out with a clinical point of view. The clinician s perspective is to devise protocols to identify, differentiate and diagnose NPH correctly. Moreover, the clinicians would like also to devise methods for effectively treating the condition of NPH. However, a clinical approach to NPH research does not provide an understanding of the reasons behind development of NPH. A proper understanding for the reasons of NPH growth could help to devise better methods for diagnosis and treatment of NPH. It was initially postulated that NPH has biomechanical causes. Though many attempts in the previous two to three decades were made to understand the biomechanics of NPH (Section 2.3), most of the attempts were not able to co-relate their findings with clinical observations. The initial attempts to model the biomechanics of NPH lacked thorough theoretical foundations i.e. they used infinitesimal deformation theory, linear 16

28 Chapter 2 Literature Review material law for the brain parenchyma as well as simplified brain geometry (Section 2.3.1). Later attempts (Section 2.3.2) at modelling the biomechanics of NPH tried to overcome some of the theoretical weakness of the earlier attempts. The workers in these later attempts understood that the brain parenchyma deformed significantly due to large ventricular dilation. As a result, finite deformation theory framework was used to describe the deformations in the brain parenchyma. However, the lack of understanding of the constitutive behaviour of the brain tissue led to the continuation of the use of linear elastic constitutive law for the brain parenchyma. Moreover, most of the attempts to understand NPH biomechanics were made on simplified brain geometry (spherical or cylindrical). In this thesis, an effort has been made to fill the existing gaps in knowledge about NPH biomechanics. A realistic 3-D model of the brain anatomy and fully non linear (finite deformation, non linear constitutive law and contact boundary condition) Finite Element procedures were used to investigate the biomechanics of NPH. The suggestion that no more than 1 mm of Hg ( Pa) transmantle pressure difference is associated with the clinical condition of NPH is investigated (Chapter 3). Computations were also carried out to investigate if soil consolidation (biphasic) based models of the brain parenchyma gave any major advantages over single phase models of the brain parenchyma for modeling NPH. Thereafter, simple experiments were also conducted using samples of lamb brain tissue to verify as to what extent the parenchyma responses are consistent with the models using soil consolidation (biphasic) theory (Chapter 4). Furthermore, the idea that pulsations observed in Intracranial Pressure (ICP) recordings of NPH patients could explain the mechanics of NPH growth was also 17

29 Chapter 2 Literature Review investigated (Chapter 5). Furthermore it is shown that computational methods are an effective tool for modelling complicated biomechanical phenomenon (Appendix A) such as needle insertion into brain tissues. 18

30 CHAPTER 3 - BIOMECHANICS OF NORMAL PRESSURE HYDROCEPHALUS * 3.1 Introduction Diagnosis of Normal Pressure Hydrocephalus (NPH) is a reoccurring problem faced by clinicians due to the overlap of symptoms and diagnostic findings between NPH and other neurodegenerative diseases (e.g. Alzheimer s; see Literature Review, Chapter 2, Section 2.1.2). Hakim and Adams (1965) and Adams et al. (1965) were the first to identify the clinical condition of Normal Pressure Hydrocephalus (NPH) (see Literature Review, Chapter 2, Section 2.1.1). Cerebrospinal Fluid (CSF) hydrodynamics approach (Czosnyka et al., 2004, Linninger et al., 2005) and analysing Intracranial Pressure (ICP) waves (Shulman et al., 1968; Marmarou et al., 1978; Czosnyka and Pickard, 2004) enhanced the diagnosis of NPH but offered limited understanding of NPH growth mechanics (see Literature Review, Chapter 2, Section 2.1.2). In this chapter, a fully non-linear, 3-D biomechanical model of NPH is presented. In previous attempts to understand NPH biomechanics, 2-D models of the brain anatomy, infinitesimal deformation theory to compute brain parenchyma deformations and linear elastic constitutive law to approximate the brain parenchyma * Portions of this chapter have been published in: Dutta-Roy, T., Wittek A., Miller, K., Biomechanical Modelling of Normal Pressure Hydrocephalus, Journal of Biomechanics 41 (10), pp , July 2008 Dutta-Roy, T., Wittek, A., Miller, K., 3-D Non-Linear Finite Element Analysis of Normal Pressure Hydrocephalus, Computational Biomechanics for Medicine II Workshop, 10 th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI-2007), Oct 2007, Brisbane, Australia, pp

31 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus material properties were used. Several studies considered the brain continuum as biphasic (solid phase: brain parenchyma; fluid phase: cerebrospinal fluid) and used coupled pore fluid diffusion and stress analysis to compute deformations of the brain parenchyma and ventricles. To model the skull-brain interaction, the outer surface of the brain was fixed to the skull. Additionally, for intra-operative image registration of brain deformation during neurosurgery, prior work used infinitesimal deformation theory and linear elastic constitutive law for the brain parenchyma as well as coupled pore fluid diffusion and stress analysis. Please refer to Chapter 2 (Literature Review, Section 2.3) for a detailed review on existing models of NPH. Due to four fold increase in ventricular volume during NPH, large strain and finite deformations occur in the brain parenchyma. Hence, to model NPH growth, use of linear elastic constitutive law (Nagashima et al., 1987; Tada et al., 1990; Kaczmarek et al., 1997; Péna et al., 1999 and Taylor and Miller, 2004) and infinitesimal deformation theory (Nagashima et al., 1987; Tada et al., 1990 and Péna et al., 1999) for the brain parenchyma contradicts known facts about the brain undergoing large deformations and brain tissue presenting non-linear constitutive behaviour. NPH computational model should include non-linear constitutive law (e.g. hyperelastic) and finite deformation formulations (Bathe; 1996) for brain parenchyma. Following Wittek et al. (2007), the simplified skull-brain interaction (Nagashima et al., 1987; Tada et al., 1990; Kaczmarek et al., 1997; Péna et al., 1999 and Taylor and Miller, 2004) needed to be updated and appropriate boundary conditions between the skull and the brain needed inclusion. 20

32 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus For valid biomechanical modelling of NPH, as well as intra-operative image registration during neurosurgery, a realistic 3-D model of the brain anatomy, appropriate mathematical model describing the deformations in the brain parenchyma and inclusion of suitable boundary conditions between the skull and the brain are required. In the present work, limitations of the previous studies listed above were overcome by using a fully non-linear (geometric, constitutive and boundary) 3-D model of the brain parenchyma. The contact between the skull and the brain outer surface constituted the non-linear boundary condition for the NPH model. Both biphasic and single phase models of the brain parenchyma were loaded with transmantle pressure difference to investigate the suggestion that no more than 1 mm of Hg transmantle pressure difference (Penn et al.; 2005 and Czosnyka; 2006) is associated with NPH. The model presented in the following sections is not intended to accurately represent the brain in various aspects of its behaviour. The goal is much more modest: to formulate the models to investigate the relationship between pressures, volumes and displacements within the brain in a reliable and efficient way. Section 3.2 describes the biomechanical model used in our simulations. Detailed description of computational model is presented in Sections 3.3 and 3.4. Results of our simulations are given in Section 3.5 followed by discussion in Section

33 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus 3.2. Biomechanical Model Continuum Mechanics: Equations Mathematical formulations are required to account for deformations, forces, stresses and strains occurring in the brain parenchyma under applied loads. The mathematical model of brain deformations used in the present work consisted of standard partial differential equations of equilibrium (solid mechanics: biphasic and single phase), non-linear constitutive model of the brain parenchyma, boundary and initial conditions. As mentioned earlier (Section 3.1), during NPH, the brain parenchyma undergoes large deformations. Hence, fully non-linear equations of continuum mechanics were required. Following Miller (2002), non-linear continuum equations for brain biomechanics are used. For both biphasic and single phase brain continuum, the stresses and strains are measured with respect to the current configuration (true stress and strain). Hence, the Cauchy stress measure (σ) and energy conjugate Almansi strain measure (ε) is used (Bathe; 1996). For biphasic continuum, the principle of virtual work (force equilibrium) used to derive the non-linear finite element formulation was given by (Biot, 1941; Bowen, 1976 and Miller, 1998): B S ij ijdv f i uidv f i uids s g uidv (3.1) V V S V 22

34 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus B where: ij ijdv is the internal virtual work, f i uidv is the virtual work done V V S by external body forces, f i uids is the virtual work done by external surface forces S and s g uidv is the virtual work done by the weight of the wetting fluid, snρ ω g V is the weight of the wetting fluid and g is gravitational acceleration. The continuity equation for the wetting fluid (equating the rate of increase of liquid mass stored at a point and the rate of mass of liquid flowing at a point) is given by (Biot, 1941 and Bowen, 1976): 1 d (J )dv n v ds V J dt (3.2) S where: 1 d (J ) dv V J dt is the time rate of change of mass of wetting liquid, n v ds is the mass of wetting liquid entering the control volume V by S crossing the surface S, J is the ratio of the volume in current configuration to the reference configuration, ρ ω is the mass density of wetting fluid, η ω is the volume ratio of wetting liquid, n is the outward normal to the surface S and v ω is the seepage velocity. The interstitial flow of fluid through the porous media is governed by Darcy s Law and is given by: 23

35 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus s v x (3.3) where: s is the saturation, n is the porosity, η ω is the volume ratio of wetting liquid, α is the porosity and κ is the permeability of the medium. For single phase continuum, no wetting fluid is present and the virtual work associated with the weight of the wetting fluid (Equation 3.1) disappears. In case of the single phase continuum, the principle of virtual work (force equilibrium) used to derive the non-linear finite element formulation was given by (Bathe, 1996 and Miller, 2002): B S ij ijdv f i uidv f i uids (3.4) V V S B where: ij ijdv is the internal virtual work, f i uidv is the virtual work done V V S by external body forces and f i uids is the virtual work done by external surface S forces. As the brain undergoes large deformation for both biphasic and single phase continuum, the current volumes V and current surfaces S over which the integration occurs (Equations ) are unknown. The volumes V and surface S are part of the solution and not a part of the input parameters. Hence, non-linear finite deformation procedures should be used to obtain the solution to Equations

36 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus Brain Mesh Figure 3.1 shows the brain mesh for a normal human brain. Person specific brain mesh (Wittek et al., 2007) was modified using Hypermesh (Altair Engineering, USA) pre-processing software and a generic mesh of a human brain was created. The brain and ventricular volume in the brain mesh were consistent with values reported by Matsumae et al. (1996) for a normal human brain (Table 3.1). Table 3.1: Brain and ventricular volumes for normal and NPH affected brain; adapted from Matsumae et. al. (1996) Case Brain Volume (cubic cm) Ventricular Volume (cubic cm) Normal brain 1188±104 27±10 NPH 1163± ±42 As the brain is approximately symmetrical, half of the brain was used in the simulations. Thus, ventricular volume in the model of a normal human was 14 cm 3 and 25

37 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus NPH developed when the ventricular volume increased from 14 cm 3 to more than 58 cm 3 (Table 3.1). Figure 3.1: Brain geometry, pressure loading and applied boundary conditions for the brain Brain Parenchyma Material Parameters A hyper-viscoelastic constitutive law proposed by Miller and Chinzei (2002) that includes stress - strain rate dependency (Miller and Chinzei, 1997 and Miller and Chinzei, 2002) and non-linear stress-strain behaviour of the brain parenchyma was modified, to account for a very long time of NPH development. The constitutive law is given by: 2 t d W [ (t ) ( d 3)] d (3.5) 26

