BIOMEDICAL DEPARTMENT Government Engineering College, Sector-28, Gandhinagar MID-SEM EXAMINATION – ANSWER KEY

Subject – Modelling and Simulation of Physiological Systems Date: 14/09/2015

Class - 5th Semester Max. Marks: 30 Time: 1𝟏

𝟐 Hour

Q-1 Write answers of any TWO of below given questions: [10]

a. Give a difference between static vs. dynamic analysis.

Ans –

Static Analysis Dynamic Analysis

It is an analysis of a system or model without

consideration of its performance parameters.

It is an analysis of a system or model that

demands the study of actual behavior and

response of the system.

Parameter needs to consider under & over

the stability boundary, so that a complete

knowledge can be acquired.

Majority of the parameters are considered to

be bounded into stable conditions of given

system.

Steady-state condition can be established

such that the system will give a predicted

value of output as applied with a predefined

input value.

Steady-state level can be established from

varying output values with application of

same input values at different period of time.

It examines all possible projection of paths

and variable values, not just those invoked

during analysis.

It reveals subtle defects or vulnerabilities

whose cause is too complex to be discovered

by static analysis.

Ex: Systolic & Diastolic Blood-Pressure

values

Ex: Systolic & Diastolic Blood-Pressure

Curves

*The difference may extend up to knowledge and applicability. The primary difference may be given

in above listed manner.

b. Derive mathematical model for regulation of glucose with steady-state analysis of

normal, Type-1 and Type-2 diabetic patients.

Ans –

The model was first proposed by Stolwijk and Hardy in 1974. The total volume of blood

and interstitial fluids to be represented by a single large compartment (~15L in a normal

adult), and that the steady-state concentration of glucose in this compartment is x (in units

of mg ml-1). For this level of x to remain constant, the total inflow of glucose into the

compartment must equal the total outflow from the compartment. Glucose enters the blood

through absorption from the gastrointestinal tract or through production from the liver.

Glucose Input Flow Rate is assumed to be QL.

1. When x is elevated beyond a certain threshold (θ), glucose is excreted by the kidneys

at a rate proportional to the gradient between x and θ:`

Renal Loss Rate = µ (x - θ) , x > θ

= 0 , x ≤ θ

2. Glucose leaves the blood to enter most cells through facilitated diffusion. In some

tissues, the rate of glucose utilization depends only on the extracellular to

intracellular concentration gradient. In most circumstances, we can ignore the

intracellular concentration.

3. In certain types of cells, such as those in muscle and adipose tissue, insulin helps to

stimulate this facilitated diffusion process. Therefore, the rate at which glucose is

taken up by these cells is proportional to x as well as to the blood insulin

concentration, y:

Tissue Utilization Rate (Insulin-dependent) = νxy

Tissue Utilization Rate (Insulin-independent) = λx

Equating the inflow to the sum of the three outflows, we obtain the following mass

balance equations for blood glucose:

QL = λx + νxy, x ≤ θ

= λx + νxy + µ (x - θ), x > θ

A strong nonlinearity in the form of the product of x and y is introduced, along with the

thresholding nonlinearity θ. The negative feedback in this control system is clearly

embedded in the characteristics.

Insulin is produced by the pancreas at a rate dependent on the plasma glucose level.

However, if x falls below a certain threshold (𝜙), insulin production ceases.

Insulin Production Rate = 0 , x ≤ 𝜙

= β (x - 𝜙), x > 𝜙

Insulin is destroyed through a reaction involving the insulinase enzyme, at a rate

proportional to its concentration in blood:

y = 0 , x ≤ 𝜙

= β/α (x - 𝜙), x > 𝜙

c. Discuss Bohr effect and Haldane effect. Derive mathematical model for regulation

of ventilation.

Ans –

Bohr Effect – CO2 and O2 gas exchange is not independent, but CO2 affects the affinity

with which O2 is bound to hemoglobin.

Haldane Effect – The level of oxygenation affects the blood CO2 concentration at any given

partial pressure.

