Feedback Control in Physiology: The Calcium Homeostatic System

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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)
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Outline

• Blood Plasma Calcium Regulation
• Calcium homeostasis in mammals
• A model for calcium homeostasis
• Hormonal interactions
• Disorders
• Conclusions

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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

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• 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

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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

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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
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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
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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

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Plasma Ca (with Parturient Paresis)
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Calcium Flow
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Mathematical Modeling of Cap
Plasma
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e error (g/l) set point (g/l) -
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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

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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

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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

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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.

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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)

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Model Prediction Vs. Actual Data
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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?

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

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• A Two Hormone Solution

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Hormonal Regulation
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Setpoint Origin The Parathyroid Glands
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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
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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

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Nonlinear Effects

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

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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

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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.
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Vcl
VT
-
Set point
e



-
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Absorption Reduction Factor
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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

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Phase Portrait for Kp5000, Ki3000
Initial condition (low clearance EP)
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Phase Portrait for Kp3000, Ki1200
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A Sufficient Condition for Breakdown

• then, will be monotonically
  decreasing, and
• for some ,

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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

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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

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• 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

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