homeostasis revisited in the genesis of stress reactivity

Homeostasis Revisited in the Genesis of Stress Reactivity

Pedram Ataee

Supervisors: Drs. G. Dumont, M. Ansermino, T. Boyce and H. NoubariDepartment of Electrical and Computer Engineering

University of British Columbia

[email protected]

June 9, 2013

Pedram Ataee (UBC) Ataee’s PhD Exam June 9, 2013 1 / 31

Overview

1 Introduction

2 Mathematical Modelling

3 Parameter Identification

4 Stability Analysis

5 Artificial Bionic Baroreflex

6 Summary and Conclusion

7 Future Work


Introduction

Theorem

Early development of mental and physical health problems in adults isassociated with an exaggerated autonomic-cardiac reactivity duringindividual’s childhood [Boyce2001].

Problem Statement

The purpose of this study is to investigate a potential autocatalytic loop(also referred as a positive feedback) leading to an exaggeratedautonomic-cardiac reactivity.


Introduction

Our Approach

We have decided

to develop a mathematical model for the autonomic-cardiacregulation,

to estimate the time-varying subject-specific model parameters foreach individual,

to create a systematic framework to investigate a potential positivefeedback.


Introduction

Challenges

Mathematical Modelling: physiology-based scheme, accuratemodelling, and minimal complexity

Parameter Identification: sensitivity, repeatability, and physiologicalconsistency as well as time-varying and subject-specific modelparameters

Stability Analysis: delayed- and nonlinear differential equation modelas well as a numerical indicator of the stability margin


Autonomic-Cardiac Regulation (Schematic Diagram)

Figure: Schematic diagram of the autonomic-cardiac regulation model.


Autonomic-Cardiac Regulation (Mathematical Model)

H(t) =βHTs

1 + γ Tp− VHTp + δH

(

H0 − H(t))

P(t) = −P(t)

R0a (1 + αTs)Ca

+H(t)∆V

Ca

.

Baroreflex

Ts = 1− σ(

P(t − τ))

and Tp = σ(

P(t))

are sympathetic modulatingfunction and parasympathetic modulating function respectively, generatedby the baroreflex control mechanism.

σ(P) = Tmin +Tmax − Tmin

1 + e−αsp(P−Psp)50 ≤ P ≤ 200.


Autonomic-Cardiac Regulation

H(t) =βHTs

1 + γ Tp

− VHTp + δH(

H0 − H(t))

P(t) = −P(t)

R0a (1 + αTs)Ca

+H(t)∆V

Ca

.

Table: Model parameters of autonomic-cardiac regulation.

Parameter Definition Nominal ValueCa arterial compliance 1.55 mlmmHg−1

R0a minimum arterial resistance 0.6 mmHgsml−1

∆V stroke volume 50 mlH0 intrinsic heart rate 100 min−1

τ sympathetic delay 3 sVH vagal tone 1.17 s−2

βH sympathetic control of HR 0.84 s−2

α sympathetic effect on Ra 1.3γ vagal damping of βH 0.2δH relaxation time 1.7 s−1


Autonomic-Cardiorespiratory Regulation

Figure: Schematic diagram of interactions between cardiovascular, respiratory andnervous systems.



Table: Respiration system impacts on VL, HR, VR, and ∆V .

Spontaneous MechanicalInhale Exhale Inhale Exhale

VL (Instantaneous Lung Volume) ↑ ↓ ↑ ↓

HR (Heart Rate) ↑ ↓ ↑ ↓

VR (Venous Return) ↑ ↓ ↓ ↑

∆V (Stroke Volume) ↑ ↓ ↓ ↑


Autonomic-Cardiorespiratoy Regulation

Blood Pressure (+Mechnical Coupling: Intrathoracic pressure)

P(t) = −P(t)

R0a (1 + αTs)Ca

+H(t)(∆V±k2VL)

Ca

.

Heart Rate (+Neuromeachnical Coupling: Lung Stretch Reflex)

H(t) =βHTs

1 + γ Tp

− (VH−k1VL)Tp + δH(

H0 − H(t))

Respiration Rate (Chemoreflex)

R(t) = k3

[

(

1 + σco2)

R0 − R(t)]

+ u(t)



Figure: Neuromechanical couplingeffects of respiration on HR and BP.

Figure: Mechanical coupling effects ofrespiration on HR and BP.


Parameter Identification (Sensitivity Analysis)

Traditional Analysis

SX (t, µj) =

∣

∣

∣

∣

X (t, µj)− X (t, µj ,0)

µj − µj ,0

∣

∣

∣

∣

×µj

X (t, µj)X = H,P

Aggregation

S(t, µj) =SH(t, µj) + SP(t, µj)

2,

Sj(t) =

√

√

√

√

√

32µj,0∑

µj=12µj,0

S2 (t, µj),

Overall Sensitivity

Sj =

√

√

√

√

tfinal∑

tinitial

S2j (t).


Sensitivity Analysis

Figure: The overall sensitivity (mean and standard deviation) of autonomic-cardiacmodel parameters over 100 sensitivity analysis runs with nominal values selected from+/-20% the associated nominal values introduced in 1.


Sensitivity Analysis

Table: Sensitivity-based parameter classification.

High-sensitivity Low-sensitivity Invariant

Psp γ H0

∆V Ca R0a

VH τ

βH αsp

α δH


Parameter Identification

J =EP + EH

2; EX =

n∑

t=0

∣

∣

∣

∣

Xs(t,M)− Xm(t)

Xm(t)

∣

∣

∣

∣

,

Experimental results from a MIMIC data (Case No.: 476).

Figure: Measured vs. model-estimatedsignals: BP, HR, and CO.

Figure: Identification results: αTs , βHTs ,and VHTp .


Parameter Identification (Tilt test)

Figure: Subject I


Stability Analysis

Delay-Free Realization

X (t) = P(t − τ) + P(t)L⇒ X (s) =

2P(s)

1 + τ2 s

⇒ X (t) =2

τ

[

2P(t)− X (t)]

Equilibrium States

W(t) =

H(t)

P(t)

X (t)

=

f1(

H(t),P(t),X (t))

f2(

H(t),P(t),X (t))

f3(

H(t),P(t),X (t))

, W(t) = 03×1


Stability Anlaysis

Lyapunov Analysis

JJJ(Wf ) =∂(f1, f2, f3)

∂(H,P ,X )

∣

∣

∣

∣

Wf

=

∂f1

∂H

∂f1

∂P

∂f1

∂X

∂f2

∂H

∂f2

∂P

∂f2

∂X

∂f3

∂H

∂f3

∂P

∂f3

∂X

W=Wf

Stability Measure (Lyapunov)

Sm = maxi=1,2,3

[

real(λi )]

Stability Measure (Empirical)

Sp =

30∑

t=1

∣

∣

∣P(t)− P(t)

∣

∣

∣

P(t)


Stability Analysis

(a) A normal condition with stress (b) A normal condition without stress

Figure: Two metrics for stability margin Sm and Sp over changes of a model parameter from50% to 200% of its nominal value for a healthy physiological condition with (i.e., a 50% lowerVH , and 100% higher βH and α comparing to their nominal values) and without (i.e., VH , βH

and α were fixed at their nominal values) stress. Sm : blue solid line, Sp: green dashed line.


Stability Anlaysis

Figure: Stability indices Sm over 2-D parameter spaces from 50% to 150% of theirnominal values for a normal physiological condition. The quantitative stability marginmetric Sm at each point of the 2-D parameter space is mapped into a pixel-intensitylevel. A higher pixel-intensity level is related to lower stability margin, and vice versa.


Autonomic-Cardiac Regulation

Figure: Schematic model of autonomic-cardiac regulation.


Artificial Bionic Baroreflex

Figure: Schematic model of an artificial bionic baroreflex.



Figure: Schematic model of the proposed artificial bionic baroreflex.

P0 Adjustment

P0(t +∆) = P0(t) + k · (BPm − BPsp)


Subject-Specific Mathematical Model

Table: Individualized nominal values of high-sensitivty parameters in 3 subjectsversus corresponding population nominal values

SubjectPopulation

I (477) II (486) III (476)

VH 0.65 1.37 2.13 1.17

βH 1.5 0.87 0.68 0.84

α 0.7 1.36 1.55 1.3

∆V 46 40 36 50



Figure: BP measurement (BP setpoint) vs. the results of the artificial bionic baroreflex(simulated BP) for individual with subject number: 477.



Figure: The calculated control signal P0 in 3 subjects.


Summary and Conclusion

A mathematical model of the atutonomic-cardiorespiratory regulation+ oscillatory pattern origins

A parameter dientification technique to monitor sympathetic andparasympathetic nerve activities

A systematic approach to stability analysis ot autonomic-cardiacregulation

A method for the design of an artificial bionic baroreflex using asubject-specific mathematical model


Future Work

To investigate the atutonomic-cardiorespiratory regulation using theproposed mathematical model

To validate the consistencey of the proposed parameter identificationresults with other markers of sympathetic and parasympatheticmarkers including HRV-based markers

To investige the occurence of positive feedback in theautonomic-cardiac regulation using the poposed systematic approach

To study the proposed method for the design an artificial bionicbaroreflex in the clinical experiments.


Publication

Journal Articles

P. Ataee, J.O. Hahn, Dumont, G.A., and W.T. Boyce. Non-Invasive Subject-SpecificMonitoring of Autonomic-Cardiac Regulation. IEEE Transactions on BiomedicalEngineering, submitted, 2012.

P. Ataee, J.O. Hahn, Dumont, G.A., Noubari H.A., and W.T. Boyce. A Model-BasedApproach to Stability Analysis of Autonomic-Cardiac Regulation. Journal of ComputerMethods and Programs in Biomedicine, submitted, 2012.

Refereed Conference Papers

P. Ataee, J.O. Hahn, C. Brouse, G.A. Dumont, and W.T. Boyce. Identification ofcardiovascular baroreflex for probing homeostatic stability. Computing in Cardiology,(37):141-144, 2010.

P. Ataee, J.O. Hahn, G.A. Dumont, and W.T. Boyce. A Systemic Approach to LocalStability Analysis of Cardiovascular Baroreflex. 33rd Annual International Conference ofthe IEEE EMBS, pages 700-703, 2011.

P. Ataee, L. Belingard, G.A. Dumont, H.A. Noubari, and W.T. Boyce.Autonomic/Cardiorespiratory Regulation: A Physiology-Based Mathematical Model. 34thAnnual International Conference of the IEEE EMBS, pages 3805-3808, 2012.

P. Ataee, G.A. Dumont, H.A. Noubari, W.T. Boyce, J.M. Ansermino. A Novel Approachto the Design of an Artificial Bionic Baroreflex. 35th Annual International Conference ofthe IEEE EMBS, accepted, 2013.


Acknowledgement

Thank you.


homeostasis revisited in the genesis of stress reactivity

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