The gas exchanger:

We assume metabolic CO2 production rate to be V̇CO2; this is the rate at which CO2 is

delivered to the lungs from the blood that is perfusing the pulmonary circulation. In the

steady state, this must equal the net flow of CO2 exiting the lungs in gas phase. The latter is

equal to the difference in volumetric fraction (or concentration) of CO2 in the air entering

(FICO2) and leaving (FACO2) the alveoli multiplied by the alveolar ventilation, V̇A. The

alveolar ventilation represents that portion of the total ventilation, V̇E that actually

participates in the gas exchange process. Part of V̇E is "wasted" on ventilating the non-gas-

a) Normal Conditions

θ = 2.5 mg ml-1

µ = 7200 ml h-1

λ = 2470 ml h-1

ν = 13900 mU-1

h-1

α = 7600 ml h-1

𝜙 = 0.51 mg ml-1

β = 0.56 mU ml mg-1

h-1

QL = 84 mg h

-1

b) Type - I Diabetes

(insulin-dependent diabetes OR

juvenile-onset OR ketone-prone)

c) Type - II Diabetes (non-insulin-dependent)

exchanging airways in the lungs; this flow is known as "dead space ventilation", V̇D. Thus

we have:

V̇A = V̇E − V̇D

and CO2 mass balance equation:

V̇CO2 = k V̇A (FACO2 - FICO2 )

In above equations, the ventilatory flow rates are generally measured in BTPS (body

temperature pressure saturated) units. While the CO2 metabolic production rate is usually

expressed in STPD (standard temperature pressure dry, i.e., at 273K and 760mmHg) units.

The constant k allows volumes and flows measured in BTPS units to be converted into

STPD units. This conversion is achieved by using the ideal gas equation:

𝑉𝑆𝑇𝑃𝐷 760

273=

𝑉𝐵𝑇𝑃𝑆(𝑃𝐵 − 47)

310

Where, the body temperature is assumed to be 37°C or 310 K and a saturated water vapor

partial pressure of 47 mmHg at that temperature. PB represents the barometric pressure under

which the gas exchange process is taking place; at sea level, this is 760 mmHg, but the value

decreases with ascent to high altitude.

k = 𝑉𝑆𝑇𝑃𝐷

𝑉𝐵𝑇𝑃𝑆=

(𝑃𝐵−47)

863

The volumetric fractions can be converted into their corresponding partial pressure, using

Dalton's law:

PICO2 = FICO2 (PB - 47) & PACO2 = FACO2 (PB - 47)

Using above equations,

PACO2 = PICO2 + 863�̇�CO2

�̇�A

Above equation shows, hyperbolic relationship between PACO2 & V̇A , is known as metabolic

hyperbola. A similar metabolic hyperbola can be deduced for O2,

PAO2 = PIO2 - 863�̇�O2

�̇�A

The respiratory controller:

The controller part of the system includes the chemoreceptors, the neuronal circuits in the

lower brain, involved in the generation of the respiratory rhythm as well as the neural drive

to breathe, and the respiratory muscles. The controller response to CO2 has been shown to

be linear over the physiological range.

In the absence of vigilance, such as during sleep, the controller output falls rapidly to zero

(i.e., central apnoea occurs) when PaC02 decreases slightly below normal awake resting

levels. Exposure to hypoxia (i.e., when Pa02 decreases below 70 mmHg) leads to an increase

in the CO2 response slope as well as the ventilator controller output.

The ventilatory controller output (V̇C) can be modelled as the sum of an 02-independent term

and a term in which there is a multiplicative interaction between hypoxia and hypercapnia:

V̇C = (1.46 + 32

𝑃𝑎𝑂2− 38.6

) (𝑃𝑎𝐶𝑂2−37), 𝑃𝑎𝐶𝑂2> 37

= 0, 𝑃𝑎𝐶𝑂2≤ 37

Q-2

Ans-

State the classification of eye movements in regards to its orientation. Describe a model

for purely horizontal saccade. On the basis of its limitation, suggest a new hypothesis

to overcome it.

