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Feedback Control in Physiology:The 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

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

10 12 14 16 18 20 220.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

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0.095

0.1

10 12 14 16 18 20 220

10

20

30

40

50

60

70

80

90

100

Plasma Ca Concentration (g/l)

Ca Clearance Rate

Parturitiontime (days)

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

Calcium poolFiltration

reabsorptionKidney

Milk, fetus

Absorption

Intestine

Secretion

Resorption

Formation

Bone

Mathematical Modeling of [Ca]p

)(1

][ clTp VVVol

Cadt

d

Vol = Plasma Volume (l)

[Ca]p = Plasma Concentration (g/l)

Ca Total Supply Rate

VT (g/day)

Intestinal Absorption

Bone Resorption

Total Ca Clearance Rate Vcl (g/day)

Milk, fetus, urine, etc.Plasma

t

clTp dVVVol

Ca0

)(1

][

k

Vcl

VT pCa][-+

e = error (g/l) = set point (g/l) - pCa][

k

Vcl

VT pCa][-+

-+ ControlSet point e

)(efVT

Standard Model

• A model describing the relation between VT and [Ca]p 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

)/()][..(1770 lgCapsV pT

(g/l)eKV pT

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

t

IpT deKeKV0

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:

k

Vcl

VT pCa][-+

-+

Set point e+

PK

IK

PI Feedback

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)

Model Prediction Vs. Actual Data

10 11 12 13 14 15 16 17 18 19 200.05

0.055

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0.065

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0.075

0.08

0.085

0.09simulation output and actual data

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?

• P feedback:

• PI feedback: Error) Error A] [Hormone (dt

dk

dt

d

]AHormone[ T V

Error A] [Hormone

A Two Hormone Solution…

]AHormone[ ]BHormone[ dt

d

Error]AHormone[

;BAT VVV ]BHormone[ ];AHormone[ BA VV

ErrorkErrorVT

Hormonal Regulation

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

The Parathyroid Gland monitors blood calcium and secretes Parathyroid Hormone (PTH) in proportion to [Ca] deficiency

Error]PTH[

]PTH[ AV

]OH 1,25[ 32DVB

BAT VVV

Setpoint Origin: The Parathyroid Glands

The Integral Term

PTH25 (OH)D 1,25 (OH)2D

]PTH[ D](OH) [1,25 2 dt

d

For a given [25 (OH)D]:

• 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

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

k

Vcl

VT pCa][-+

-+e

+PK

IK

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

Source: R.C. Daniel, “Motility of the Rumen and Abomasum During Hypocacemia,” Can. J Comp Med 1983.

Rumen Contractions

Abumasal Contractions

Normal

DuringHypocalcemia

Normal

DuringHypocalcemia

Set point

k

Vcl

VT pCa][-+

-+e

+PK

IK (.)fx

Absorption Reduction Factor

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

calcium concentration in g/l

mot

ility

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

Ip KandK

Ip KandK

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

10

20

30

40

50

60

70

80

90

100

x1

x2Phase portrait for Kp=5000 and Ki=3000

Phase Portrait for Kp=5000, Ki=3000

Equilibrium(high clearance rate)

Initial condition (low clearance EP)

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

10

20

30

40

50

60

70

80

90

100Phase portrait for Kp=3000 and Ki=1200

x1

x2

Phase Portrait for Kp=3000, Ki=1200

Equilibrium(high clearance rate)

Initial condition (low clearance EP)

A Sufficient Condition for Breakdown

then, will be monotonically decreasing, and

for some ,

)(

]))(()([1

1

.

2

121

.

1

xrKx

VxrKsatxxfvol

x

I

clpS

.0)( , some

for and decreasinglly monotonica be will)0()(Then

,:0 where.

)(

If.)0( and ,)0( Suppose

110

1

210

2101

Txx

T

ttx

vol

SV

rKx

volxf

VS

xxrxx

cl

I

cl

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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