Feedback Control in Physiology The Calcium Homeostatic System

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 3

• 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 4

Calcium Regulation in the Cow • Constant plasma concentrations of calcium are easily maintained during periods on nonlactation (daily need is typically less than 20 g/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 5

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 12 14 16 Parturition 18 20 22 time (days) 6

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 7

Plasma Ca (with Parturient Paresis) 8

Calcium Flow Milk, fetus Filtration Formation Bone Calcium pool Resorption Kidney reabsorption Secretion Absorption Intestine 9

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 Vol = Plasma Volume (l) [Ca]p = Plasma Concentration (g/l) 10

Vcl VT + - 11

Vcl Set point e + - Control VT + - e = error (g/l) = set point (g/l) - 12

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 13

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 14

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: 15

PI Feedback Vcl Set point e + - + VT + - 16

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. 17

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 19

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? 20

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

A Two Hormone Solution… 22

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 OH 2 D 3) Hormone stimulates calcium absorption from the intestine Bone resporption and intestinal absorption account for the entire calcium supply 23

Setpoint Origin: The Parathyroid Glands 24

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 OH 2 D 25 (OH)D PTH 1, 25 (OH)2 D For a given [25 (OH)D]: 25

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 26

Nonlinear Effects • The supply of calcium from the bone cannot be increased indefinitely in response to an increases in [PTH] Set point + e - VT + Vcl + - 27

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 28

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. 29

Set point + e - VT + Vcl + - x 30

Absorption Reduction Factor 31

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 the linear model lead to smaller undershoot in 32

Phase Portrait for Kp=5000, Ki=3000 Equilibrium (high clearance rate) 33 Initial condition (low clearance EP)

Phase Portrait for Kp=3000, Ki=1200 Equilibrium (high clearance rate) Initial condition (low clearance EP) 34

A Sufficient Condition for Breakdown 35

Summary & Future Work • Calcium homeostasis is achieved through integral feedback. Integral action is realized by the dynamic interaction among 1, 25 (OH)2 D 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

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