NONLINEAR DYNAMICS AND CHAOS. Facilitating medical diagnosis. Medical classifications

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1 LECTURE : BIOMEDICAL MODELS NONLINEAR DYNAMICS AND CHAOS Patrick E McSharry Systems Analysis, Modelling & Prediction Group patrick@mcsharry.net Tel: Medical diagnostics Dynamical diseases Heart rate variability Cardio-vascular model Ventilation control model Electroencephalogram (EEG) Epileptic seizure detection and anticipation Trinity Term 27, Weeks 3 and 4 Mondays, Wednesdays & Fridays 9: - : Seminar Room 2 Mathematical Institute University of Oxford Nonlinear dynamics and chaos c 27 Patrick McSharry p. Nonlinear dynamics and chaos c 27 Patrick McSh Facilitating medical diagnosis Medical classifications RR BP I( RR, BP,τ) Signal processing: noise reduction filtering artefact removal feature extraction v tilt begins v tilt ends MODEL x(t+) = F[x(t), a] Estimation of clinically relevant parameters in the model for each distinct physiological state Time varying medical diagnostics providing intelligent monitoring: trends, alarms and visualisation Physiology & Medical knowledge Non linear analysis: information theory Bayesian analysis synchronisation deterministic/stochastic Aid clinical decision making Multi variate linear & non linear analyses Nonlinear dynamics and chaos c 27 Patrick McSharry p.3 Temperature ( o C): Heathy adult: 37 Hypothermia: < 3 Hyperthermia: > 4 Heart rate (beats per minute): Average: 7-72 Normal range: -9 Newborn range: 3- Total number of heart beats is approximately constant during a lifetime Blood pressure is greatest when the heart ventricle contracts (systole) and lowest when ventricle relaxes (diastole) Systolic pressure tends to rise with age Diastolic pressure is a better indicator of hypertension Blood pressure: Systolic (mmhg) / Diastolic (mmhg) Optimal: <2 / < 8 Normal: <3 / 8 High normal: 3-39 / 8-9 Hypertension: 4 / >9 Nonlinear dynamics and chaos c 27 Patrick McSh

2 Workout intensity Dynamical diseases Target heart rate: Formula for the maximum heart rate (HRmax) of the average person is: HRmax = 22 - age Lower threshold is: HRlow =.6 HRmax Upper threshold is: HRupper =.9 HRmax Example: A 2-year old (HRmax = 9) should exercise between 7 and 7 beats per minute. Estimate of resting heart rate (HRrest): Calculate your pulse for one minute on three successive mornings after waking up Define HRrest as the average of these three numbers Reserve heart rate range: Karvonen Formula gives heart rate reserve as: HRres = HRmax - HRrest Exercise between HR = HRrest +. HRres and HR2 = HRrest +.8 Hrres Suppose the 2-year old had a HRrest of 7, then HRres = 9-7 = 2. This gives HR = 7 +.(2) = 33 and HR2 = 7 +.8(2) = 76 Nonlinear dynamics and chaos c 27 Patrick McSharry p. Many of the complicated regulatory mechanisms in the human body rely on the interacti between different processes It is difficult to make an accurate diagnosis based on a few average values Currently, clinicians have long records (days) of high-frequency data: Electrocardiogram ECG: (Heart rate, QT-intervals) Electroencephalogram EEG: (Sleep analysis, epilepsy, vigilance) Blood pressure: (hypotension, hypertension) Respiration: (respiratory illnesses) Blood gases (oxygen, carbon dioxide) It is possible to think of changes between health and disease as dynamical transitions. Bifurcation diagrams show transitions between cyclic and chaotic fluctuations (ordered a complex behaviour) Epilepsy: transition from irregular behaviour to ordered (synchronous) activity Congestive heart failure: transition from complex (multi-fractal) behaviour to less variabl (mono-fractal) activity Complexity is often associated with healthy dynamics - the ability to react to new conditi and provide adequate regulation Regularity is often associated with disease and failure to respond to perturbations Nonlinear dynamics and chaos c 27 Patrick McSh Cardiovascular system DeBoer model Heart rate, systolic and diastolic blood pressure and respiration Power spectrum of RR intervals HF peak at.3 Hz: Respiration, Respiratory Sinus Arrhythmia (RSA) LF peak at. Hz: Mayer waves Control mechanisms Sympathetic activity (increases heart rate) Parasympathetic (vagal) activity (decreases heart rate) Barorecptor detects drops in pressure and increases heart rate Heart rate variability: LF/HF ratio Power [sec 2 /Hz] < > < LF > HF Mayer RSA baroreceptors blood pressure afferent X Central Nervous System β symp α symp peripheral resistance cardiac output vagal (fast) RR interval cardiac pacemaker Notation at heart beat i: Systolic pressure: S i Diastolic pressure: D i RR interval: I i respiration Nonlinear dynamics and chaos c 27 Patrick McSharry p.7 Nonlinear dynamics and chaos c 27 Patrick McSh