Classification of eye movements:

The eye movements are mainly classified in to two types:

[5]

(1) Eye movements according to visualization

(2) Eye movements according to its orientation

Considering eye movements according to its orientations, they are further classified in to

three main types:

(1) Horizontal eye movement

(2) Vertical eye movement

(3) Torsional eye movement

Model for Horizontal Eye movement:

The first quantitative saccadic eye movement model, illustrated in following schematic

was published by Westheimer in 1954. Based on visual inspection of a recorded 20º

saccade, and the assumption of a step controller, Westheimer proposed the following

second order model:

In order to get detailed analysis of peak velocity and time, the transfer function for above

mentioned equation is given by:

Where 𝜔𝑛 = √𝐾

𝐽 & 𝜁 =

𝐵

2√𝐾𝐽 ; based on 20º saccade Westheimer estimated 𝜔𝑛 =

120 𝑟𝑎𝑑/𝑠𝑒𝑐 & 𝜁 = 0.7

With input τ(s) = 𝛾

𝐾

Where

Tp was found 37ms

Outcomes of Westheimer’s horizontal eye movement model:

The saccade system was not linear because the peak velocity-saccade magnitude plot

was nonlinear. Thus, the input was not an abrupt step function.

Westheimer’s second-order model proves to be an adequate model for saccades of

all sizes if a different input function is assumed.

Limitation of Westheimer’s horizontal eye movement model:

This model provided a satisfactory fit to the eye position data for a saccade of 20◦,

but it was not for saccades of other magnitudes.

Hypothesis to overcome limitation of Westheimer model:*

Westheimer’s model holds true only for first 20 degree of purely horizontal, more

complex model need to develop in order to analyze saccadic characteristics for

torsional movements.

Higher order system need to develop for more detailed analysis.

MARKS DISTRIBUTION: Correct Classification – 1 Mark; Proper labelled diagram

with equation – 3 Marks; Limitation & Hypothesis – 1 Mark *Hypothesis may vary as per

student’s knowledge.

Q-3

Ans-

What do you mean by cardio vascular variability? Suggest techniques to analyze

spontaneous variability.

Cardiovascular variability - Beat-to-beat fluctuations in the duration of the cardiac cycle,

arterial blood pressure, and cardiac output are well-known phenomena. Which postulates

that the spontaneous variability in heart rate and blood pressure results from chaotic

behavior occurring in the baro-reflex control system. The Heart rate variability (HRV) is the

physiological phenomenon of variation in the time interval between heartbeats. It is

measured by the variation in the beat-to-beat interval.

The SA node receives several different inputs and the instantaneous heart rate or RR interval

and its variation are the results of these inputs. The main inputs are the sympathetic and the

parasympathetic nervous system (PSNS) and humoral factors. Respiration gives rise to

waves in heart rate mediated primarily via the PSNS, and it is thought that the lag in the

baroreceptor feedback loop may give rise to 10 second waves in heart rate (associated with

Mayer waves of blood pressure), but this remains controversial.

Factors that affect the input are the baroreflex, thermoregulation, hormones, sleep-wake

cycle, meals, physical activity, and stress. Decreased PSNS activity or increased SNS

activity will result in reduced HRV. High frequency (HF) activity (0.15 to 0.40 Hz),

especially, has been linked to PSNS activity. Activity in this range is associated with the

respiratory sinus arrhythmia (RSA), a vaguely mediated modulation of heart rate such that

it increases during inspiration and decreases during expiration. Less is known about the

physiological inputs of the low frequency (LF) activity (0.04 to 0.15 Hz). Though previously

thought to reflect SNS activity, it is now widely accepted that it reflects a mixture of both

the SNS and PSNS.

[5]

Techniques to analyze spontaneous variability:

The most widely used methods can be grouped under time-domain and frequency-domain.

Other methods have been proposed, such as non-linear methods.