38 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus n t k [ 1 g (1 e ) (3.6) 0 k k 1 where: W is the potential function, λ i are the principal stretches, μ 0 is the instantaneous shear modulus in undeformed state, τ k are the characteristic times, g k are the relaxation coefficients and α is the material coefficient which can assume any real value without any restrictions. Constants for the brain tissue constitutive model as proposed by Miller and Chinzei (2002) are given in Table 3.2. The values for material constants given in Table 3.2 are valid for relatively high strain rates such as those present during surgical intervention. Table 3.2: Material constants for the constitutive model of the brain tissue; adapted from Miller and Chinzei (2002) Instantaneous Response 0 =842 Pa Characteristic Time Relaxation Coefficients k=1 1 =0.5 sec g 1 = k=2 2 =50 sec g 2 =

39 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus NPH typically develops over 4 days (Nagashima et al., 1987) and load application on the brain was infinitesimally slow, resulting in zero strain rates for Equation 3.5. Hence, limiting case of the relaxation function (Equation 3.6) was considered (Taylor and Miller, 2004): lim n n t k gk(1 e ) = g k (3.7) t k 1 k 1 The relaxed hyperelastic shear modulus was given by modified form of Equation 3.6: n 0[1 g ] (3.8) k k 1 Using n=2 and values given in Table 3.1, the relaxed hyperelastic shear modulus is: = Pa (3.9) Also, due to slow load application, the strain rate dependency of the hyperviscoelastic constitutive law (Miller and Chinzei, 2002 and Equation 3.5) could be excluded (Taylor and Miller, 2004). Hence, hyperelastic (Ogden, 1984) constitutive model for the brain parenchyma was used: W 2 ( ) ( J el 1) 2 (3.10) D1 28

40 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus where: W is the potential function, λ i s are the principal stretches, μ is the relaxed shear modulus, α is the material coefficient which can assume any real value without any restrictions, J el is the elastic volume ratio and D 1 is the material coefficient related to the initial bulk modulus (K 0 ).. The value μ and α were Pa (Equation 3.9 and Taylor and Miller, 2004) and -4.7 (Miller and Chinzei, 2002) respectively. The brain parenchyma was considered to be homogenous and isotropic for the simulations (Ozawa et al., 2001 and Miller et al., 2005). In the present approach, properties of the modelled brain tissue are effectively averaged over characteristic length scale in this case approximately 1 cm (see e.g. Miller at al. 1997, 2002). The constitutive model of the brain tissue (Equation 3.5 and Equation 3.10) contains contributions of all constituents of brain tissue, including the contribution of blood vessels, in the sense of volumetric averaging with the characteristic scale of approximately 1 cm 3. These modelling issues were explained in earlier publications (e.g. Miller et al., 1997; Miller, 1999; Miller et al.; 2000; Miller and Chinzei, 2002; Taylor and Miller, 2004 and Miller et al., 2009). Biphasic Model of Brain Parenchyma In previous NPH growth studies, authors (Hakim, 1971; Nagashima et al., 1987; Kaczmarek et al., 1997; Péna et al., 1999; Stastna et al., 1999; Stastna et al., 1999; Tenti et al., 1999; Tenti et al., 2000; Taylor and Miller, 2004; Drapaca et al., 2005; Sivaloganathan et al., 2005; Sivaloganathan et al., 2005; Sivaloganathan et al., 2005; Smillie et al., 2005; Sobey and Wirth, 2006; Wirth and Sobey, 2006; Drapaca et al., 29

41 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus 2006 and Momjian et al., 2008) treated the brain as a sponge like structure which consisted of a solid matrix of neurons and neuroglia with extracellular space (voids) between them filled with Cerebrospinal Fluid (CSF). The brain parenchyma was biphasic due to presence of neurons and neuroglia (solid or porous phase) and CSF (fluid phase) at each point in the continuum. Following the above previous studies and Miller (1998), coupled pore fluid diffusion and stress analysis (Biot, 1941) was performed to study the interaction between the CSF (fluid phase) and brain parenchyma (solid or porous phase) when loaded by transmantle pressure difference. In the model, Poisson s ratio (υ) of 0.35 (Nagashima et al., 1987; Kaczmarek et al., 1997; Péna et al., 1999 and Miller et al., 2005) for the solid phase (brain parenchyma) was used, with relaxed hyperelastic material properties (μ = Pa; Section 3.2.3). The initial void ratio of the brain parenchyma was 0.2 (Nagashima et al., 1987 and Syková, 2004) with permeability of 1.59 X 10-7 m/sec (Kaczmarek et al., 1997). The brain parenchyma was fully saturated with CSF. The fluid phase (CSF) was incompressible, non-viscous with mechanical properties of water. Darcy s law (Bowen, 1976) modelled the flow of CSF within the brain parenchyma. Single Phase Model of Brain Parenchyma To investigate the effect of brain parenchyma compressibility on the transmantle pressure difference in single phase model, the brain parenchyma was modelled as incompressible (Miller and Chinzei, 1997; Miller, 2001 and Miller and Chinzei, 2002), nearly incompressible and compressible by varying Poisson s ratio (υ). 30

42 E Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus A hyperelastic constitutive law (Ogden, 1984) with Poisson s ratio (υ) of 0.5, and 0.35 (Kaczmarek et al., 1997 and Miller et al., 2005) and relaxed hyperelastic shear modulus (section 3.2.3) were used. The material properties of the brain parenchyma for different Poisson s ratio (υ) are summarised in Table 3.3. Table 3.3: Material properties for incompressible, nearly incompressible and compressible brain; E (relaxed Young s Modulus) and K (relaxed Bulk Modulus) are E calculated from E 2 (1 ), K, μ is taken from section 3.2.3) 3(1 2 ) Case (Pa) E (Pa) K (Pa) Incompressible Nearly Incompressible Compressible Brain Model Loading Transmantle pressure difference (P trans ) was applied on the ventricular surface (Figure 3.1), to load both single and biphasic models. There was no pressure acting on the outer surface of the brain. It is widely believed that at most 1 mm of Hg ( Pa) transmantle pressure difference is associated with the clinical condition of NPH 31

43 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus (Penn et al., 2005 and Czosnyka, 2006). Accordingly, a transmantle pressure difference (P trans1 ) of 1 mm of Hg ( Pa) was firstly applied on both single and biphasic models to verify this view. Table 3.4: Loading applied to single and biphasic model of brain parenchyma Load Case Model Load Case 1 Load Case 2 Load Case 3 (P trans1 ) (P trans2 ) (P trans3 ) Single Phase 1 mm of Hg ( Pa) 1.69 mm of Hg ( Pa) Pressure required to produce NPH Biphasic 1 mm of Hg ( Pa) 1.69 mm of Hg ( Pa) - Since this transmantle pressure difference (P trans1 ) did not produce NPH, we increased it to P trans2. At P trans2 transmantle pressure difference, the biphasic model did not converge due to excessive distortions of some elements in the complicated brain mesh. Therefore, the same transmantle pressure difference (P trans2 = Pa=1.69 mm of Hg) load was applied to incompressible (υ=0.5) and nearly incompressible single phase model (υ=0.49) and compared to ventricular volume that was obtained for the biphasic model. As P trans1 and P trans2 transmantle pressure difference load did not produce the condition of NPH, the single phase continuum model was loaded with 32

44 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus higher transmantle pressure difference (P trans3 ) which could produce NPH. The three load cases are summarised in Table 3.4. Biphasic Model of Brain Parenchyma Even though material strain rate effects were absent due to the use of the hyperelastic constitutive law for the brain parenchyma, rate effects due to the relative motion between the solid (brain parenchyma) and liquid (CSF) phases were present. The time period of pressure load application was important. The transmantle pressure difference (P trans ) was applied over a period of 4 days (Nagashima et al., 1987). Single Phase Model of Brain Parenchyma As hyperelastic constitutive law for the brain parenchyma was used, material strain rate effects were absent and the pressure load application time for static solution of the single phase model was unimportant. The time period of transmantle pressure difference (P trans ) application was arbitrarily taken to be 10 seconds. For both, single and biphasic model of the brain parenchyma, the transmantle pressure difference (P trans ) was applied using a polynomial (Waldron and Kinzel, 1999). This polynomial ensured vanishing derivatives at the beginning and end of the loading, minimising possible dynamic effects. 33

45 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus Brain Model Boundary Conditions Due to approximate symmetry of the brain about the mid-sagittal axis, half of the brain was used in the simulations. Nodes on Plane 1 (Figure 3.1) had symmetric boundary conditions in the YZ plane (no motion allowed for X translation) applied to them. As the brain was resting in the skull, all the nodes at the bottom of the brain in Y and Z translation (Figure 3.1) were constrained. At the top of the brain, the sub-arachnoid s space was accounted for, by introducing a 3 mm gap between the brain outer surface and the skull. To constrain the nodes on the outer surface of the brain within the skull, a frictionless, finite sliding, nodeto-surface penalty contact between the brain and the skull (ABAQUS/Standard 2004) was used. This follows previous studies by Miller et al. (2000) and Wittek et al. (2005, 2007 and 2009). The contact boundary condition is non-linear because an iterative method would be required to establish whether the brain is in contact with the skull or not even if the rest of the model were linear (i.e. infinitesimal deformation and liner material properties). For modelling brain deformations, boundary conditions are one of the main sources of nonlinearity. The other sources of non-linearity are finite deformations and non-linear material properties. 34

46 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus Biphasic Model of Brain Parenchyma In addition to the above, boundary conditions for the pore pressure variable needed to be defined for the biphasic model. The pore pressure on the ventricular surface was set equal to the transmantle pressure difference (P trans1 or P trans2 ). The pore pressure on the outer surface of the brain was set to 0 Pa. 3.3 Computational Model Biphasic Model of Brain Parenchyma 5858 porohyperelastic type C3D20P (20-node brick, pore pressure) (ABAQUS/Standard, 2004) and 89 type C3D10 (10-node quadratic tetrahedron) (ABAQUS/Standard, 2004) elements were used for the biphasic model. In order to accurately represent the complex brain geometry, tetrahedral elements completed the mesh in places where the quality of the hexahedral elements was poor Single Phase Model of Brain Parenchyma Incompressible/nearly incompressible brain parenchyma model consisted of 5858 type C3D20H (20-node quadratic brick, hybrid) (ABAQUS/Standard, 2004) and 89 type C3D10H (10-node quadratic tetrahedron, hybrid) (ABAQUS/Standard, 2004) elements type C3D20 (20-node quadratic brick) (ABAQUS/Standard, 2004) and 89 type C3D10 (10-node quadratic tetrahedron) (ABAQUS/Standard, 2004) elements 35