3 Effective pressure Baroreflex S l 2 Baroreflex [ms / mmhg] sympathetic vagal 2 i 7 i 6 i i 4 i 3 i 2 i i time [beats] S S i = S + 8atan[(S i S )/8] Controls RR interval through vagal (fast) and β-sympathetic I i = G v S i + G βf(s, τ β ) + c Controls peripheral resistance R i using time constant T i = R i C through α-sympathetic T i = c T G α F(S, τ α ) Nonlinear dynamics and chaos c 27 Patrick McSharry p.9 Nonlinear dynamics and chaos c 27 Patrick McSha Myocardium, Windkessel and Noise DeBoer model equations Myocardium Length of previous RR interval controls strength of the ventricular contraction reflected in the pulse pressure: P i = γi i + c 2 Starling s law states that increased filling of the ventricles after a longer interval leads to a more forceful contraction Respiration is also incorporated P i = γi i + c 2 + Asin(2πf resp t i ) Windkessel Windkessel properties of the arterial tree states that the new diastolic pressure D i depends on the previous systolic pressure S i, on the length of the preceding interval I i, and on the value of the time constant T i : D i = c 3 S i exp( I i /T i ) Noise Noise sampled from a Gaussian distribution was added to the RR interval equation and the pulse pressure: I i = G v S i + G βf(s, τ β ) + c + η (I) with η (I) N(, 2ms) P i = γi i + c 2 + Asin(2πf resp t i ) + η (P) with η (P) N(, 2mmHg) Operation points S = 2 mmhg D = 7 mmhg I = 8 ms T =,42 ms Equation flow: S i S i I i t i T i D i+ P i+ S i+ S i = S + 8atan[(S i S )/8] I i = G v S i + G βf(s, τ β ) + c + η (I) t i+ = t i + I i T i = c T G α F(S, τ α ) D i+ = c 3 S i exp( I i /T i ) P i+ = γi i + c 2 + Asin(2πf resp t i+ ) + η (P) S i+ = D i+ + P i+ Nonlinear dynamics and chaos c 27 Patrick McSharry p. Nonlinear dynamics and chaos c 27 Patrick McSha

4 DeBoer model time series DeBoer model power spectra 3 S i 2 8 D i 7 7 s 2 /Hz RR intervals (mmhg) 2 /Hz Systolic pressure P i I i.8.7 T i 4 Nonlinear dynamics and chaos c 27 Patrick McSharry p.3 (mmhg) 2 /Hz Diastolic pressure (mmhg) 2 /Hz Pulse pressure Nonlinear dynamics and chaos c 27 Patrick McSha Modelling real signals Physiological condition monitoring Whittam et al. have demonstrated that the DeBoer model is capable of fitting the spectral peaks of the RR and systolic pressure calculated from physiological signals recorded from a group of normal subjects. S, D, I, T and respiration from recordings were fed into the DeBoer model. I LF, I HF = LF & HF power peaks of I S LF, S HF = LF & HF power peaks of S α LF = p I LF /S LF, α HF = p I HF /S HF, Baroreflex Sensitivity: BRS = α Original DeBoer model: G v = α, G α = α and G β = 2α Modified model: G v = α HF, G α = G β = α LF. The model places constraints on the dynamics by encoding the nonlinear interactions between the input physiological signals. For given recordings S i, D i, I i and respiration, the model parameters (G v, G α, G β ) wh best fit the data can be determined using a Maximum Likelihood approach. Variations in the physiological signals will yield changes in the model parameter space. Employing HRV, or indices obtained from the power spectrum (e.g. LF/HF ratio) implies underlying linear assumption Employing the model implies a nonlinear assumption Nonlinear dynamics and chaos c 27 Patrick McSharry p. Nonlinear dynamics and chaos c 27 Patrick McSha