1. Time-domain methods

These are based on the beat-to-beat or Spontaneous variability, which are analysed to give

variables such as:

The standard deviation of NN (SDNN) intervals

Root Mean Square of Successive Differences (RMSSD)

standard deviation of successive differences (SDSD)

Estimated breath cycle (EBC)

2. Frequency-domain methods

Frequency domain methods assign bands of frequency and then count the number of

Spontaneous variability that match each band. Power spectral density (PSD), using

parametric or nonparametric methods, provides basic information on the power distribution

across frequencies. One of the most commonly used PSD methods is the discrete Fourier

transform. Methods for the calculation of PSD may be generally classified as nonparametric

and parametric. In most instances, both methods provide comparable results. The

advantages of the nonparametric methods are (1) the simplicity of the algorithm used (fast

Fourier transform [FFT] in most of the cases) and (2) the high processing speed, while the

advantages of parametric methods are (1) smoother spectral components that can be

distinguished independent of preselected frequency bands, (2) easy postprocessing of the

spectrum with an automatic calculation of low- and high-frequency power components with

an easy identification of the central frequency of each component, and (3) an accurate

estimation of PSD even on a small number of samples on which the signal is supposed to

maintain stationarity. The basic disadvantage of parametric methods is the need of

verification of the suitability of the chosen model and of its complexity (that is, the order of

the model).

Q-4 Write answers of below given questions. [10]

a. Enlist the experiments to measure saccade characteristics and also enlist various

saccadic parameters.

Ans- Experiments to measure saccade characteristics:

(1) EOG

(2) Video Occulography

(3) Scleral search coil

(4) Infrared Occulopathy

Saccadic Parameters:

(1) Latent Period (LP)

(2) Time to peak velocity

(3) Peak Velocity (PV)

(4) Saccade Duration (SD)

b. Discuss the result plots of simple lung mechanics model for 15 & 60 breaths/min.

Ans - At 15 breaths/min:

At this relatively low frequency, the volume waveform is more in phase with

Pao; the airflow, Q, shows a substantial phase lead relative to Pao.

This demonstrates that lung mechanics is dominated by compliance effects at

low frequencies.

At 60 breaths/min:

Peak airflow, Q, clearly increases, while tidal volume is decreased. Q has

become more in phase with Pao while volume displays a significant lag.

Thus, resistive effects have become more dominant at the higher frequency.

Conclusion:

The changes in peak Q and tidal volume with frequency demonstrate the phenomenon

of frequency dependence of pulmonary resistance and compliance, i.e. the lungs appear

stiffer and less resistive as frequency increases from resting breathing.

c. Write merits of distributed and lumped parameter models with appropriate

examples.

Ans- Lumped parameter

1. Single block can be derived, so that, it is very quickly designable.

2. The model is very simple and easy to understand.

3. Model optimization based on variable inputs can be easily acquired.

4. Transfer function can be very easily modified, so that, the ratio of output to input

can be optimized.

5. Ex. – A car as whole(Lumped) system. We can modify, modulate and manipulate

only the input (external) parameters, i.e. velocity, acceleration, angle of freedom,

maneuverability etc.

Distributed Parameters

1. Model is diversified and thus single parameter control can be acquired.

2. Micro-modifications can be done and thus model itself can be optimized.

3. More parameters can be included after isolated system identification.

4. Ex. – A car as individual(Distributed) system components. We can modify,

modulate and manipulate even the system parameters, i.e. Gearbox size, engine

cylinder volume, fuel tank size, suspension system etc., for better performance.

d. Give difference between steady state operating point & steady-state operating

level with an appropriate engineering system example.

Ans-

steady state operating point steady-state operating level The change in systems’ parameters will

cause the system to deviate initially, but

after some time the system will be settling

at one particular point and that point is said

to be steady state operating point.

The change in systems’ multiple

parameters will cause the system to

deviate initially, but after some time the

system will be settling at one particular

level and that point is said to be steady

state operating level.

Ex: Blood Pressure, Heart Rate Ex: Blood Glucose-Insulin concentration

Level

e. Discuss the effects of Hypoxia for inhalation of 12% of O2 (PIO2 = 86 mmHg)

Ans- The subject is inhaling only 12% of oxygen, thus PIO2 = 86 mmHg and PICO2 is left

to zero. Exposure to hypoxia tends to produce an additional drive to breathe, the added

ventilation blows off CO2, and consequently, the lower PAC02 acts to offset the hypoxic-

induced drive. As a result, the ventilation changes from its normoxic conditions. The graph

of PAO2 vs PACO2 shows completely linear relationship of PACO2 over the range of PAO2

values. It is stabilized on 39 as shown in figure.

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * *