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Feedback Control in Physiology: The Calcium Homeostatic System

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Title: Feedback Control in Physiology: The Calcium Homeostatic System


1
Feedback Control in PhysiologyThe Calcium
Homeostatic System
Mustafa Khammash Dept. of Electrical Computer
EngineeringIowa State University, Ames,
Iowa Joint work withHana El-Samad, Jesse Goff
(NADC)
2
Outline
  • Blood Plasma Calcium Regulation
  • Calcium homeostasis in mammals
  • A model for calcium homeostasis
  • Hormonal interactions
  • Disorders
  • Conclusions

3
Physiological Role of Calcium
  • Maintain the integrity of the skeleton.
  • Control of biochemical processes
  • Intracellular
  • Activity of a large number of enzymes
  • Conveying information from the surface to the
    interior of the cell
  • Extracellular
  • Muscle and nerve function
  • Blood clotting

4
  • The biochemical role of Calcium requires that its
    blood plasma concentrations be precisely
    controlled
  • Normal concentration of about 9 mg/dl must be
    maintained within small tolerances despite
  • variations in dietary calcium levels
  • variation in demand for calcium
  • Humans and other mammals have an effective
    feedback mechanism for regulating plasma
    concentration of calcium Cap

5
Calcium Regulation in the Cow
  • Constant plasma concentrations of calcium are
    easily maintained during periods on nonlactation
    (daily need is typically less than 20g/day)
  • An especially large loss of plasma calcium to
    milk takes place during lactation (up to 50
    g/day)
  • Most animals adapt to the onset of lactation

6
Plasma Ca Concentration (g/l)
0.1
0.095
0.09
0.085
0.08
0.075
0.07
0.065
0.06
0.055
0.05
10
12
14
16
18
20
22
time (days)
Ca Clearance Rate
100
90
80
70
60
50
40
30
20
10
0
10
12
14
16
18
20
22
time (days)
Parturition
7
Parturient Paresis
  • In some cows, the calcium regulatory mechanism
    fails to meet the increased calcium demands
  • These animals become severely hypocalcemic
  • Results in disruption of muscle and nerve
    function
  • Leads to recumbency
  • The clinical syndrome is referred to as
    Parturient Paresis (Milk Fever)
  • It affects 6 of the dairy cows in the US

8
Plasma Ca (with Parturient Paresis)
9
Calcium Flow
10
Mathematical Modeling of Cap
Plasma
11
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12
e error (g/l) set point (g/l) -
13
Standard Model
  • A model describing the relation between VT and
    Cap is given by Ramberg et al.
  • Source Ramberg, Johnson, Fargo, and
    Kronfeld, Calcium homeostasis in cows, with
    special reference to parturient hypocalcemia,
    Am. J. Physiol. , 1984.
  • This is Proportional Feedback

14
Deficiencies in the Standard Model
  • From basic principles of control theory,
    proportional feedback alone cannot explain
  • The observed zero steady-state error (Perfect
    Adaptation)
  • The shape of the time response of Cap following
    increased Calcium clearance at calving

15
Integral Feedback
  • In order to account for the zero state-state
    error integral feedback must be present.
  • When combined with Proportional Feedback,
    Integral Feedback will account for
  • The zero steady-state error in response to Ca
    clearance
  • The second order shape of the Cap time response
  • We propose the feedback

16
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17
Implications of PI Feedback
  • At any given time, the calcium supply rate VT is
    not dictated only by the level of calcium
    deficiency at that time.
  • Supply rate depends on both the level and
    duration of calcium deficiency prior to and until
    the time of interest.
  • Understanding the dynamics of the system is
    unavoidable.

18
Model vs. Experiment
  • Data from two groups of normal lactating dairy
    cowsaround the day of calving (NADC)
  • One group was used to determine model parameters
  • The model prediction was compared against data
    from the larger second group (20 animals)

19
Model Prediction Vs. Actual Data
20
How Do Cows Integrate?
  • Our model was arrived at through necessity
    arguments
  • Is there a plausible physiological basis?
  • Given that calcium is hormonally regulated, what
    is the mechanism through which integration is
    realized?

21
  • Can a single hormone be at work?
  • P feedback
  • PI feedback

22
  • A Two Hormone Solution

23
Hormonal Regulation
24
Setpoint Origin The Parathyroid Glands
25
The Integral Term
  • Two forms of Vitamin D 25 (OH)D and 1,25
    (OH)2 D
  • PTH activates 25 (OH)D in the kidney to form
    1,25 OH2 D

For a given 25 (OH)D
26
Understanding Parturient Paresis
  • In normal animals, a linear model was adequate
    for describing observed regulatory response
  • However, the linear model alone cannot account
    for
  • Breakdown in Ca seen in cows with Parturient
    Paresis
  • Recovery after Calcium IV infusion

27
Nonlinear Effects
  • The supply of calcium from the bone cannot be
    increased indefinitely in response to an
    increases in PTH

28
Absorption Nonlinear Effects Rumen Motility
  • When Cap is significantly reduced, the
    processes responsible for intestinal absorption
    will be impacted
  • The net result is a slowing of intestinal
    absorption when it is most needed
  • A clear example is the impact of reduced plasma
    calcium levels on rumen motility

29
Hypocalcemia Affects Motility
Rumen Contractions
Normal
During Hypocalcemia
Abumasal Contractions
Normal
During Hypocalcemia
Source R.C. Daniel, Motility of the Rumen and
Abomasum During Hypocacemia, Can. J Comp Med
1983.
30
Vcl
VT
-
Set point
e



-
31
Absorption Reduction Factor
32
Exploring the Model Properties
  • With both nonlinear effects included, calcium
    break-down does take place
  • Breakdown depends on the saturation level,
    absorption reduction function, and the linear
    model parameters
  • Fixing the nonlinear elements, breakdown depends
    entirely on the values of
  • Larger values of lead to
    smaller undershoot in the linear model

33
Phase Portrait for Kp5000, Ki3000
Initial condition (low clearance EP)
34
Phase Portrait for Kp3000, Ki1200
35
A Sufficient Condition for Breakdown
  • then, will be monotonically
    decreasing, and
  • for some ,

36
Summary Future Work
  • Calcium homeostasis is achieved through integral
    feedback. Integral action is realized by the
    dynamic interaction among 1,25 (OH)2D and PTH
  • Sequence of discovery Perfect adaptation
    ?necessity of integral action ? specific action
    at molecular level
  • The dynamic interactions give a new perspective
    on calcium homeostasis disorders and disease
    trajectories
  • Future work
  • Osteoporosis
  • Other homeostatic mechanisms, e.g. blood sugar,
    diabetes

37
Control Theory in Biological Systems
  • Feedback regulation mechanisms are ubiquitous
  • Bring out the dynamic nature of biochemical
    interactions
  • Explain interactions in the context of
    regulation
  • Pathologic behavior when systems operate at their
    extremes. Capturing the dynamics will
  • lead to better understanding of the trajectory of
    disease
  • suggest more effective courses of treatment

38
  • Identify functional biological modules
  • Reveal structural constraints on the dynamics
  • Structural constraints impose functional
    requirements on biological modules
  • Easier to understand/predict the function of
    sub-modules
  • New understanding of the behavior of biological
    subsystems
  • Notions such as robustness, adaptation,
    amplification, isolation, and nonlinearity are
    required for a deeper understanding of biological
    function
  • Many similarities with engineering systems
  • Ask the right questions

39
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40
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