5 Medical applications Control of ventilation by blood CO 2 levels Do the model parameters cluster for normal subjects? Check that intra-subject data sets give similar results Compare variation/similarity between intra-subject and inter-subject populations Perturbation (stress) tests: Valsalva maneuver: performed by attempting to forcibly exhale while keeping the mouth and nose closed. Causes blood pressure to rise and forces the heart to respond by beating more slowly. Mueller maneuver: performed by attempting to forcibly inhale while keeping the mouth and nose closed. Opposite to Valsalva. Isometric handgrip Tilt tests Lower body pressure decreases Comparison with HRV for condition monitoring Nonlinear dynamics and chaos c 27 Patrick McSharry p.7 Mackey & Glass. Oscillation and Chaos in Physiological Control Systems, Science 97: (977) Negative feedback system: deviations in the state variable from a steady state value ten be minimised by the feedback. This usually tends to force the system to a steady-state, may also produce oscillations. x denotes the partial pressure of CO 2 in the blood C 2 is produced by body tissues at a constant rate λ under constant conditions CO 2 is eliminated from the body by ventilation V which is a monotonic increasing functi arterial CO 2 levels at some time τ in the past Delay τ is due to blood transit time from brain stem (ventilation is determined by chemoreceptors) to the lungs (where C 2 is eliminated) x n V (x) = V m θ n +x n and parameters θ and n determine the shape of the response curve Denote the value of x at some time τ in the past by x τ. Assume rate of removal of CO 2 is proportional to x times V (x τ ) ẋ = λ αxv (x τ ) Increases (decreases) in arterial CO 2 levels lead to increases (decreases) in ventilation which in turn leads to decreases (increases) in arterial CO 2 levels Nonlinear dynamics and chaos c 27 Patrick McSha Ventilation steady-state analysis Ventilation time series At the steady-state ẋ =, let x = x, V = V and define S = dv (x )/dx Empirical measurements: λ = 6 mmhg/min, V m = 8 litres/min, τ =.2 min Steady-state values:x = 4 mmhg, V = 7 litres/min, S = 4 litres/min mmhg Given x, V, S and V m, can infer: α = λ/x V θ n = x n (V m V )/V n = x S V m /V (V m V ) Steady-state is stable if S < πv /2λτ Destabilised (Hopf bifurcation) by: (i) increasing S, τ or λ or (ii) decreasing V. Steady-state becomes unstable, oscillation in x and V with period 4τ V m V Ventilation Arterial CO 2 concentration x Volume [litres/min] Volume [litres/min] time [min] time [min] Top: S = 7.7 litres/min mmhg, Bottom: S = litres/min mmhg Nonlinear dynamics and chaos c 27 Patrick McSharry p.9 Nonlinear dynamics and chaos c 27 Patrick McSha

6 Scalp Electroencephalogram (EEG) Scalp EEG during epileptic seizure Channel Spatial geometry of the -2 electrode system: Prefrontal (F p ), Frontal (F), Central (C), Temporal (T), Parietal (P), Occipital (O), and Ear or mastoid (A). 3 2 Nonlinear dynamics and chaos c 27 Patrick McSharry p Nonlinear dynamics and chaos c 27 Patrick McSha Moving variance Linear analysis A/ B/ (a) (b) Varianace C/ 2 6 f [Hz] 4 2 log P D/ (c) D/2 F/ (d) f [Hz] log S F/2 τ [secs].. ACF G/ Nonlinear dynamics and chaos c 27 Patrick McSharry p.23 Nonlinear dynamics and chaos c 27 Patrick McSha

7 Novelty detection with MDPE Random & Deterministic processes Multi-Dimensional PDF Evolution Choose centres in m-dimensional state space Centres define an m-dimensional partitions Calculate distribution of points in these partition Compare distributions using χ 2 -test Given a learning period which represents normality, compare subsequent windows of data McSharry et al., Medical & Biological Engineering & Computing 4(4): (22) p(x i ) x i (a) i.2... (b) x i.8.6 ρ k.4.2 (c) 2 4 k x i+ (d) x i z i (e) i.2. (f).8 (g) (h) p(z i )..6 ρ k.4 z i+..2 z i 2 4 k z i Nonlinear dynamics and chaos c 27 Patrick McSharry p.2 Nonlinear dynamics and chaos c 27 Patrick McSha MDPE applied to synthetic signal Variance analysis Frontal Frontal 4 Parietal 3 Parietal 4 A/ s i = β i z i + β i x i (a) B/ s.9 (b) C/ β..8.2 µ (d).7 D/ σ 2. (c). D/2 4. (e) 8 PSD E/ (f) ACF.2 E/2 MDPE (g) F/ Nonlinear dynamics and chaos c 27 Patrick McSharry p.27 Nonlinear dynamics and chaos c 27 Patrick McSha

8 MDPE analysis Intracranial EEG: Correlation density (Martinerie, 998) Frontal Frontal 4 Parietal 3 Parietal 4 A/ 4 B/ Varianace 3 C/ 2 D/.6 C(r ).4.2 D/2 E/.6 C(r ).4.2 E/ F/ Nonlinear dynamics and chaos c 27 Patrick McSharry p.29 Nonlinear dynamics and chaos c 27 Patrick McSha Correlation density versus variance 2 EEG Variance C(r ) / 8/ /.4 C(r ) Variance McSharry et al. Nature Medicine 9(3): (23) McSharry et al. IEEE Trans. Biomed. Eng. (): (23) Nonlinear dynamics and chaos c 27 Patrick McSharry p.3