Feedback Control in Physiology:
The Calcium Homeostatic System
Mustafa Khammash
Dept. of Electrical & Computer Engineering
Iowa State University, Ames, Iowa
Joint work with
Hana El-Samad, Jesse Goff (NADC)
1
Outline
• Blood Plasma Calcium Regulation
•
•
•
•
Calcium homeostasis in mammals
A model for calcium homeostasis
Hormonal interactions
Disorders
• Conclusions
2
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
• 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
[Ca]p
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
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
Parturition
18
20
22
time (days)
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
Plasma Ca (with Parturient Paresis)
Calcium Flow
Milk, fetus
Filtration
Formation
Bone
Calcium pool
Resorption
Kidney
reabsorption
Secretion
Absorption
Intestine
Mathematical Modeling of [Ca]p
Ca Total Supply Rate
VT (g/day)
Total Ca Clearance Rate
Vcl (g/day)
Intestinal Absorption
Plasma
Milk, fetus, urine, etc.
Bone Resorption
d
1
[Ca] p 
(VT  Vcl )
dt
Vol
Vol = Plasma Volume (l)
[Ca]p = Plasma Concentration (g/l)
t
1
[Ca] p 
(VT  Vcl )d

Vol 0
Vcl
VT
+
-
k
[Ca] p
Vcl
Set point
VT
e
+
-
Control
+
-
k
e = error (g/l) = set point (g/l) - [Ca] p
VT  f (e)
[Ca] p
Standard Model
• A model describing the relation between VT and [Ca]p is
given by Ramberg et al.:
VT  1770 ( s. p.  [Ca] p )
(g / l)
Source: Ramberg, Johnson, Fargo, and Kronfeld, “Calcium homeostasis in cows, with special
reference to parturient hypocalcemia,” Am. J. Physiol. , 1984.
• This is Proportional Feedback
VT  K p e
(g/l)
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 [Ca]p following
increased Calcium clearance at calving
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 [Ca]p time response
• We propose the feedback:
t
VT  K p e  K I  e d
0
PI Feedback
Vcl
KP
Set point
e
+
-
+
KI 
VT +
-
k
[Ca] p
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.
Model vs. Experiment
• Data from two groups of normal lactating dairy cows
around 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)
18
Model Prediction Vs. Actual Data
simulation output and actual data
0.09
0.085
0.08
0.075
0.07
0.065
0.06
0.055
0.05
10
11
12
13
14
15
16
17
18
19
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?
Can a single hormone be at work?
VT
• P feedback:

[Hormone A]
[Hormone A]

d
[Hormone A]
• PI feedback:
dt
Error

d
(Error  k Error)
dt
A Two Hormone Solution…
VT  VA  VB ; VA
[Hormone A]

d
[Hormone B]
dt
VT

 [Hormone A];
VB
Error

[Hormone A]
Error  k  Error
 [Hormone B]
Hormonal Regulation
The Parathyroid Gland monitors blood
calcium and secretes Parathyroid Hormone
(PTH) in proportion to [Ca] deficiency
PTH stimulates renal calcium
reabsorption and bone resorption
(1,25 OH2 D3) Hormone stimulates
calcium absorption from the intestine
Bone resporption and intestinal absorption
account for the entire calcium supply
[PTH ]
VA
VB

Error
 [PTH ]
 [1,25 OH D ]
2
VT  VA  VB
3
Setpoint Origin: The Parathyroid
Glands
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
25 (OH)D
PTH
1,25 (OH)2D
For a given [25 (OH)D]:
d
[1,25 (OH) 2 D]
dt

[PTH ]
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
Nonlinear Effects
• The supply of calcium from the bone cannot be increased
indefinitely in response to an increases in [PTH]
Set point
+
e
-
VT
KP
KI 
+
Vcl
+
-
k
[Ca] p
Absorption Nonlinear Effects:
Rumen Motility
• When [Ca]p 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
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.
Set point
+
e
-
VT
KP
+
KI 
x
Vcl
+
-
k
f (.)
[Ca] p
Absorption Reduction Factor
1.1
1
0.9
0.8
motility
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.02
0.03
0.04
0.05
0.06
0.07
calcium concentration in g/l
0.08
0.09
0.1
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 K p and K I
• Larger values of K p and K I lead to smaller undershoot in
the linear model
Phase Portrait for Kp=5000, Ki=3000
Phase portrait for Kp=5000 and Ki=3000
100
90
80
70
x2
Equilibrium
(high clearance rate) 60
50
40
30
20
10
0
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
x1
Initial condition (low clearance EP)
Phase Portrait for Kp=3000, Ki=1200
Phase portrait for Kp=3000 and Ki=1200
100
90
80
70
x2
Equilibrium
(high clearance rate) 60
50
40
30
20
10
0
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
x1
Initial condition (low clearance EP)
A Sufficient Condition for
Breakdown
1
x1 
[ f ( x1 ) x2  sat S ( K p (r  x1 ))  Vcl ]
vol
.
.
x2  K I (r  x1 )
Suppose x1 (0)  x10  r , and x2 (0)  x. If
 S  Vcl
Vcl  S
vol. 
where
0




:

,
2
KI r
vol
x
 
Then x1 (t ) (t  0) will be monotonica lly decreasing and for
 f ( x10 ) 
x10
some T 
, x1 (T )  0.

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
36
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
37
• 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
38