47 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus constituted the compressible brain parenchyma model. As with the biphasic model, in order to accurately represent the complex brain geometry, tetrahedral elements completed the mesh in places where the quality of the hexahedral elements was poor. Elements of type C3D20H and C3D10H do not exhibit volumetric locking and therefore can be confidently used for an incompressible/nearly incompressible continuum (e.g. brain). For both biphasic and single phase models, the skull model consisted of 1006 type R3D4 (4-node, bilinear quadrilateral, 3-dimensional rigid) (ABAQUS/Standard, 2004) elements. 3.4 Finite Element Solver ABAQUS-Standard non-linear implicit finite element code (ABAQUS/Standard, 2004) was used to obtain the solution for the NPH biomechanical model. The code accounts for constitutive, geometric (finite deformation) and contact non-linearities. The STATIC (ABAQUS/Standard, 2004) procedure was used to solve single phase model and the SOILS (ABAQUS/Standard, 2004) procedure for the biphasic model. Wu et al. (1998) validated the use of the SOILS procedure in ABAQUS for hydrated biphasic tissues. 36

48 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus In the SOILS procedure, permeability of the porous medium is a function of saturation and void ratio. For the biphasic model, the brain parenchyma (porous medium) is fully saturated with CSF. The permeability of the porous medium (brain parenchyma) is deformation and void ratio dependent. (ABAQUS/Standard, 2004). 3.5 Computational Results Ventricular Volume Load Case 1: Transmantle pressure difference (P trans1 ) equivalent to 1 mm of Hg ( Pa) Biphasic Model of Brain Parenchyma Ventricular volume computed when a transmantle pressure difference of Pa (P trans1 ) loaded the biphasic model of brain parenchyma was 36.6 cm 3. P trans1 load did not produce NPH. Single Phase Model of Brain Parenchyma Ventricular volumes for incompressible, nearly incompressible and compressible brain parenchyma (Section , Single phase model of brain parenchyma) under P trans1 load are given in Table 3.5. As can be seen from Table 3.5, the level of brain tissue compressibility has limited influence on the ventricular volumes produced. As in the case of the biphasic model, P trans1 failed to produce the condition of NPH. 37

49 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus Table 3.5: Ventricular volume when transmantle pressure difference (P trans1 ) equivalent to 1 mm of Hg ( Pa) is applied to single phase and bi phase brain parenchyma. NPH develops when ventricular volume increased from 14 cm 3 to 58 cm 3. Case Poisson s Ratio ( ) Ventricular Volume (cm 3 ) Incompressible Nearly Incompressible Compressible Bi-phase Ventricular volumes summarised in Table 3.5 above are similar to the ventricular volume obtained for biphasic model of the brain parenchyma under P trans1 load. Load Case 2: Transmantle pressure difference (Ptrans2) equivalent to 1.69 mm of Hg ( Pa) Biphasic Model of Brain Parenchyma As P trans1 = Pa failed to produce the condition of NPH (Table 3.5), a higher transmantle pressure difference (P trans2 ) was used to load the brain parenchyma. For the biphasic case, due to excessive distortion in a few elements of the complicated brain mesh (quality of these elements could not be improved due to complicated brain 38

50 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus geometry), the solution failed to converge under transmantle pressure difference load higher than 1.69 mm of Hg ( Pa). Table 3.6: Ventricular volume produced when transmantle pressure difference (Load Case 2 (P trans2 ) = Pa= 1.69 mm of Hg) is applied to incompressible/nearly incompressible single and biphasic model. NPH developed when ventricular volume increased from 14 cm 3 to 58 cm 3. Case Poisson s Ratio ( ) Load Case 2-P trans2 Ventricular Volume (cm 3 ) Biphasic mm of Hg ( Pa) Incompressible mm of Hg ( Pa) Nearly Incompressible mm of Hg ( Pa) Single Phase Model of Brain Parenchyma Since the biphasic brain parenchyma is fully saturated with CSF in its undrained condition, it can be considered as incompressible or nearly incompressible continuum. Therefore, to compare the results, the same transmantle pressure difference (P trans2 =1.69 mm of Hg= Pa) was used to load incompressible (Poisson s ratio υ=0.5) and nearly incompressible single phase (Poisson s ratio υ=0.49) brain parenchyma. The ventricular 39

51 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus volumes produced under this load for incompressible and nearly incompressible single phase and biphasic brain parenchyma are given in Table 3.6. It can be seen clearly from the results that a transmantle pressure difference (P trans2 ) higher than 1.69 mm of Hg would be required to produce NPH for the biphasic model. Load Case 3: Transmantle pressure difference (P trans3 ) required for clinically producing the condition of NPH Single Phase Model of Brain Parenchyma Table 3.7: Transmantle pressure difference (Load Case 3 P trans3 ) required to produce NPH in single phase brain parenchyma model. NPH developed when ventricular volume increased from 14 cm 3 to 58 cm 3 Case Poisson s Ratio ( ) Load Case 3-P trans3 Incompressible mm of Hg ( Pa) Nearly Incompressible mm of Hg ( Pa) Compressible mm of Hg ( Pa) 40

52 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus P trans1 did not produce NPH for the single phase model (Table 3.5). The transmantle pressure difference was then increased from P trans1 to P trans3 to produce NPH (Table 3.7). Table 3.7 clearly shows that for all cases, transmantle pressure difference higher than 1 mm of Hg ( Pa) was required to produce NPH. 3.6 Discussions Ventricular Volume In undrained condition, the biphasic brain parenchyma is fully saturated with CSF. Therefore, it was considered to be incompressible or nearly incompressible continuum. Hence, the ventricular volume obtained for the biphasic model was compared with that for incompressible/nearly incompressible single phase model, under 1 mm of Hg ( Pa) transmantle pressure difference (P trans1 ) load. The ventricular volume for biphasic brain parenchyma (36.6 cm 3 ) was 5.1% and 4.2% higher compared to incompressible (υ=0.5) and the nearly incompressible (υ=0.49) single phase model of the brain parenchyma respectively (Table 3.5). For biphasic and single phase incompressible/nearly incompressible model of the brain parenchyma, P trans1 load did not produce NPH (Section 3.2.3). As P trans1 did not produce NPH in the biphasic model, the transmantle pressure difference was increased in an attempt to produce NPH. The results could be obtained only for transmantle pressure difference (P trans2 ) of up to Pa (1.69 mm of Hg) load. 41

53 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus Under transmantle pressure difference higher than P trans2, the biphasic model did not converge due to excessive distortions of some elements in the complicated brain mesh. This pressure load (P trans2 = Pa=1.69 mm of Hg) did not produce NPH for the biphasic model. As mentioned above, the biphasic brain parenchyma in its undrained condition is incompressible or nearly incompressible. Therefore, the same transmantle pressure difference (P trans2 = Pa=1.69 mm of Hg) load was applied to incompressible (υ=0.5) and nearly incompressible single phase model (υ=0.49) and compared to ventricular volume that was obtained for the biphasic model. For the incompressible (υ=0.5) and nearly incompressible (υ=0.49) single phase model, ventricular volume was 4.2% and 6% higher respectively when compared to the biphasic model (Table 3.6). In the results presented above, it is clearly seen that there are no significant differences in computed ventricular volumes between biphasic and incompressible/nearly incompressible single phase model of the brain parenchyma under equal load. Hence, no major advantage would be gained by using a biphasic model for the brain parenchyma. It is proposed that for modelling NPH, single phase model of the brain parenchyma is adequate. The single phase model simplifies the mathematical description of the NPH model and results in significant reduction of computational time (483 minutes for the single phase incompressible model compared to 2921 minutes for the biphasic model on Pentium 4, 3.2 GHz, 2 GB RAM personal computer), while appropriately representing the brain constitutive properties for NPH investigation. 42

54 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus Under P trans1 = Pa load applied to single phase brain model, ventricular volume was 7% and 6% higher for the compressible brain parenchyma (υ=0.35) compared to incompressible (υ=0.5) and the nearly incompressible brain parenchyma (υ=0.49) respectively (Table 3.5). Although the relaxed bulk modulus (K ) for the compressible parenchyma was significantly lower than incompressible and nearly incompressible cases (Table 3.3), brain compressibility exerted limited influence on the ventricular volumes produced. This could be explained by the inclusion of realistic boundary conditions which comprised of the sub arachnoid s space (3 mm gap between the brain outer surface and skull) and the skull-brain interaction modelled using contact boundary conditions (Miller et al., 2000 and Wittek et. al, 2005, 2007). P trans1 load deformed the brain because the brain outer surface displaced outwards. The skull-brain contact included in the model checked the displacement of the brain outer surface and limited further movement after it displaced through a distance equal to sub arachnoid s space (3 mm). Therefore, even for low Poisson s ratio (υ) of 0.35 (Table 3.5), 1 mm of Hg ( Pa) transmantle pressure difference failed to produce the clinical condition of NPH (Section 3.2.3) As P trans1 did not produce the condition of NPH, the model was loaded with higher transmantle pressure difference (P trans3 ) which could produce NPH (Table 3.7). Incompressible and nearly incompressible models required 17% and 13% higher transmantle pressure difference respectively when compared to the compressible model. Brain compressibility had influence on the transmantle pressure difference required to produce NPH. 43

55 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus Figure 3.2: Transmantle pressure difference v/s ventricular volume for single phase (incompressible, nearly incompressible and compressible) and biphasic brain model It can be seen from the results presented above, that we required a minimum transmantle pressure difference of 1.76 mm of Hg ( Pa) to produce NPH for compressible (υ=0.35) single phase model of the brain parenchyma (Table 3.7). Based on transmantle pressure difference-ventricular volume plot shown in Figure 3.2, it could be argued that the biphasic model would require a load higher than 1.76 mm of Hg ( Pa) transmantle pressure difference to produce NPH. Penn et al. (2005) and Czosnyka (2006) have measured and reported that transmantle pressure difference ( Pa) of unto 1 mm of Hg is associated with NPH. Despite using various constitutive models (single phase and biphasic) and varying Poisson s ratio (υ) to a value as low as 0.35, we could not produce NPH for 1 mm of Hg ( Pa) transmantle pressure difference load. 44

56 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus The limitation of modelling and computer simulation work on brain biomechanics conducted to date, including this thesis, is that vasculature is not explicitly included in the model. However, the computational model presented in this chapter implicitly models the vasculature as the constitutive law of the brain parenchyma includes contribution of the blood vessels (Miller et al., 2000 and Miller et al., 2009). The constitutive law used here (Miller and Chinzei, 2002) treats the brain parenchyma as a continuum and was developed by subjecting cylindrical samples of the brain parenchyma (the blood vessels were not removed from them) to tension and compression. This is a standard approach when attempting to identify effective mechanical properties of complex materials (see e.g. Germanovich and Dyskin, 1994). It is also worth noting that Gefen and Margulies (2004) showed experimentally that there is no effect of perfusion on brain mechanical properties Periventricular Lucency In a healthy human brain, tightly packed ependymal cells line the ventricular cavity (Standring S., 2004). This limits the extracellular space, keeping permeability of brain parenchyma low and results in minimal CSF diffusion into the brain parenchyma. As the region of brain parenchyma around the ventricular surface stretches under pressure load (P trans1 or P trans2 ), it may disrupt the brain parenchyma. This resulted in an increase of void ratio (Figure 3.3) and permeability (Section 3.4), which suggested swelling of extracellular space. 45

57 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus Figure 3.3: Void ratio distribution for biphasic model [loading pressure: 1 mm of Hg ( Pa)] The pressure gradient between ventricles and sub arachnoid s space, combined with increased extracellular space (Figure 3.3) and permeability (Section 3.4, Finite Element Solver), facilitated increased seepage of CSF through the brain parenchyma (Figure 3.4). As a result, CSF accumulated in the extracellular space around the ventricular surface thereby promoting brain oedema. Mori et al. (1980) observed the brain oedema as Periventricular Lucency in CT scans of NPH affected patients. 46

58 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus Figure 3.4: Total fluid volume ratio distribution for biphasic model [loading pressure: 1 mm of Hg ( Pa)] Our interpretation of modelling results is consistent with Péna et al. (1999) and Taylor and Miller (2004). Biphasic brain parenchyma model appears to explain the phenomenon of Periventricular Lucency. The same cannot be explained by single phase model of the parenchyma. Nevertheless, further criticism of biphasic modelling approach is given in Chapter 4. 47

59 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus 3.7 Summary In this chapter a realistic 3-D model of the brain to model NPH was presented. The limitations of previous studies on NPH were overcome by using a fully non-linear (geometric, constitutive and boundary) model of the brain parenchyma. Finite deformation formulations were used to compute the deformations of the brain parenchyma. An Ogden type, hyperelastic constitutive model for the brain parenchyma was used. The brain parenchyma was treated both as biphasic and single phase continuum. The sub-arachnoid s space was accounted for, by introducing a 3 mm gap between the brain outer surface and the skull. To constrain the nodes on the outer surface of the brain within the skull, a frictionless, finite sliding, node-to-surface penalty contact was introduced between the brain and the skull. The brain parenchyma was loaded with transmantle pressure difference of 1 mm of Hg ( Pa) to investigate the suggestion that no more than 1 mm of Hg transmantle pressure difference (Penn et al.; 2005 and Czosnyka; 2006) is associated with NPH. The work presented in this chapter showed that there is no significant advantage gained by treating the brain parenchyma as biphasic continuum for computing ventricular volume. To model NPH, it is proposed that the brain parenchyma be treated as nearly incompressible single phase continuum. The single phase model for the brain parenchyma simplifies the mathematical description of the NPH model and reduces the computational time (483 minutes for the single phase incompressible model compared to 2921 minutes for the biphasic model on Pentium 4, 3.2 GHz, 2 GB RAM personal computer). 48

60 Chapter 3 - Biomechanics of Normal Pressure Hydrocephalus According to the results, transmantle pressure difference of unto 1 mm of Hg ( pa) is not sufficient to produce the condition of NPH. To the best of our knowledge, the present work is the first conclusive demonstration of this using modelling and simulation techniques, adding strength to the previous, purely experimental studies (Penn et al., 2005). This suggests that the hypothesis of a purely mechanical basis for NPH growth needs to be revised. 49

61 CHAPTER 4 - MODELLING THE BRAIN STRUCTURE FOR NORMAL PRESSURE HYDROCEPHALUS: BI-PHASE OR SINGLE PHASE? 4.1 Introduction Over the years, a lot of effort has gone into investigating various phenomena affecting the brain using computational biomechanics (e.g. hydrocephalus and tumour growth, drug delivery, intra-operative image registration, surgical simulation etc.). A mathematical framework for the model of the brain tissue is needed to analyse these events. Generally, bi-phase (soil consolidation) theory [e.g. intra-operative neuro-image registration, Normal Pressure Hydrocephalus modelling etc.] or single phase continuum theory [e.g. needle insertion mechanics, surgical simulation, intra-operative neuro-image registration, injury biomechanics etc.] is used to model the brain parenchyma (please see: Chapter 2, Literature Review, Section 2.1 and 2.3; Chapter 3, Biomechanical Model of Normal Pressure Hydrocephalus; Bilston et al., 2001; Brands et al., 2004; Donnelly and Medige, 1997; Hrapko et al., 2006, Hrapko et al., 2009; King, 2000; Lunn et al., 2006; Miga et al., 1999; Mendis et al., 1995; Nicolle et al., 2004; Ning et al., 2006; Paulsen et al., 1999; Platenik et al., 2002; Shen et al., 2006; Takhounts et al., 2003; Wittek et al., 2007; Wittek et al., 2008; Yang et al., 2003; Zhang et al., 2001; Zhang et al., 2001; Zhang et al., 2002; Zhang et al., 2004 etc.). Portions of this chapter have been submitted for review in JSME International Journal Series A: Journal of Solid Mechanics and Material Engineering: Dutta-Roy, T., Miller, K.,, Wittek, A., On Appropriateness of Modelling Brain Parenchyma as Biphasic Continuum

62 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus Recently, Franceschini et al. (2006) claimed to have presented direct experimental evidence to support the hypothesis that brain tissue obeyed soil consolidation theory and hence was biphasic. However, in the previous chapter (Chapter 3, Biomechanical Model of Normal Pressure Hydrocephalus), it was proposed that the brain parenchyma be treated as nearly incompressible single phase continuum for modelling Normal Pressure Hydrocephalus (NPH) as treating the brain parenchyma as biphasic continuum for computing ventricular volume during NPH had no significant advantages. In this chapter, simple experiments were conducted using samples of lamb brain tissue to verify as to what extent the parenchyma responses are consistent with the models using soil consolidation theory (Biot, 1941, Bowen, 1976; Miller, 1998; Taylor and Miller, 2004; Franceschini et al., 2006 and Cheng and Bilston, 2007). The results indicated that the brain parenchyma behaviour is incompatible with this theory. Section 4.2 describes the experimental setup, experimental procedure and results of the experiments. Detailed discussion is given in Section Experiments Specimen Preparation A lamb brain (weight: 104 grams) was obtained from the butcher. The brain was a by-product of commercial slaughter process. The age of the animals at slaughter was 51

63 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus approximately 6 months. There was a delay of around 12 hours between the slaughter of the lamb and the time we conducted the experiments. The brain was not frozen at any time during the procedure. Region of brain surface without Pia mater Pia mater present on brain surface Figure 4.1: Brain surface after removal of Pia The Pia mater was carefully teased out from the Sulci features on the brain surface using two pairs of DuPont s Swiss Tweezers Number 7. Thereafter, with gentle manipulations, the Pia mater was torn from the brain surface (Figure 4.1). An approximately cylindrical sample (diameter ~ 30mm and height ~ 20mm) was cut out of 52

64 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus Transparent plastic tube of length 85 cm (2) Die with inserted cylindrical sample (1) Figure 4.2: Assembled experimental setup with 85 cm CSF solution column applying pressure on cylindrical brain sample inserted into the die. Blue colouring agent (India ink) was mixed with the CSF solution for photography purpose. In the original experiments, no colouring agent was present. 53

65 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus the region of the brain from which Pia mater was removed, using a sharp cylindrical punch and scalpel Experimental Setup The experimental setup is shown in Figure 4.2 (previous page). It consisted of a cylindrical die (1) and transparent plastic tube of length 85 cm (2). Knife Edge Taper O Ring Figure 4.3a: Top view of the cylindrical die 54

66 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus Wire Mesh Figure 4.3b: Bottom view of the cylindrical die A taper was provided near the base of the cylindrical die. Also a knife edge was machined on its base (Figures 4.3a, 4.3b). To ensure proper sealing between the transparent plastic tube and cylindrical die, an O ring was machined into the die (Figures 4.3a, 4.3b). A wire mesh (mesh size: 2mm) was attached to the bottom of the die. 55

67 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus Experimental Procedure Artificial CSF solution was prepared and the chemical composition was 148 mm NaCl, 3 mm KCl, 1.4 mm CaCl 2.2H 2 0, 801 mm MgCl 2.6H 2 0, 800 mm Na 2 HPO 4.7H 2 0 and 225 mm NaH 2 PO 4 dissolved in double distilled water. Cylindrical brain sample Die Figure 4.4: Cylindrical sample inserted into the die 56

68 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus The cylindrical brain sample was inserted into the die (Figure 4.4). It was insured that the brain cylindrical sample did not deform much, while inserting it into the die. Thereafter, the transparent plastic tube was snapped fixed to the die. The experimental setup was placed in a beaker. CSF solution was poured into the transparent plastic tube. Consequently, the cylindrical brain sample inserted into the die was subjected to pressure from a column of CSF solution in the transparent plastic tube (Figure 4.2). Table 4.1: Height of artificial CSF column applied on the brain sample Load Case Height of artificial CSF Column (cm) Comments (Milhorat, 1972) Load Case 1 10 cm or 981 Pa Normal CSF pressure in ventricles Load Case 2 20 cm or 1962 Pa CSF pressure in ventricles during Normal Pressure Hydrocephalus Load Case 3 85 cm or Pa CSF pressure in ventricles during High Pressure Hydrocephalus 57

69 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus As the Pia Mater was torn from the brain surface, the brain parenchyma was exposed to the artificial CSF solution (Figure 3a). The ventricular surface of the brain parenchyma sample was exposed to the wire mesh on the bottom end of the die (Figure 3b). Under exerted CSF solution pressure, the taper and knife edge in the cylindrical die combined with the sticky character of the brain, to form a water tight seal between the brain sample and the die. This effectively split the die into two separate compartments. Three separate heights of CSF solution column were applied to the cylindrical brain sample inserted in the die and are summarised in Table 4.1. Each height of the CSF solution column was applied for a period of 120 mins. This time frame was chosen to prevent deterioration of the brain tissue sample, yet allow enough time for the CSF solution from the tube to percolate through the brain tissue. Tight seal between the die and transparent plastic tube was obtained due to the O ring machined into the die. No CSF solution leakage resulted from the transparent plastic tube-die connection due to the tight seal Results and their Comparison to Consolidation Theory Predictions For all the three load cases, check was made after 120 minutes to observe any leakage of CSF solution into the beaker through the brain tissue. No CSF solution leakage through the brain tissue was observed. Also, no change in the height of the CSF solution column in the transparent plastic tube was observed. 58

70 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus Soil consolidation theory (ABAQUS/Standard, 2004) was used to model hydrocephalus (Nagashima et al., 1987; Kaczmarek et al., 1997; Péna et al., 1999 and Taylor and Miller, 2004 and Momjian et al., 2008) and intra-operative image registration of brain deformation during neurosurgery (Miga et al., 1999; Paulsen et al., 1999; Miga et al., 2000; Platenik et al., 2002 and Lunn et al., 2006). In our case, for a cylindrical brain sample subjected to pressure from a column of CSF solution undergoing no deformation, the soil consolidation theory (ABAQUS/Standard, 2004) reduces to Darcy s Law (ABAQUS/Standard, 2004). Darcy s law is given by (ABAQUS/Standard, 2004): snv k x (4.1a) u z g (4.1b) where: s is the saturation, n is the porosity of the medium, v ω is the seepage velocity (m/sec), k is the hydraulic conductivity of the medium (m/sec), φ is the piezometric head (m), u ω is the pressure of wetting fluid (Pa), ρ ω is the density of the wetting fluid (kg/m 3 ), z is the elevation above some datum (m) and g is the magnitude of the gravitational acceleration (m/sec 2 ) which acts in the direction opposite to z. There are various values for the hydraulic conductivity (k) of the brain parenchyma given in the literature as summarised in Table 4.2. The sample was approximately cylindrical with a diameter ~ 30 mm and height ~ 20 mm. For load case 59

71 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus 3 (section 2.3: 85 cm column of CSF solution), the total volumetric flow of CSF solution for different hydraulic conductivity (as given in literature: Table 4.2) through the cylindrical sample over 2 hours was calculated (Equations 4.1a and 4.1b) and is summarised in Table 4.2. Table 4.2: Total volume flow of CSF solution through the cylindrical brain parenchyma sample Hydraulic Conductivity (m/sec) Total CSF volume flow in 2 hours (ml) 1.37 X 10-7 (Smillie et al., 2005) X 10-7 (Kaczmarek et al., 1997) 2.42 X (Franceschini et al., 2006) 8.11 X 10-8 (Cheng and Bilston, 2007) Discussions As mentioned earlier, a number of researchers have used biphasic (soil consolidation) continuum theory to model the brain parenchyma. (Nagashima et al., 60

72 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus 1987; Kaczmarek et al., 1997; Miller, 1998; Miga et al., 1999; Paulsen et al., 1999; Péna et al., 1999; Miga et al., 2000; Platenik et al., 2002; Lunn et al., 2006; Taylor and Miller, 2004 and Momjian et al., 2008). For the biphasic theory (soil consolidation), Darcy s Law (Equation 4.1a and 4.1b) models the flow of wetting liquid through the solid phase. Simple calculations using Darcy s Law (section and Table 4.2), showed that in 2 hours, at-least some amount of CSF solution should have leaked through the cylindrical brain sample. As shown in Section 4.2.4, no percolation of CSF solution through the brain tissue into the beaker was seen. The experimental observations are in direct opposition to the total volumetric flow of CSF solution as predicted by Darcy s Law (Table 4.2). Therefore, it can be argued that soil consolidation theory is not in agreement with lamb brain tissue behaviour. It can be noted here that the extremely low value of hydraulic conductivity used by Franceschini et al. (2006), essentially prevents any substantial flow through the solid matrix and therefore makes the biphasic model almost equivalent to single phase models at physiological pressure loads. The criticism may arise that in our in-vitro experiments brain cells could have swollen, consequently significantly decreasing the permeability that may be present invivo. We attempted to minimise this possible effect by using artificial CSF for pressure loading. It may be worth noting however that cell swelling has not been considered by other experimenters either. In particular recent works by Franceschini et al. and Cheng and Bilston did not consider cell swelling and still concluded, in opposition to results presented here, that the brain tissue in-vitro was biphasic. 61

73 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus Furthermore, Abbott (2004) presents a review for evidence of bulk blow of CSF through the brain parenchyma. The review showed that there are multiple mechanisms (pressure gradient, concentration gradient etc.) for bulk flow through the brain parenchyma and pin-pointing the exact mechanism is controversial. Moreover, it should be noted that flow rates of μlmin-1gbrain-1 for rats and μlmin-1gbrain-1 for rabbits were recorded. These rates could be regarded as bulk flow and be significant for some phenomenon associated with the brain (drug therapy, immune surveillance and inflammation, stem cell therapies etc.). But for phenomenon such as modelling Normal Pressure Hydrocephalus (NPH), brain shift etc, such extremely low flow rates are insignificant. This further strengthens our argument that extremely low value of hydraulic conductivity (Table 2), essentially prevents any substantial flow through the solid matrix and therefore makes the biphasic model almost equivalent to single phase models at physiological pressure loads. A further criticism of our work is that the interstitial gap in the brain tissue is typically in the order of a few nanometres and as a result the capillary forces are high. Therefore, high capillary forces would not let the artificial CSF flow through the brain parenchyma sample. However, the interstitial gap in clayey soils is in the order of a few angstroms (Mitchell, 1993) and no capillary effect is seen. Fluid passes through the clayey soil, albeit at a very slow rate. Furthermore, soil consolidation theory is still used to understand and predict the deformation of the clayey soil and the effective stresses in it due to fluid flow. 62

74 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus In conclusion, the experiments showed that the lamb brain tissue behaviour is inconsistent with the soil consolidation theory (Biot, 1941 and Bowen, 1976). If there is a need to model fluid flow within the brain parenchyma, for applications such as drug delivery or nutrient transfer, using principles of transport phenomena (Bird et al., 1960) should be considered. Further, in light of the findings presented above, the explanation of the phenomena of Periventricular Lucency (Chapter 3, Biomechanics of Normal Pressure Hydrocephalus, Section 3.6.2; Taylor and Miller, 2004 and Dutta-Roy et al., 2008) based on biphasic model of the brain parenchyma appears to have been accidental and probably needs to be discarded. 4.4 Summary In this chapter, simple experiments were conducted using samples of lamb brain tissue to verify as to what extent the brain parenchyma responses are consistent with the models using soil consolidation theory. Approximately cylindrical sample (diameter ~ 30mm and height ~ 20mm) were obtained from a lamb brain. The brain was a byproduct of commercial slaughter process and obtained from the butcher. Prior to acquiring the approximately cylindrical samples from the brain parenchyma, the Pia mater was carefully teased out from the Sulci features and was torn from the brain surface with gentle manipulations. The experimental setup consisted of a cylindrical die and transparent plastic tube of length 85 cm. The cylindrical brain sample was inserted into the die and the 63

75 Chapter 4 Nature of Brain Structure for Modelling Normal Pressure Hydrocephalus transparent plastic tube was snapped fixed to the die. An O ring was machined into the die to ensure proper sealing between the transparent plastic tube and cylindrical die. The experimental setup was placed in a beaker. Artificial CSF solution was made and poured into the transparent plastic tube. As a result, the brain sample in the die was subjected to pressure from a column of CSF solution. Three separate heights of CSF solution column (10 cm, 20 cm and 85 cm) were applied to the cylindrical brain sample inserted in the die. No CSF solution leakage through the brain tissue was observed for all the three load cases. Hydraulic conductivity (k) as given in the literature (1.37 X X m/sec) was used in the Darcy s Law to calculate the volume of CSF solution flow through the cylindrical samples after 120 minutes. Our experimental observations (no flow of CSF through the cylindrical brain sample) are in direct opposition to the total volumetric flow of CSF solution as predicted by Darcy s Law. Therefore, it can be argued that soil consolidation theory is not in agreement with lamb brain tissue behaviour. Alternative mathematical formulations (e.g. transport theory) maybe used to model the flow of fluid through the brain parenchyma. Further, the explanation of Periventricular Lucency based on biphasic continuum (Chapter 3, Biomechanics of Normal Pressure Hydrocephalus, Section 3.6.2) needs to be discarded. 64

76 CHAPTER 5 CAN VARIATIONS IN BRAIN PARENCHYMA STIFFNESS DUE TO PULSATILE BLOOD FLOW EXPLAIN NORMAL PRESSURE HYDROCEPHALUS? 5.1 Introduction A lot of effort has been made to understand the phenomenon of Normal Pressure Hydrocephalus (Literature Review, Chapter 2; Biomechanics of Normal Pressure Hydrocephalus, Chapter 3), but it continues to perplex clinicians. Enhanced diagnosis methodologies for Normal Pressure Hydrocephalus (NPH) have evolved (Literature Review, Chapter 2, Section 2.3) but these methods offer limited understanding of the biomechanics behind NPH growth. It is widely believed that NPH has biomechanical causes (please see Chapter 2, Literature Review, Section 2.3 and Chapter 3, Biomechanics of Normal Pressure Hydrocephalus), but these models are unable to co-relate their findings with clinical observations (Chapter 3, Biomechanics of Normal Pressure Hydrocephalus and Dutta- Roy et al., 2008). Clinical techniques used to enhance the diagnosis of NPH (Shulman et al., 1968; Marmarou et al., 1978; Czosnyka and Pickard, 2004; Czosnyka et al., 2004; Linninger et al., 2005 etc) indicated that the pulsations observed in Intracranial Pressure (ICP) recordings of NPH patients could explain the growth of NPH. Therefore, in this chapter, a fully non-linear (geometric, constitutive and boundary conditions) 3-D model Portions of this chapter have been published in: Dutta-Roy, T., Wittek, A., Miller, K., Can Vascular Dynamics Cause Normal Pressure Hydrocephalus?, Computational Biomechanics for Medicine IV Workshop, 12 th MICCAI, Sept 2009, London, pp

77 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? of the brain parenchyma was used to investigate the suggestion that variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle) contributed to NPH growth. The biomechanical and computational models are discussed is Sections 5.2 and 5.3 respectively. The results of our calculations are summarised in Section 5.5 followed by discussions in Section Biomechanical Model The biomechanical model is detailed in Chapter 3 (Biomechanics of Normal pressure Hydrocephalus, Section 3.2) and hence is not repeated here. As shown previously in Chapter 3 (Biomechanics of Normal Pressure Hydrocephalus, Section 3.5.1, Tables 3.5 and 3.6 and Dutta-Roy et al., 2008), that under equal transmantle pressure difference load, there were no significant differences between the computed ventricular volumes for biphasic and incompressible/nearly incompressible, single phase model of the brain parenchyma. Consequently, in this chapter, the brain parenchyma was modelled as nearly incompressible, single phase continua (Poisson s Ratio (υ) = 0.49). The relaxed shear modulus (μ ) of the brain parenchyma was Pa (Chapter 3, Biomechanics of Normal Pressure Hydrocephalus, Section 3.2.3, Table 3.3; Miller and Chinzei, 2002; Taylor and Miller, 2004 and Dutta-Roy et al., 2008). 66

78 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? 5.3 Computational Model The computational model is detailed in Chapter 3 (Biomechanics of Normal pressure Hydrocephalus, Section 3.3) and hence is not repeated here. Only the computational model for implementing the variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle) is given in the subsequent section Brain Parenchyma Model As mentioned earlier, clinical techniques used to enhance the diagnosis of NPH (Shulman et al., 1968; Marmarou et al., 1978; Czosnyka and Pickard, 2004; Czosnyka et al., 2004; Linninger et al., 2005 etc) indicated that the pulsations observed in Intracranial Pressure (ICP) recordings of NPH patients could explain the growth of NPH. Hence, the suggestion that variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle) contributed to NPH growth was investigated. During systolic part of the cardiac cycle, the heart pumps in blood through the vascular tree whereas in the diastolic part, the blood recedes through it. As a result, the brain parenchyma is stiffer during systolic part as compared to the diastolic part. This effect was implemented by a time varying shear modulus. 67

79 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? Figure 5.1: Methodology used for implementing time varying shear modulus 68

80 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? Ogden type, hyperelastic constitutive model of the brain parenchyma (Chapter 3, Biomechanics of Normal Pressure Hydrocephalus, Section 3.2.3; Miller and Chinzei, 1997; Miller and Chinzei, 2002 and Dutta-Roy et al., 2008) was implemented using the HYPERELASTIC function (ABAQUS/Standard, 2004). It does not provide for variation of the shear modulus with time. However, it allows for variation of shear modulus as a function of the field variable associated with temperature (ABAQUS/Standard, 2004). This property of the HYPERELASTIC function was used to implement a time varying shear modulus. As a first step, the temperature field variable was varied with time using a fully coupled thermal-stress analysis [COUPLED TEMPERATURE DISPLACEMENT procedure (ABAQUS/Standard, 2004)]. The thermal conductivity, expansion and heat flux going into and out of the brain parenchyma was set to zero. Hence, no thermal stresses were generated as the temperature field variable was changed with time and the thermo-elastic response of the brain parenchyma was negligible. As a result of the fully coupled thermal-stress analysis, a time varying temperature field distribution was setup. Thereafter, the brain parenchyma material properties (shear and bulk modulus) was specified for each value of the temperature field distribution (Table 5.1) using the HYPERELASTIC function (ABAQUS/Standard, 2004). As the temperature field increased with time and the shear modulus varied with temperature, a time varying shear modulus was achieved with this technique. The method is summarised as a block diagram in Figure

81 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? Table 5.1: Time varying shear modulus function and computed ventricular volumes under 1 mm of Hg transmantle pressure difference; μ = Pa is the relaxed shear modulus (Chapter 3, Section 3.2.3; Dutta Roy et al., 2008 and Taylor and Miller, 2004), ω is the heart rate frequency (72 beats/min or 7.5 rad/sec) Time varying shear modulus [The relaxed shear modulus ( ) was Load application modified according to the cardiac cycle taken from Linninger et al., 2005] time (secs) period π 1 π Case 1: μ μ μ [1.3 sin(ω t ) cos(2ωt )] average = 236 Pa, maximum = 314 Pa, minimum = 155 Pa π 1 π Case 2: μ μ μ [1.3 sin(ω t ) cos(2 ωt )] average = 236 Pa, maximum = 314 Pa, minimum = 155 Pa π 1 π Case 3: μ μ 0.5μ [1.3 sin(ω t ) cos(2ωt )] average = 196 Pa, maximum = 235 Pa, minimum = 155 Pa π 1 π Case 4: μ μ 0.25μ [sin(ω t ) cos(2ωt )] average = 156 Pa, maximum = 206 Pa, minimum = 105 Pa π 1 π Case 5: μ μ 0.5μ [sin(ω t ) cos(2ωt )] average = 157 Pa, maximum = 257 Pa, minimum = 55 Pa 70

82 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? As no study currently exists on the effects of blood flow (according to the cardiac cycle) on brain parenchyma material properties, a parametric investigation was conducted by varying the brain parenchyma relaxed shear modulus. The relaxed shear modulus (μ ) was modified according to the cardiac cycle [taken from Linninger et al., (2005)] (Table 5.1) and a time varying shear modulus (Figure 5.2) was produced. Figure 5.2: Sinusoidal time varying shear modulus (Table 5.2; Linninger et al., 2005). For Cases 1 5, the heart beat frequency is 72 beats/min. Hence the shape of the function remains the same. Only the relaxed shear modulus average and amplitude varies. 71

83 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? 5.4 Finite Element Solver ABAQUS-Standard non-linear implicit finite element code (ABAQUS/Standard, 2004) was used to obtain the solution for the NPH biomechanical model. The code accounts for constitutive, geometric (finite deformation) and contact non-linearities. Material strain rate effects are absent for the hyperelastic material law that was used for the brain parenchyma. But to model the variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle), the relaxed shear modulus (μ ) was varied with time (Table 5.1 and Figure 5.2). Thus, a quasi-static solution was needed and the VISCO (ABAQUS/Standard, 2004) procedure was used to solve the model of NPH. As mentioned earlier, the brain parenchyma was loaded with 1 mm of Hg transmantle pressure difference (Penn et al., 2005; Czosnyka, 2006 and Dutta-Roy et al., 2008) and the load was applied over the time periods summarised in Table 5.1. NPH generally develops over 4 days (Nagashima et al., 1987). However, computing our model over the time period taken for development of NPH would require enormous computational resources. As strain rate dependency was not included in our model, the results of our computations should not differ if we apply the pressure load over 4 days, 100 minutes or 60 seconds. To confirm this, the load was applied over 600 seconds and 60 seconds for Cases 1 and 2. For Cases 1 and 2 (Table 5.1), there was practically no difference between computed ventricular volumes. Hence for Cases 3-5 (Table 5.1), the transmantle pressure difference load was applied over 60 seconds. 72

84 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? Table 5.2: Computed ventricular volumes for time varying shear modulus. NPH developed when ventricular volume increased from 14 cm 3 to 58 cm 3. Time varying shear modulus [The relaxed shear modulus ( ) was modified according to the cardiac cycle taken from Linninger et al., 2005] Computed Ventricular Volume (cm 3 ) Case 1: average = 236 Pa, maximum = 314 Pa, minimum = 155 Pa 32.7 Case 2: average = 236 Pa, maximum = 314 Pa, minimum = 155 Pa 32.6 Case 3: average = 196 Pa, maximum = 235 Pa, minimum = 155 Pa 34 Case 4: average = 156 Pa, maximum = 206 Pa, minimum = 105 Pa 43.7 Case 5: average = 157 Pa, maximum = 257 Pa, minimum = 55 Pa

85 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? 5.5 Computational Results Computed ventricular volumes for time varying shear modulus are presented in Table 5.2. NPH develops when the ventricular volume increased from 14 cm 3 to more than 58 cm 3 (Matsumae et al., 1996). 1 mm of Hg transmantle pressure difference cannot produce NPH for Cases 1-4 (Table 5.2), even with variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle) taken into account. For Case 5, NPH was produced. 5.6 Discussions For cases 1-4 (Table 5.2), 1 mm of Hg transmantle pressure difference load did not produce NPH even with variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle) taken into account. The period of the cardiac cycle is multiple orders of magnitude smaller than the time taken for clinical development of NPH (hour to days). Even after applying the transmantle pressure difference load quickly, over a smaller time scale (60 seconds), NPH could not be produced (Table 5.2, Cases 1-4). However for Case 5 (Table 5.1), as the modified relaxed shear modulus varied sinusoidally, it decreased to a low value (55 Pa) for short time periods. This combined with increasing transmantle pressure difference load produced NPH. If a biological phenomenon exists which reduces the brain parenchyma material parameters to a value as low as 55 Pa, even for a short periods of time, then development of NPH by purely mechanical means is plausible. 74

86 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? A transmantle pressure difference of unto 1 mm of Hg ( Pa) could not produce the clinical condition of NPH, even with variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle) taken into account. Following cases 1-4, it can be argued that it is highly unlikely for a phenomenon which occurs many times a minute (in this case: variation in brain parenchyma stiffness due to the cardiac cycle), to be able to influence an event such as NPH which presents itself over a time scale of hours to days. The results presented in the current chapter, taken together with Chapter 3 (Biomechanics of Normal Pressure Hydrocephalus) and Dutta- Roy et al. (2008) adds strength to the suggestion that the hypothesis of a purely mechanical basis for NPH growth needs to be revised. 5.7 Summary Previous mechanical based models investigating the biomechanics of NPH growth are unable to co-relate their findings with clinical observations. Clinical techniques used to enhance the diagnosis of NPH indicate that there are abnormal Intracranial Pressure (ICP) recordings in patients with NPH. Hence, a fully non-linear (geometric, constitutive and boundary conditions) 3-D model of the brain parenchyma was used to investigate the suggestion that variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle) contributed to NPH growth. A person specific brain mesh was modified and a generic mesh of a human brain was created. The brain parenchyma was modelled as a nearly incompressible, single phase continuum. A hyperelastic constitutive law and finite deformation theory 75

87 Chapter 5 Can Variations in Brain Parenchyma Stiffness Due to Pulsatile Blood Flow Explain Normal Pressure Hydrocephalus? described the deformations within the brain parenchyma. The relaxed shear modulus was modified according to the cardiac cycle to produce a time varying shear modulus. As no study currently exists on the effects of blood flow (according to the cardiac cycle) on brain parenchyma material properties, a parametric investigation was conducted by varying the brain parenchyma relaxed shear modulus. It is widely believed that no more than 1 mm of Hg ( Pa) transmantle pressure difference is associated with NPH. Hence, the brain parenchyma was loaded with 1 mm of Hg transmantle pressure difference to investigate this suggestion. Fully non-linear, implicit, quasi-static finite element procedures in the time domain were used to obtain the deformation of the brain parenchyma and the ventricles. The results of the simulations showed that 1 mm of Hg ( Pa) did not produce the clinical condition of NPH, even with variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle) taken into account. However, if a biological phenomenon exists which reduces the brain parenchyma material parameters to a value as low as 55 Pa, even for a short period of time, then development of NPH by mechanical means is plausible. The work in the current chapter showed that it is highly unlikely for a phenomenon, such as the variation of brain parenchyma stiffness due to the cardiac cycle, which occurs many times a minute to be able to influence an event such as NPH which presents itself over a time scale of hours to days. This result further adds strength to the hypothesis that a purely mechanical basis for NPH growth needs to be revised. 76

88 CHAPTER 6 CONCLUSIONS AND FUTURE WORK Computational mechanics and Finite Element Method (FEM) are powerful and reliable tools for understanding complex physical phenomena including biological and medical events. The viability of FEM was shown in the Literature Review (Chapter 2) and especially in the Section 2.2 (Biomechanical Modelling: An Example) where it was used to investigate the mechanics of needle insertion into brain tissues. The computed force-displacement relationship showed good agreement with experimental results. Further, it confirms that to accurately model complex medical phenomena, subject specific fully non-linear biomechanical computational model is needed. The mathematical and computational basis used to model needle insertion into brain tissue was further applied to investigate the mechanics of Normal Pressure Hydrocephalus (NPH). A realistic 3-D model of the brain geometry and fully non-linear Finite Element (FE) procedures were used to investigate the growth of NPH (Chapter 3, Biomechanics of Normal Pressure Hydrocephalus). Clinicians generally accept that at most 1 mm of Hg transmantle pressure difference ( Pa) is associated with the condition of NPH. The computational results from the fully non-linear model of NPH showed that transmantle pressure difference of 1 mm of Hg ( Pa) did not produce NPH for either single phase or biphasic model of the brain parenchyma. A minimum transmantle pressure difference of mm of Hg ( Pa) was required to produce the clinical

89 Chapter 6 Conclusions and Future Work condition of NPH. This suggested that the hypothesis of a purely mechanical basis for NPH growth needs to be revised. Additionally, it was also showed that under equal transmantle pressure difference load, there were no significant differences between the computed ventricular volumes for biphasic and incompressible/nearly incompressible single phase model of the brain parenchyma. As a result, there was no advantage gained by using a biphasic model for the brain parenchyma. It is proposed that for modelling NPH, nearly incompressible single phase model of the brain parenchyma was adequate. Clinical techniques used to enhance the diagnosis of NPH indicated that the pulsations observed in Intracranial Pressure (ICP) recordings of NPH patients could explain the growth of NPH. Therefore, a fully non-linear 3-D model of the brain parenchyma was used to investigate the suggestion that variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle) contributed to NPH growth. The computational results showed that 1 mm of Hg transmantle pressure difference cannot produce NPH even with variation in brain parenchyma stiffness due to blood flow (according to the cardiac cycle) taken into account. However, if a biological phenomenon exists which reduces the brain parenchyma material parameters to a value as low as 55 Pa, even for short periods of time then development of NPH by purely mechanical means is plausible. Following the computational results, it can be argued that it is highly unlikely for a phenomenon which occurs many times a minute (in this case: variation in brain parenchyma stiffness due to the cardiac cycle), to be able to influence an event such as NPH which presents itself over a time scale of hours to days. This result further adds strength to the hypothesis that a purely mechanical basis for NPH growth needs to be revised. 78

90 Chapter 6 Conclusions and Future Work To fully investigate and understand the mechanics of needle insertion into brain tissues, computing needle force for the whole duration of the procedure is required. The work presented in this thesis focused on maximum deformation of the brain and the Pia Mater just before failure of Pia Mater. Computing needle forces during the entire insertion process would require modelling of puncture (i.e. mechanical failure) of at least the Pia Mater and the brain tissue. This would require in-depth experimental studies to determine the failure mechanisms and criteria for the brain tissue and Pia Mater. To model the failure of Pia Mater and brain tissue, robust computational techniques (e.g. meshless methods) which can deal with topological changes of the brain model (e.g. cutting, insertion, suturing etc) encountered during needle insertion need to be developed further. Even though there is progress in clinical research on identification, differentiation, diagnosis and treatment of Normal pressure Hydrocephalus (NPH); the cause for NPH growth has not been established. Several generation of researchers postulated that NPH has mechanistic causes. The development of NPH by purely mechanical means is plausible if there is a biological phenomenon which reduces the brain parenchyma material parameters to a value as low as 55 Pa, even for short periods of time. This possibility should also be investigated further. It is shown in this thesis that the hypothesis of a purely mechanical basis for NPH growth needs to be revised. Therefore, alternative theoretical framework is required to analyse the growth of NPH. It is suggested that the biochemical basis (using transport theory) for NPH growth be further explored. Recently, transport theory was 79

91 Chapter 6 Conclusions and Future Work used to model infusion studies in NPH patients. It is shown in this thesis that consolidation (biphasic) theory is not in agreement with lamb brain tissue behaviour. Alternative mathematical formulations need to be used to model the flow of fluid through the brain parenchyma. It is suggested that transport theory be used to analyse the flow of fluid through brain parenchyma. The coupling of transport phenomena with non linear deformation equations of solid mechanics could be a powerful tool for analysis of the reasons behind NPH growth. Alternatively, the possibility of treating the mechanics of NPH growth as a Fluid- Structure Interaction (FSI) problem should also be investigated. The production and flow of CSF within the Central Nervous System (CNS) is currently an active field of research using Computational Fluid Dynamics (CFD). The coupling of CSF flow and deformations within the CNS (using equations of solid mechanics) gives rise to a FSI problem. This approach could also help in explaining the mechanics of NPH growth and should be investigated further. 80

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113 APPENDIX A BIOMECHANICAL MODELLING: AN EXAMPLE A.1 Biomechanical Modelling: An Example 1 A.1.1 Mechanics of Needle Insertion into Swine Brain: Background Advantages of surgical robots and manipulators are well recognised in clinical and technical communities. These include precision and accuracy as well as potential for remote surgery. Precision and accuracy are key factors when performing the most common neurosurgical procedure: needle insertion into the brain. Planning needle insertion is seemingly simple: a needle is aimed at an anatomical target whose location can be determined from pre-operative radiological images (e.g., magnetic resonance images or MRI). In reality, reaching the target requires advanced surgical skills. This is because of the change of target position caused by brain tissue deformation under the load imposed by the needle. Thus, trajectory planning for the manipulators performing a surgical procedure, such as needle insertion into the brain, must include information about the current, i.e. intraoperative organ deformation. Although 1 Portions of this section have been published in: Wittek, A., Dutta-Roy, T., Taylor, Z., Horton, A., Washio, T., Chinzei, K. and Miller, K., Subject Specific Non-Linear Biomechanical Model of Needle Insertion into Brain, Computer Methods in Biomechanics and Biomedical Engineering, 2008, 11 (2), Wittek A., Dutta-Roy T., Miller K., Biomechanics of Needle Insertion into Brain Using Non Linear Biomechanical Model, Proceedings of the 7 th International Symposium on Computer Methods in Biomechanics and Biomedical Engineering, 22 nd - 25 th March 2006, Antibes, Cote D Azur, France, pp Dutta-Roy, T., Wittek A., Taylor, Z., Chinzei, K., Washio T., Miller, K., Towards Realistic Surgical Simulation: Biomechanics of Needle Insertion into Brain, Proceedings of the 16 th CISM-IFToMM Symposium on Robot Design, Dynamics and Control, RoManSy 2006 Symposium, Warsaw University of Technology, Warsaw, Poland, 20 th - 24 th June 2006, pp

114 such information can be obtained using intraoperative MRI, currently tens of seconds are required to produce and analyse a new set of images. One possible method of dealing with these delays could be predicting organ deformation by means of computer simulations. When organ deformation can be predicted and the tissue mechanical properties are known, interaction force between the end-effecter (i.e. needle or other surgical tool) and the organ surface during surgery can also be computed. Thus, the simulation can be used to predict reaction forces acting on the needle as well as to provide realistic force and tactile feedback for virtual reality surgical simulators. Realistic simulations cannot be done without thorough understanding of the biomechanics of needle insertion. Yet, most of the previous studies on needle insertion have been focused on haptics & visual feedback (Brett et al., 1997; DiMaio and Salcudean, 2002; DiMaio and Salcudean, 2003; DiMaio and Salcudean, 2005; Gerovich et al., 2004; Spicer et al., 2004 etc.) and surgery planning (O Leary et al., 2003; Alterovitz et al., 2005; Heverly et al., 2005 etc.). These studies have two major drawbacks: the use of linear elasticity and infinitesimal deformation theory for computing tissue deformations and of phenomenological models for computing force acting on the needle. With the exception of Simone and Okamura (2002) and Washio and Chinzei (2004), only a few studies attempted to understand the biomechanics of needle insertion. Despite experimental evidence (Miller and Chinzei, 1997; Miller and Chinzei, 2001; Miller, 2002; Miller et al., 2005) that the brain and other soft tissues exhibit nonlinear constitutive behaviour and the obvious fact that during needle insertion procedure, the tissue undergoes large deformations, studies conducted by Simone et al. (2002) and 103

115 O Leary et al. (2003) rely on linear stress-strain relationship and assume infinitesimal deformations when calculating organ deformations and the needle reaction forces. Notable exceptions to the above are Nienhuys (2003) [non-linear finite deformation analysis to predict deformation of soft objects subjected to cutting] and Heverly et al. (2005) [fracture mechanics approach to model needle insertion]. The work presented in this section employs finite deformation procedures and a non-linear constitutive model of the brain tissue to calculate the forces acting on the needle during insertion. This gives higher reliability for the calculated forces. Subsequent parts of the present Section deal with needle insertion experiments into swine brain (Section A.1.2), establishing biomechanics of needle insertion (Section A.1.3), computational model for insertion simulation (Section A.1.4), results and validation of the computational model (Section A.1.6) and detailed discussions (Section A.1.7) A.1.2 Biomechanics of Needle Insertion: Experiments 2 Four swine brains were obtained from a company specialising in extracting organs from commercially slaughtered animals (Tokyo Shibaura Zooki, Tokyo, Japan). The animals were six months old at the time of slaughter and the brains were by-product of the slaughter process. The brain weights are summarised in Table A.1. The needle insertions were conducted on all the four brains. Preliminary experiments were 2 The experiments were conducted at the facilities of Surgical Assist Technology Group, AIST, Tsukuba, Japan. The needle insertion experiments were designed and instrumented by Dr Toshikatsu Washio and Dr Kiyoyuki Chinzei at the Surgical Assist Technology Group, AIST, Tsukuba. I was a part of the team which conducted the experiments and analysed the results. 104

116 conducted on brain number 1. Therefore, only the results of needle insertion on brain numbers 2, 3 and 4 were analysed. Table A.1 Weight of Swine brain used for needle insertion experiments Brain No. Mass [gms]

117 Figure A.1 Swine brain placed in a saline solution filled container The brain was glued to a transparent glass plate and constrained by two custom made moulds ( paper clay coated with water-resistive agent) as shown in Figure A.1. In order to simulate the in vivo conditions, the plate-brain assembly was placed in a container filled with saline solution (temperature: 37 C). 106

118 Figure A.2a Needle insertion: experimental set up Figure A.2b Needle tip geometry A custom built apparatus (Figure A.2) was used to conduct the needle insertion into the swine brains (Surgical Assist Technology Group, AIST, Tsukuba, Japan). The apparatus consisted of a linear motion table with a force sensor and a stationary base (Figure A.2a). The container with the brain was placed on the stationary base. The needle was attached to the force sensor on the linear motion table. The force along the longitudinal direction of the needle axis was measured. As a very stiff needle was used, no needle deformations were observed. A solid stainless needle with a diameter of 1.15 mm and conical tip (Figure A.2b) was used. The insertion speed was kept constant at 5 mm/sec. The needle displacement 107

119 was measured using a laser range scanner. For both the needle force and displacement, the sampling rate was 100 Hz. Figure A.3 Typical needle insertion force time history recorded in this study. Point N1 indicates time when the needle touches the Pia Mater enveloping the brain. P1 indicates the part of the force time history preceding the time when the needle punctures the Pia Mater. P2 and P3 indicate parts of the force time history following the Pia Mater puncture, and are associated with the Pia Mater and brain springback. P4 indicates the part associated with further progress of needle into the brain The general behaviour of needle force time histories obtained in this study is summarised in Figure A.3. The mechanics of needle insertion into very soft organs was determined by Washio and Chinzei (2004) in their experiments on swine liver and spleen. The behaviour of the needle-force time history obtained in this study was consistent with the one reported by Washio and Chinzei (2004). Following Washio and Chinzei (2004), the general needle force-time history is explained as follows: At point 1 in Figure A.3, the 108

120 needle touched the organ surface and organ deformation started. This was accompanied by an increase in force acting on the needle. As the needle moved forward further, the force continued to increase since the membrane (i.e. Pia Mater in case of the brain) surrounding the organ stretched and the organ deformed. Consequently, elastic energy is stored in the membrane. At point 2 in Figure A.3, the membrane ruptured accompanied by a drop in force acting on the needle tip. The elastic energy stored in the membrane was released. This was followed by springback of both the organ (brain) and the membrane (Pia Mater). After springback, the needle proceeded further into the organ. Therefore, based on results by Washio and Chinzei (2004), the needle insertion was divided into four steps (Figure A.3): 1. Maximum deformation of the brain and the Pia Mater just before failure of Pia Mater 2. Failure of the pia 3. Springback of the pia and the brain 4. Further insertion of the needle into the brain. In this work, only step 1 is modelled as steps 2 to 4 would require extensive experimental study to determine the failure mechanism/mechanisms of the Pia Mater and brain tissues. 109

121 A.1.3 Biomechanical Model of Needle Insertion Continuum Mechanics: Equations For single phase continuum, the principle of virtual work (force equilibrium) used to derive the non-linear finite element formulation was given by (Bathe, 1996 and Miller, 2002): B S ij ijdv f i uidv f i uids (A.1) V V S B where: ij ijdv is the internal virtual work, f i uidv is the virtual work done V V S by external body forces and f i uids is the virtual work done by external surface S forces. Miller (2002) has detailed the required non-linear continuum equations for brain biomechanics. As the brain undergoes non-linear deformation for the single phase continuum, the current volumes V and current surfaces S over which the integration occurs (Equation A.1) are unknown. The volumes V and surface S are part of the solution and not a part of the input parameters. Hence, non-linear finite deformation procedures should be used to obtain the solution to Equation A

122 Swine Brain Mesh The brain geometry for the model was constructed from a set of magnetic resonance images (MRI) of the swine brain. When conducting MRI scans, the brain was glued to the glass plate and constrained using two paper clay moulds. Thus the model geometry accurately represented the swine brain during the needle insertion. The brain mesh (Figure A.4) was constructed using techniques detailed by Wittek et al. (2007). Figure A.4a General view of the swine brain mesh Figure A.4b Refined mesh graft in the needle insertion area In order to accurately model the needle-brain interactions, the element size when discretising the brain should not exceed the needle diameter. As the needle diameter diameter was 0.55 mm, over million elements would be required to discretise the entire brain. This would result in extremely long computation times. Therefore, only a graft of dimensions 6x6x12 mm in direct neighbourhood (Figure A.4b) of the needle insertion 111

123 point was discretised using elements of size 0.5 mm. Elements of size of up to 3.5 mm were used for the remaining part of the brain (Figure A.4a). Tied contact interface (Hughes et al., 1976) was applied to connect the graft with remaining part of the model. The Pia Mater is a thin tissue membrane surrounding the brain. As the swine brain surface was covered with Pia Mater, it was also taken into account while developing the model (Figure A.4a). Swine Brain Parenchyma and Pia Mater Material Parameters The effects of cerebrospinal fluid flow in the brain were disregarded as the time scale of this flow is many times larger than that of the surgical procedures. Consequently, the brain tissue was treated as an incompressible (Poisson s ratio υ=0.5) single phase continuum. During needle insertion, the brain tissue undergoes large deformations. Therefore, to model the swine brain parenchyma, Ogden type hyperviscoelastic constitutive model as proposed by Miller and Chinzei (2002) was used. The Ogden type hyper-viscoelastic constitutive model includes strain rate dependency and non-linear stress-strain behaviour. The constitutive law is given by: 2 t d W [ (t ) ( d 3)] d (A.2) n t k [ 1 g (1 e ) (A.3) 0 k k 1 112

124 where: W is the potential function, λ i are the principal stretches, μ 0 is the instantaneous shear modulus in undeformed state, τ k are the characteristic times, g k are the relaxation coefficients and α is the material coefficient which can assume any real value without any restrictions. The values of the constants for brain tissue constitutive model as given in Equations A.2 and A.3 are reported by Miller and Chinzei (2002) and are valid for relatively high strain rates such as those present during surgical intervention. However, the property of biological tissues varies between individuals. Therefore, subjectspecific data must be used when modelling surgical procedure. The methodology for determining subject-specific values for all the constants in Equations A.2 and A.3 is detailed in Miller and Chinzei (1997) and would require extensive experimental study. However, the characteristic times ( k s), the relaxation coefficients (g k s), the material coefficient ( and the number of terms (n) in Prony series (Equation A.3) describes the shape of stress strain rather than the stress magnitude. This implies that they are unlikely to exhibit appreciable variation between subjects. Hence, only the subjectspecific instantaneous shear modulus (μ 0 ) was determined. The remaining constants (i.e. n=2, =-4.7, 1 =0.5 s, 2 =50 s, g 1 =0.450 and g 2 =0.365) in Equations A.2 and A.3 were taken from Miller and Chinzei (2002). The instantaneous shear modulus (μ 0 ) was determined by conducting unconfined compression experiments (Miller and Chinzei, 2002) on cylindrical samples of the swine brain. The design and the protocol of the unconfined brain experiments are detailed by Miller and Chinzei (1997), Miller and Chinzei (2002) and Miller (2002). 113

125 The reader may refer to the above mentioned references for deeper understanding of the experimental procedure and methodology used for obtaining the instantaneous shear modulus (μ 0 ) from recorded stress-strain curves. From the unconfined brain compression experiments, the instantaneous shear modulus (μ 0 ) of 210 Pa was obtained. Constants for the subject specific swine brain tissue constitutive model (Equations A.1 and A.2 and Miller and Chinzei, 2002) are given in Table A.2. Table A.2: Material constants for the constitutive model of the swine brain tissue μ 0 =210 Pa Instantaneous Response Characteristic Time Relaxation Coefficients k=1 τ 1 =0.5 sec g 1 = k=2 τ 2 =50 sec g 2 = The current knowledge of the mechanical properties of Pia Mater is very limited. In Finite Element models, Pia Mater is typically represented as a linear-elastic membrane (Zhang et al., 2001 and Sigal et al., 2004). However, the experimental literature clearly illustrates non-linear stress stress relationship for the Pia Mater (Aimedieu et al., 2004). As there is insufficient knowledge of the mechanical properties of Pia Mater and to the best of the author s knowledge this is the first attempt to model the non-linear behaviour of the Pia Mater, a simplified Neo-Hookean type constitutive law was used. Hence, Mooney-Rivlin type hyperelastic material model (Mooney, 1940) was used to describe the behaviour of the Pia Mater and is given by: 114

126 C C W 1 (I 3) 2 1 (I2 3) 2 2 (A.3) where W is the potential function, I 1 and I 2 are invariants of right Cauchy deformation tensor, C 1 and C 2 are material constants such that shear modulus ( = 2(C 1 + C 2 ). It is assumed that C 1 =C 2, which yields: C1 C2 4 (A.4) The Pia Mater Young s modulus (E Pia ) was taken as 78 kpa - consistent with values used for brain models in impact biomechanics (Zhang et al., 2001; Zhang et al., 2002 and Takhounts et al., 2003). Taking into account that the Pia Mater and brain tissues are virtually incompressible (i.e. Poisson s ratio υ=0.5), the shear modulus for the Pia Mater ( is given by: E Pia 26kPa (A.5) 2(1 ) Swine Brain Model Loading As in the present study, the focus is on understanding the biomechanics of needle insertion into the brain rather than the effects of specific type of needle or needle tip shape on the forces acting on the needle, the needle was not directly modelled. The brain model was loaded through the prescribed motion of four nodes that would be in contact with the needle during the actual insertion (Figure A.5). The sum of forces at these nodes was a close approximation of the reaction force between the brain surface and the needle. 115

127 Figure A.5 Loading through prescribed nodal velocity (5 mm/sec) Following the needle insertion experiments (Section A.1.2, Biomechanics of Needle Insertion: Experiments), a constant velocity of 5 mm/sec was used to prescribe the nodal motion. The velocity was applied for 2 secs, which corresponded to needle insertion depth of 10 mm. This depth was selected as preliminary needle insertion experiments indicated that the needle punctures the Pia Mater at an insertion depth of 8-11 mm. Swine Brain Model Boundary Conditions During the needle insertion experiments, the bottom surface of the brain was glued to a glass plate. Hence, all nodes defining this surface very rigidly constrained (Figure A.6). 116

128 Figure A.6 Boundary conditions for the brain model when modelling the needle insertion. Nodes defining the surfaces contacting the glass plate and clay moulds were rigidly constrained The brain was laterally supported using two paper clay moulds (Figure A.1) during the needle insertion experiments; the brain surface tends to stick to these moulds. Therefore, the nodes on the brain surface in contact with the moulds were also rigidly constrained (Figure A.6). 117

129 A.1.4 Computational Model of Needle Insertion Swine Brain Parenchyma and Pia Mater Incompressible swine brain parenchyma model (Figure A.4) consisted of constant stress type solid elements (8-noded under-integrated bricks [hexahedra with a single Gauss point and linear shape functions]) (LS-DYNA, 2003). The constant stress type solid element (8-noded under-integrated bricks [hexahedra with a single Gauss point and linear shape functions]) (LS-DYNA, 2003) do not exhibit volumetric locking and therefore can be confidently used for an incompressible continuum (e.g. brain). 8- noded under-integrated bricks [hexahedra with a single Gauss point and linear shape functions] (LS-DYNA, 2003) tend to exhibit non-physical deformation modes known as hourglassing modes (Belytschko, 1983). To prevent formation of such modes, Flanagan-Belytschko type stiffness-based hourglass control (Flanagan et al., 1981) was used. To connect the graft section to the remaining part of the swine brain parenchyma model (Figure A.4a), tied contact interface (Hughes et al., 1976) was applied. The Pia Mater surrounding the brain was discretised using 4567 Belytschko- Tsay shell elements (Belytschko et al., 1984; LS-DYNA, 2003) with one integration point across the thickness, i.e. these elements had no bending stiffness and behaved as a membrane. The Belytschko-Tsay shell elements were attached (through node sharing) to the hexahedral elements forming the brain outer surface. The Pia Mater thickness was 118

130 taken to be 0.5 mm (Jequier and Jequier, 1999) when modelling needle insertion into swine brain. A.1.5 Finite Element Solver The model was computed using LS-DYNA (Livermore Software Corporation, Livermore California, USA) (LS-DYNA, 2003) non-linear finite element code. This code uses explicit time integration (time step: secs) and accounts for both the constitutive and geometric (i.e. finite deformation formulations are used) nonlinearities. A.1.6 Computational Results Figure A.7 presents the calculated force-displacement relationship from computational model (Section A.1.3, Biomechanical Model of Needle Insertion) and compares it with the force displacement relationship obtained experimentally when inserting the needle into swine brain. The computed force displacement relationship was almost in the middle of an envelope of the experimental results obtained when inserting the needle into brain no. 4 (Figure A.7a), i.e. the brain for which the subjectspecific properties of brain tissue were determined. The agreement was very good even for the needle displacement as large as 10 mm. 119

131 Experiments Model Displacement [mm] Figure A.7a Computed Force displacement relationship validated against needle insertion experiment [Swine Brain Number 4 (7 insertions)] Experiments: Average Experiments: Average +/-S.D. Model Displacement [mm] Figure A.7b Computed Force displacement relationship validated against needle insertion experiments [3 Swine Brains (18 insertions)] Note for Figures A.7a and A.7b: The brain tissue properties for the computational model were determined using tissue samples obtained from Swine Brain Number 4 (Swine Brain Parenchyma and Pia Mater Parameters). 120

132 The predicted needle force displacement relationship was also very close to the average relationship determined from all the needle insertions conducted in this study (i.e. 18 insertions into three brains) (Figure A.7b), which indicates consistency in our experimental and modelling results. Overall deformation of the model under needle insertion depth of 8 mm is summarised in Figure A.8. It indicates large localised deformation of the brain tissue in the area of needle insertion. This is very similar to the localised deformations during needle insertion experiments. Figure A.8 shows that the assumption of infinitesimally small deformations used to calculate the forces acting during needle insertion (Brett et al., 1997; Simone et al., 2002; DiMaio and Salcudean, 2002; O Leary et al., 2003; DiMaio and Salcudean, 2003; DiMaio and Salcudean, 2005; Gerovich et al., 2004; Spicer et al., 2004; Alterovitz et al., 2005; Heverly et al., 2005 etc.) would not produce reliable results. Figure A.8 Needle insertion simulation: model cross section at 8 mm insertion depth 121