A control engineering model of calcium regulation

A Control Engineering Model of Calcium Regulation

Christopher R. Christie, Luke E. K Achenie, and Babatunde A. Ogunnaike

Department of Chemical Engineering (C.R.C., L.E.K.A.), Virginia Polytechnic Institute and StateUniversity, Blacksburg, Virginia 24060; and Department of Chemical Engineering (B.A.O.), University ofDelaware, Newark, Delaware 19716

Context: A control engineering perspective provides a framework for representing importantmechanistic details of the calcium (Ca) regulatory system efficiently. The resulting model facilitatesthe testing of hypotheses about mechanisms underlying the emergence of known Ca-relatedpathologies.

Objective: The objective of this work is to develop a comprehensive computational model that willenablequantitativeunderstandingofplasmaCaregulationundernormalandpathologicalconditions.

Design: Ca regulation is represented as an engineering control system where physiological sub-processes are mapped onto corresponding block components (sensor, controller, actuator, andprocess), and underlying mechanisms are represented by differential equations. The resultingmodel is validated with clinical observations of induced hypo- or hypercalcemia in healthy subjects,and its applicability is demonstrated by comparing model predictions of Ca-related pathologies tocorresponding clinical data.

Results: Our model accurately predicts clinical responses to induced hypo- and hypercalcemia inhealthy subjects within a framework that facilitates the representation of Ca-related pathologies interms of control system component defects. The model also enables a deeper understanding of theemergence of pathologies and the testing of hypotheses about related features of Ca regulation—forexample, why primary hyperparathyroidism and hypoparathyroidism arise from “controller defects.”

Conclusions: The control engineering framework provides an efficient means of organizing thesubprocesses constituting Ca regulation, thereby facilitating a fundamental understanding of thiscomplex process. The resulting validated model’s predictions are consistent with clinically observedshort- and long-term dynamic characteristics of the Ca regulatory system in both healthy anddiseased patients. The model also enables simulation of currently infeasible clinical tests andgenerates predictions of physiological variables that are currently not measurable. (J Clin Endo-crinol Metab 99: 2844–2853, 2014)

Calcium (Ca)-dependent physiological processes re-quire that, for normal human bodily functions, total

extracellular fluid (ECF) Ca concentration be maintainedwithin a narrow range, 2.45 � 0.25 mM (1, 2). This ho-meostatic requirement is met through the various pro-cesses constituting the body’s Ca regulatory system. Dis-orders in the Ca regulatory mechanisms cause abnormalhormonal secretion and contribute to chronic imbalancesin plasma Ca levels, often leading to many other long-term

physiological problems (2, 3). For example, enlargedparathyroid glands (PTGs) lead to oversecretion of PTH,causing hypercalcemia and excessive bone loss.

Human Ca homeostasis is achieved through the coor-dinated actions of four organs: the PTG, kidneys, bone,and intestines. Ca transfer between the plasma and thekidneys, intestines, and bone is regulated by the secre-tion of three hormones: calcitriol (CTL), calcitonin, andPTH (1, 2). Figure 1A shows a schematic and describes

Abbreviations: ADH, autosomal dominant hypoparathyroidism; CaR, Ca-sensing receptor;CB, osteoclast-to-osteoblast; CTL, calcitriol; ECF, extracellular fluid; ECS, engineering con-trol system; FBH, familial benign hypocalciuric hypercalcemia; HHM, humoral hypercalce-mia of malignancy; HoPT, hypoparathyroidism; PHPT, primary hyperparathyroidism; PTG,parathyroid gland.

O R I G I N A L A R T I C L E

E n d o c r i n e R e s e a r c h

2844 jcem.endojournals.org J Clin Endocrinol Metab, August 2014, 99(8):2844–2853 doi: 10.1210/jc.2013-3451

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the mechanisms of hormonal Ca and phosphate (PO4)regulation.

Engineering control systems (such as a residentialhome’s heating, ventilation, and air conditioning system)consist of four basic elements: 1) the sensor receives rele-vant information about the variable of interest (room tem-perature) from 2) the controlled process (the room in ques-tion), and transmits this information to 3) the controller(the wall-mounted thermostat), which determines andsends appropriate corrective action signals to 4) the actu-

ator (the compressor and fan) for implementation of theprocess (4). The biological control system responsible forCa regulation may be represented similarly by identifyingthe physiological equivalents of each engineering controlsystem (ECS) component as follows: 1) the sensor is theCa-sensing receptor (CaR); 2) the controlled process is theplasma Ca and PO4 pools; 3) the controller is the PTG; and4) the actuators are the kidneys, bone, and intestines. Thisleads to the block diagram representation of Ca regulationin Figure 1B, where each block represents a control system

Figure 1. A, Schematic of physiological components and mechanisms involved in Ca homeostasis. CaRs located on the surface of the PTG detectsub-basal Ca levels (hypocalcemia) and stimulate the production and secretion of PTH through a complex intracellular pathway (1). In the kidneys,high levels of PTH induce two distinct responses: 1) production of CTL (1,25 dihydroxyvitamin D) via the hydroxylation of 25-hydroxyvitamin D (2);and 2) increased Ca reabsorption in the renal tubules (3). Finally, increased levels of PTH, in concert with CTL, stimulate increased bone cellproliferation, causing a net bone resorption and subsequent transfer of Ca from bone to the plasma. CTL, in turn, inhibits PTH secretion byinhibiting PTG growth and increases intestinal Ca absorption (1–3, 5). The resulting net transfer of Ca from the kidneys, bone, and intestines tothe plasma increases plasma Ca to basal levels (normocalcemia). High Ca levels (hypercalcemia) produce the opposite response in each organ.Phosphate regulation: Supranormal PO4 levels (hyperphosphatemia) stimulate PTH secretion, which causes increased bone resorption (net transferof PO4 to plasma) and reduced renal PO4 reabsorption (increased renal PO4 excretion). CTL production is induced by the high PTH levels butinhibited by hyperphosphatemia (6, 7); the net effect is a reduction in CTL levels, leading to reduced intestinal PO4 absorption (8). The overallresult is a return to normal plasma PO4 levels (normophosphatemia). B, Corresponding ECS block diagram representation of plasma calciumregulation. The connecting lines between the individual blocks represent information flow (ion/hormone concentrations or ion transfer rates)between the blocks.

doi: 10.1210/jc.2013-3451 jcem.endojournals.org 2845

component connected by lines representing the informa-tion emanating from and conveyed to each component,according to the mechanistic description of the physiolog-ical processes represented by the block in question.

From this perspective, we observe that computationalmodels of human Ca regulation in the literature have beencast in the form of a series of graphical interpolations (9)or a single monolithic system of differential equations (10,11). Alternatively, organizing the complete Ca regulatorysystem into interconnected functional components pro-duces a modular representation that is easier to validatewith clinical data module-by-module; it also results in anoverall system that is easier to analyze holistically andfrom which greater fundamental insight can be derived.

Materials and Methods

The dynamic behavior of PTH, CTL, bone cell proliferation,ECF Ca, and phosphate are represented mathematically in theform of nonlinear ordinary differential equations derived frommaterial balances applied to each of the component subprocessesas depicted in Figure 1B. The resulting equations, relevant pa-rameters, and initial conditions are summarized in Tables 1 and2 and Supplemental Tables 1–3. A detailed mechanistic descrip-tion of each component subprocess along with the general as-sumptions and other considerations follow.

1) All hormones and ions are uniformly distributed in theECF, and their concentrations are identical to that in the plasma.

2) Consistent with receptor theory and physiological dose-response characteristics (12, 13), the rates of secretion, produc-tion, or proliferation and all hormone/ion dependencies are de-scribed by the logistic function, H (x), given in Equation 1. Theinput variable, x, is the stimulus used to elicit the response, H; Ax

and Bx are, respectively, the minimum and maximum rates; Sx isthe midpoint (the value of x corresponding to a half-maximalresponse); mx is the slope of the curve at Sx.

3) Phosphate regulation is based on the description in Ref. 11.4) The effect of calcitonin is ignored because its role in Ca

regulation is not as important as that of PTH and of CTL (1, 2).5) Least-squares optimization is employed in estimating the

parameter values, and all model simulations are carried out usingMATLAB® and Simulink®.

6) Component defects are implemented as first-order dy-namic changes over time in the relevant parameter values.

Sensor: G protein-coupled CaRsCa regulation begins with the detection of Ca ions that bind

to an amino-terminal domain located on the surface of the CaR.The CaRs are attached to the surface of the parathyroid (PT) cellvia coupling to G proteins, which transmit the detected Ca signalto the cell (14). In some families, hereditary mutations of the CaRgene result in familial benign hypocalciuric hypercalcemia(FBH)—a loss-of-function mutation characterized by elevatedPTH, low Ca excretion, and mild-to-high hypercalcemia; or au-tosomal dominant hypoparathyroidism (ADH)—a gain-of-sense mutation, characterized by low PTH, high urinary Ca ex-cretion, and hypocalcemia (15).

In our model, the two processes of CaR sensing of plasma Ca,Ca(p) and G protein transmission of the signal, are combined andare represented by a single, linear static relationship betweenCa(p) and the sensor output, Ca(s), where the constant of pro-portionality is the sensor gain, Ksens. For a healthy individual,Ksens is unity; for those with genetic mutations in the CaR, thegain-of-function mutation is represented as an increase in Ksens

for ADH, or for FBH, missense mutation is represented as adecrease in Ksens.

Controller: the PTGsPT cells have a 2-fold response to changes in Ca(p): 1) cellular

proliferation; and 2) increased PTH production (4), jointly re-

Table 1. Model Equations

Logistic Equation PO4�i� � �PO4�i�PO4meal (9)H�x� � �Ax � Bx�/�1 � �x/Sx�

mx� � Bx (1) Actuator - KidneysProcess–Plasma dCTL/dt � �H1�PTH� � H�PTHrP��H�PO4� � kCTLCTL (10)

dCa� p�/dt � Ca�i� � Ca�b� � Ca�u� � Ca�d� (2)Ca�u� �

V �Ca� p��0.1 � 0.09H2�PTH��, Ca� p� � Cathr

��CauCa� p� � �Cau�, Ca� p� Cathr� (11)

dPO4� p�/dt � PO4�i� � PO4�b� � PO4�u� � PO4�ic� (3)A PO4�u� �GFR

V�PO4uPO4� p�

Sensor–CaR on PTG Actuator - Bone

Ca�s� � KsensCa� p� (4)Ca�b� � Cab�

�1 � �Cab� � �Cab�H�OC��RANKL/OC��Cab�

�Ca� p�

Ca� p�0��1 � �Cab� � �Cab�OB

OBo�� (13*)A

Controller–Parathyroid Gland (PTG)�p � �b�PTH � PTHrP

PTHo��b� PTH � PTHrp

PTH � PTHrp � PTHo�� PS� (14*)C

dPTH/dt � H�Ca�s��PTG � kPTHPTH (5) PO4�b� � �PO4bCa�b� (15)

�PTG

PTGo��PTGmax � PTG�� PTGTPTG

� � �1 � PTG�� PTG� PTGTPTG

� � �1 � PTG�� (6)B dPO4�ic�/dt � kaPO4icPO4� p� � kbPO4icPO4�ic� (16)A

TPTG� � 1 � tanh��CTL�CTL � CTLo�� (7)B Model Disturbance–PTHrP production

Actuator - Intestines dPTHrP/dt � V�RPTHrp � kPTHrPPTHrP� (17)

Ca�i� �Cameal

V�H�CTL� � �Cai�

AEquation taken from (8); BTPTG represents the inhibitory effect of CTL of PTG growth; the equations are taken from (7); CEquation modified from(14).

*The detailed description of the bone cell proliferation model for determining RANKL, OB, and OC is provided in (14).

Subscripts denoting the different Ca or PO4 pools; (b) bone; (d) disturbance; (i) intestine; ((ic) intracellular; (p) plasma; (s) sensor; (u) urine.

2846 Christie et al Control eng Model of Ca Regulation J Clin Endocrinol Metab, August 2014, 99(8):2844–2853

sulting in an overall increase in PTH secretion. Brown (30) firstproposed a reverse sigmoid relationship to represent the com-bined Ca-mediated PTG response. However, as shown in Figure2A, the change in PTH as a function of Ca(p) is asymmetric: it isgenerally higher under conditions of hypocalcemia (Figure2A) and lower under hypercalcemia (Figure 2A) (1, 4, 21). Weuse a model proposed in Ref. 31 to capture this asymmetry.Estimates of the unknown parameters in the response func-tion, H(Ca(s)), are determined from published clinical data onPTH responses to induced hypo- and hypercalcemia in healthysubjects (24, 32, 33).

Increased levels of CTL are known to inhibit PT cellular pro-liferation both in vivo and in vitro (34)—a form of regulatorynegative feedback. Although the mechanisms by which this oc-curs are not yet fully understood (35), a cell-surface receptor-likemechanism has been proposed (4). The overall inhibition of PTcellular proliferation by CTL is therefore represented as in Ref.(7). Additionally, consistent with the literature (3, 21), the PTHhalf-life in the plasma (related to the rate of PTH decay, kPTH) isset at 1.3 minutes.

Actuator: kidneys—CTL productionCirculating vitamin D3 undergoes 25-hydroxylation (a loosely

regulated process in the liver), forming 25-hydroxyvitamin D3. In

thekidneys,25-hydroxyvitaminD3 thenundergoesPTH-andPO4-controlled renal hydroxylation to form the physiologically activeCTL (1,25-dihydroxyvitamin D3) (6, 7). In the model, we assumean infinite supply of CTL precursors, so that hepatic hydroxylationmay be ignored. On the other hand, we represent renal hydroxy-lation as a product of the PTH- and PO4-dependent logistic func-tions, estimating the associated parameters from clinical data (19,26,27).Additionally, aCTLhalf-lifeof8hours (20)wasused in thecalculation of the CTL elimination rate.

Actuator: kidneys—renal calcium reabsorption“Ca excretion” in the kidneys is defined as the difference

between the filtered load (nonprotein-bound Ca or solubleCa) and the amount reabsorbed. The filtered Ca is computedas the net difference between total plasma Ca and insolubleCa. About 80 –90% of the filtered Ca is reabsorbed by pro-cesses linked to sodium reabsorption (36), and approximately10% of the filtered Ca is further reabsorbed by PTH-depen-dent processes (1). Additionally, the Ca reabsorption is lim-ited by the capacity of the renal tubules (1). The net effect isa maximum tubular reabsorptive capacity for calcium (18);this is represented as “Ca threshold” (Cathr), the concentra-tion of Ca(p) above which there is no additional renal reab-

Table 2. Parameter Estimates

Parameter Value Units Source Parameter Value Units Source

Ca(p) 0 1.715E � 01 mmol (17) PTG 8.500E-01 Unitless (10)CTL0 1.260E � 03 pmol (17) ACa(s) 4.641E � 02 pmol�h�1 LSPE (24)OB0 7.282E-04 pM cells (16) BCa(s) 6.276E � 03 pmol�h�1 LSPE (24)PO4(ic)0 4.516E � 04 mmol Calculated mCa(s)A �3.000E � 01 Unitless LSPE (24)PO4(p)0 1.680E � 01 mmol (11) mCa(s)B �2.500E � 02 Unitless LSPE (24)PTG0 5.000E-01 Unitless Estimated mCa(s)m �1.500E � 02 Unitless LSPE (24)PTH0 5.526E � 01 pmol (17) mCa(s)S1 1.800E � 01 mmol LSPE (24)Cameal 9.158E-01 mmol�h�1 Calculated mCa(s)S2 1.500E � 01 mmol LSPE (24)Cathr 3.108E � 01 mmol LSPE (18) ACTL 4.150E-01 Unitless LSPE (25)GFR 6.000E � 00 L�h�1 (19) BCTL 0.000E � 00 Unitless LSPE (25)ka,PO4ic 5.180E � 01 h�1 (11) mCTL �8.416E � 00 Unitless LSPE (25)kb,PO4ic 1.927E-02 h�1 (11) SCTL 1.306E � 03 pmol CalculatedkCTL 8.660E-02 h�1 (20) A1,PTH 3.228E � 00 h�1 LSPE (26, 27)kPTH 3.199E � 01 h�1 LSPE (21) B1,PTH 1.000E � 00 h�1 LSPE (26, 27)kPTHrP 6.932E � 00 h�1 LSPE (22) m1,PTH �2.000E � 01 Unitless LSPE (26, 27)Ksens 1.000E � 00 Unitless Estimated S1,PTH 9.823E � 01 pmol LSPE (26, 27)PO4,meal 5.695E-01 mmol�h�1 Calculated A2,PTH 1.045E � 00 Unitless LSPE (1)PS 1.500E � 02 Unitless Calculated (16) B2,PTH 0.000E � 00 Unitless LSPE (1)PTGmax 1.000E � 00 Unitless Estimated m2,PTH �6.520E � 00 Unitless LSPE (1)V 1.400E � 01 L (11) S2,PTH 5.186E � 01 pmol LSPE (1)�Cau 3.146E-01 Unitless LSPE (18) APO4 1.034E � 00 Unitless LSPE (19)�PO4u 5.545E-02 Unitless Calculated BPO4 4.182E-01 Unitless LSPE (19)�Cau �5.710E � 00 mmol LSPE (18) mPO4 1.242E � 01 Unitless LSPE (19)�Cab 6.038E-01 Unitless LSPE (28) SPO4 1.860E � 01 mmol LSPE (19)�b 2.907E � 00 Unitless LSPE (28) AOC 1.271E � 00 Unitless LSPE (28)�Cab 1.500E-01 Unitless LSPE (28) BOC 9.951E-01 Unitless LSPE (28)�Cai 1.500E-01 Unitless LSPE (23) mOC �1.188E � 00 Unitless LSPE (28)�CTL 2.143E-03 pmol�1 Calculated (10) SOC 9.615E-01 pM cells/pM cells LSPE (28)�PO4b 4.640E-01 Unitless (10) APTHrP �4.630E-01 h�1 LSPE (27)�PO4i 7.000E-01 Unitless (11) BPTHrP 5.280E-01 h�1 LSPE (27)�PTG 7.500E-02 h�1 LSPE (10) SPTHrP 1.534E � 00 pM LSPE (27)Cab 7.973E-01 h�1 LSPE (28) mPTHrP 5.888E � 00 Unitless LSPE (27)

RPTHrP 1.04E � 01 pM h�1 LSPE (29)

Source indicates: (x), values taken directly from the reference (x); LSPE (x), values determined using least-squares parameter estimation and dataprovided in reference (x); Calculated (x), values calculated from data provided in reference (x); Calculated, values determined assuming steady-stateinitial conditions; Estimated, values of 0.5 or 1 chosen as reference.

sorption. Finally, for normal bodily function, a minimum Caexcretion rate of approximately 1 mmol/d is necessary (1).

Renal excretion is represented in the model by a piecewisecontinuous curve where below the Ca threshold, tubular Careabsorption is 90% passive and 10% PTH-dependent andabove the threshold, Ca excretion varies linearly with Ca(p)

levels. Parameter values are determined from literature data(18).

Actuator: intestines—calcium absorptionThe rate of Ca absorption that matches the rate of renal

excretion at steady state (1) is regulated through both activeand passive pathways. Passive absorption via the paracellularpathway accounts for 8 –23% of total absorption, whereastranscellular active absorption is CTL-dependent (1, 23, 36).Therefore, we assume that passive absorption accounts for anaverage of 15% of total intestinal Ca absorption. Parametervalues for CTL-dependent active absorption are determinedfrom data published by Heaney et al (25).

Actuator: bone cellproliferation, formation, andresorption

The bone cell proliferation modelused in this work is based on Ref. 16. Toincorporate this bone model into ouroverall model, the PTH and PTHrP con-centrations in the fractional PTH recep-tor occupation expression (�P) were nor-malized as shown in Equation 14.Finally, the expression linking bone cellproliferation to calcium flux is takenfrom Ref. 11 with parameters estimatedusing clinical data (28).

Model disturbance: PTHrPproduction

To capture tumor-related effects on Caregulation adequately, the PTHrP produc-tion must be included in the model. PTHrPis important in many physiological pro-cesses in utero and neonatally, includingCa control (37), but it has no effect on cal-ciumcontrol in thehealthyadult (3).How-ever, its production by some cancerouscells causes abnormal responses in the Caregulatory system (38) (Figure 1B). Al-though PTHrP activates PTH receptors,thus effecting responses identical to thoseof PTH in the kidneys and bone, it stimu-lates less CTL production than does PTH(3, 27).

Estimates of unknown parametersassociated with plasma PTHrP levelsare based on the following data: PTHrPhas a half-life of 2 to 8 minutes (22);average basal PTHrP level in healthysubjects is 1.34 pM (39), and it variesbetween 2.4 and 51.2 pM in patientswith humoral hypercalcemia of malig-nancy (HHM) (29). In addition, be-cause PTHrP action is identical to that

of PTH in the bone and renal tubules but dissimilar in CTLproduction, PTHrP concentration is added to the PTH con-centration term in 1) the PTH-mediated CTL secretion term,H2(PTH) in Equation 11; and 2) the fractional PTH receptoroccupation expression (�P) in Equation 14; finally, an inde-pendent term for PTHrP-mediated CTL production is addedin Equation 10.

Process: ECF calcium and phosphateCa(p) and PO4(p) concentrations in the ECF are determined

from the net ion transfer rates from the kidneys, intestines, bone,and any “disturbances” introduced via iv infusions.

Results

In what follows, the basic model is first validated againstclinical data from healthy subjects. Subsequently, the utilityof the validated model is demonstrated through a systematic

0.8 0.9 1 1.1 1.2

a�ve

Plasma Ca rela�ve to baseline

HCaHoCaCombined

5 10 15 20

ela�

Time (hrs)

5 5.5 6 6.5 7

Time (hrs)

CaPTHClin - CaClin - PTH

5 15 25 35

Time (hrs)

Ca PTHCTL CauCB

Time (hrs)

CaPTHCa - ClinPTH - Clin

5 15 25

Time (hrs)

Ca PTHCTL CauCB

Figure 2. A, PTH secretory response to plasma Ca during both hypercalcemia (HCa) andhypocalcemia (HoCa). Our model captures the asymmetry with the indicated “combined” curve. B–F,Response of healthy subjects to induced hypo- and hypercalcemia. B, Calcium response to 2-hourinfusion of sodium EDTA (28). C, Ca and PTH response during 2-hour infusion of sodium citrate (24).D, Model prediction of Ca, PTH, CTL, urinary Ca, and CB ratio response during and after 2-hourinfusion of sodium citrate. E, Ca and PTH response during 2-hour infusion of calcium gluconate (32).F, Model predictions of Ca, PTH, CTL, urinary Ca, and CB ratio response during and after 2-hourinfusion of calcium gluconate.

comparison of model predictions of various Ca-related pa-thologies against corresponding physiological data.

An importantattributeof themodel is its ability topredictthe dynamic responses of CTL, renal Ca excretion, bone cellproliferation (as indicated by the osteoclast-to-osteoblast[CB] ratio), and other such physiological variables that areoften unavailable for direct measurement but which provideadditional insight into the overall performance of the Caregulatory system. For ease of comparison, the simulationresults are presented not in their usual clinical units but aschanges relative to healthy baseline values.

Model validationThe datasets used for model validation are from clini-

cally induced hypo- and hypercalcemia in healthy sub-jects, in response to infusions of EDTA, sodium citrate, orcalcium gluconate (24, 28, 32). The corresponding modelsimulations are obtained by representing the experimentalstimulus as an appropriate Ca(p) disturbance with magni-tude and duration matching those of the clinical infusion.The results are shown in Figure 2.

Figure 2B shows scaled measurements of the averageplasmaCaresponseofagroupofhealthypatients toa2-hourinfusion of sodium EDTA followed by a 10-hour recovery(28), with the corresponding model simulation superim-posed. Similarly, as shown in Figure 2C, we are able to pre-dict the PTH response during a 2-hour sodium citrate infu-sion (24) accurately. In response to the same sodium citrateinfusion, Figure 2D shows the corresponding dynamic pro-filesofCTL,renalCaexcretion(Ca(u)), andCBratio—quan-tities not reported in the study possibly because they are un-available by measurement. Induced hypocalcemia causes aninitial rapid increase inPTHwithacorresponding increase inCTL and reduced Ca excretion. Upon removal of the distur-bance, the subsequent increase inplasmaCacausesagradualreduction in PTH with a consequent increase in renal excre-tion, Ca(u). Owing to its slower dynamics, CTL levels con-tinue to rise long after PTH levels have peaked. Once nor-mocalcemia is achieved, the inhibitory effect of high CTLlevels on PTH secretion dominates the Ca-dependent PTHsecretory response, thus causing PTH to fall to sub-basallevels until CTL returns to baseline. Finally, PTH inducesincreased bone cell proliferation (CB ratio) resulting in netbone resorption; because bone cell proliferation is a slow pro-cess, the CB ratio takes a long time to return to basal levels.

In response to hypercalcemia induced by 2-hour cal-cium gluconate infusions in healthy subjects (32), Figure2E shows how our model simulation accurately predictsclinical observations of PTH and Ca. Figure 2F shows ourmodel prediction of the associated responses of the clini-cally unreported Ca excretion, CTL, and CB profiles dur-ing infusion and recovery. The increased Ca inhibits PTH

secretion and leads to lower CTL secretion, greater Caexcretion, and negligible bone cell proliferation.

With the preceding results validating both our proposedmodeling approach and the resulting model of Ca regulationitself, we are now in a position to use the modeling frame-work to investigate Ca-related pathologies as control systemcomponent defects. Understanding a particular pathology interms of an appropriate control system component failureprovidesauniqueperspective fromwhicheffective treatmentprocedure (targeted to the identified component defect) maybe postulated and subsequently tested.

Model utility: investigation of diseased states ascontrol system component defects

FBH and ADH (sensor defects)FBH, characterized by elevated PTH, low Ca excretion,

and mild-to-high hypercalcemia (15), is due to a missensemutation of the CaR gene. From a control system perspec-tive, therefore, FBH arises from a sensor defect. To investi-gate the phenomenon within our control system framework,the loss of receptor sensitivity, characteristic of FBH, is rep-resentedbyareduction in the sensorgain (Ksens). Inmatchingthe simulation results to the baseline biochemistry of FBHpatients reported in clinical studies (40), it was necessary toemploy a 12% reduction in Ksens occurring gradually overtime according to the trajectory indicated in Figure 3A. Asimulation of the consequences of this alteration is shown inFigure 3, B and C. The reduced CaR sensitivity causes higherPTH secretion, which in turn induces increased CTL secre-tion (indicating increased intestinal Ca absorption) and hy-pocalciuria (Figure3B), leading toanoverall increase inCa(p)

(Figure 3C), precisely as observed in FBH patients (15). Ingeneral, as CaR sensitivity (sensor gain) decreases (ie, in-creasingseverityofFBH), there isacorresponding increase inPTH secretion, which leads to increased CTL secretion andreduced Ca excretion. Additionally, so that our simulationresults match the clinical data of induced hypercalcemia inFBH subjects (Figure 3D) (40), we increased the maximum(Bca) and minimum (Aca) PTH secretory rate parameters inH(Ca(s)).Suchincreasesare indicativeofPTGgrowthandareconsistent with clinical findings of enlarged PTGs in somecases of FBH (15, 41).

Conversely, but in conceptually similar fashion, lowPTH, hypercalciuria and hypocalcemia, the pathophysi-ology associated with ADH (a gain-of-function mutationof the CaR gene), can be reproduced from the controlsystem model by simulating the sensor defect as increasedsensitivity of the CaR. A 12% increase in Ksens occurringgradually as indicated in Figure 3E results in an expectedreduced PTH secretion, which in turn causes a similarreduction in CTL levels and increased Ca excretion (Figure3F), ultimately causing a reduction in Ca(p) (Figure 3G).

Primary hyperparathyroidism (PHPT) and hypopara-thyroidism (HoPT) (controller defects)

In PHPT, adenomatous or hyperplastic PTGs secrete ex-cess PTH, which induces increased renal Ca reabsorption,bone resorption, and intestinal Ca absorption, leading tohypercalcemia (2, 42). Because the PTG is the controller inthe control system framework, PHPT therefore arises from acontroller defect. This phenomenon is represented in ourmodel by increasing Bca and Aca and reducing the slope(mCa(s)) to achieve a 3-fold increase in PTH levels at steadystate (Figure 3H). The result is an increase in CTL levels, a

reduction in renal excretion (Figure 3I), a significant increasein CB ratio (higher bone resorption) (Figure 3I), and ulti-mately a 30% increase in Ca(p) (Figure 3J).

Conversely, HoPT, characterized by inappropriatelylow or no PTH secretion, reduced CTL, hypercalciuria,and hypocalcemia (43) is reproduced in simulation by de-creasing Aca and Bca to obtain a 25% decrease in PTHfrom initial basal levels (Figure 3L). The result is lowerCTL secretion, increased Ca excretion, and a lower CBratio (net bone formation) (Figure 3M), all combining toproduce a 30% reduction in Ca(p) (Figure 3N).

0 250 500

Time (hrs)

0 250 500

Time (hrs)

PTHCTLCau

0 250 500

Time (hrs)

CaPTH(Sim)PTH (Clin)

0 250 500

Time (hrs)

0 250 500

Time (hrs)

PTH CTLCau

0 250 500

Time (hrs)

0 1000 2000

Time (hrs)

0 1000 2000

Time (hrs)

0 1000 2000

Time (hrs)

CaPTH(Sim)PTH (Clin)

0 400 800

Time (hrs)

0 400 800

Time (hrs)

CTL CauCB

0 400 800

Time (hrs)

A B C D

H I J K

Figure 3. Simulations of pathologies as ECS component defects. A–D, FBH (sensor defect)–reduction in sensor gain. A, Sensor gain change trajectory.B, PTH, CTL, urinary Ca (Cau) responses. C, Plasma Ca. D, Model prediction and clinical observation of Ca and PTH response to 2-hour clinical infusion ofcalcium gluconate in FBH (39). E–G, ADH (sensor defect)–increase in sensor gain. E, Sensor gain change trajectory. F, PTH, CTL, urinary Ca (Cau)responses. G, Plasma Ca response. H–K, PHPT (controller defect)–increase in PTH secretion. H, PTH change trajectory. I, CTL, urinary Ca, and CB ratioresponses. J, Plasma Ca response. K, Model prediction and clinical observation of Ca and PTH response to 2-hour clinical infusion of calcium gluconate inPHPT (39). L–N, HoPT (controller defect)–decrease in PTH secretion. L, PTH change trajectory. M, CTL, urinary Ca, CB ratio responses. N, Plasma Ca.

HHM—external factor: actuator disturbanceHHM arises because of tumorous secretion of PTHrP,

which induces increased bone resorption and reduced Caexcretion leading to hypercalcemia. The resulting hyper-calcemia causes reduced PTH secretion, which in turnleads to lower CTL levels (44). Because PTHrP does notplay a role in normal Ca regulation, it is an externalfactor; and because PTHrP is detected by and inducesresponses in both the bone and kidneys (actuators),from a control system perspective, HHM occurs due toan external factor disturbing the actuator. The phenom-enon is introduced in our framework through the ad-dition of PTHrP-secreting tumor cells (Figure 1B) and issimulated by increasing the PTHrP production rate,RPTHrP, to obtain a 5-fold increase in PTHrP levels (Figure4A). The result is lower PTH and CTL levels, reduced renalCa excretion, and increased CB ratio (net bone resorption)(Figure 4B), with the net effect of an increase in plasma Calevels (Figure 4C).

Vitamin D deficiency (actuator defect)VitaminDdeficiencycausesa reduction in theproduction

of CTL, which in turn results in increased PTH secretion,

bone resorption, and normocalcemia (45). This phenome-non corresponds to a defect in the very first of the four ac-tuator blocks in Figure 1B, simulated in our model, by re-ducing the production of CTL in the kidneys to zero asshown in Figure 4D, essentially decommissioning thisactuator. The consequent elimination of the inhibitoryeffect of CTL on PTH secretion causes a 40% increasein PTH levels that leads to: 1) increased Ca reabsorptionin the kidneys; and 2) net bone resorption through a10% increase in the CB ratio (Figure 4E), with an overalleffect of a marginal increase (2%) in plasma Ca levels(Figure 4F).

Discussion and Conclusions

Based on the premise that effective regulation of physiolog-ical processes is achieved by biological control systems thatpossess direct analogs with ECSs, we have proposed a mod-eling and analysis framework for plasma calcium regulationin the form of an ECS. Such an approach achieves two im-portant objectives: 1) the collection of complex subprocesses

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Figure 4. Simulations of pathologies as component defects in the ECS Ca model. A–C, HHM (process disturbance)–PTHrP production. A, PTHrP.B, PTH, CTL, urinary Ca, and CB responses. C, Plasma Ca response. D–F, Vitamin D deficiency (actuator defect)–no production of CTL. D, CTL. E,Calciotropic response of PTH, urinary Ca, and CB ratio. F, Plasma Ca response. G–I, CTL missense in the PTG (controller defect). G, Detected CTL inthe PTG. H, Calciotropic response of PTH, CTL, urinary Ca, and CB ratio. I, Plasma Ca response.

subtending Ca regulation can be organized into intercon-nectedfunctionalmodules, therebyfacilitatingmodelingandanalysis of the entire system; and 2) pathologies can be un-derstood in terms of ineffective regulation caused by identi-fiable component defects/failures. The unknown parameterswithin each functional block were estimated using data fromrelevant clinical tests of healthy subjects; and the overallmodel was validated with additional clinical data from in-duced hypo- and hypercalcemia. We have shown that theresultingmodel is capableofpredictingboth short- and long-term physiological changes in the Ca homeostatic environ-ment, under both healthy and pathological conditions.

ThePTHoversecretioncharacteristicofPHPTismodeledby an increase in the maximum (Bca) and minimum (Aca)PTH secretory rate parameters. Our simulation results forPHPT(Figure3,H–K)areconsistentwith thegeneral clinicalobservations of the disease (2, 42), except that the hypocal-ciuria predicted by our model is consistent with less than25% of PHPT cases (46). To predict the PTH response toclinically induced hypercalcemia (2-h calcium gluconate in-fusion) in PHPT subjects (Figure 3K) (40) more accurately, itwas necessary also to reduce the hypercalcemic slope(mCa(s)A). The relative change to the slope is therefore anindicator of decreased PTH secretory response to changes inCa(p) in PHPT compared to the healthy model (40, 47) andpossibly an indicator of the degree of PHPT.

In their review, Diaz et al (5) hypothesized that a cell-surface receptor-like mechanism drives CTL-mediated PTHsecretion; therefore, we investigated the effect of a missensein this receptor on Ca regulation by implementing a 95%reduction in the PTG-detected CTL (Figure 4G)—a control-ler defect. The resulting calciotropic response (increasedPTH, CTL, and CB ratio; reduced urinary Ca and hypercal-cemia; Figure 4, H and I) is similar to that resulting from adecrease inKsens (FBH).However, themagnitudeoftheeffectis markedly lower for CTL missense. This finding suggeststhat the pathological effects of receptor defects (Ca or CTL)on the PTG are similar, differing only in the magnitude of therespective effects on PTH secretion. Note, therefore, that ourECSmodelofCaregulationmakes it possible todifferentiatepathologies that may otherwise be confounded.

Althoughqualitativelyaccurate ingeneral,quantitatively,the model predictions of pathological conditions do not al-ways have an exact match with all clinical observations ofCa,PTH,CTL,urinaryCa,and/orCBratio.This is likelydueto the fact that our model simulations of pathologies arebased on individual component defects in the “healthy”model: 1) without accounting for the possible effects of ex-ternal hormones/ions on Ca regulation under pathologicalconditions; and 2) assuming that the pathologies in questionmanifest as defects isolated to a single component with no“interactions” from or on other subprocesses.

The deterministic model provided herein does not ac-count for the stochasticity that exists in the Ca regulatorysystem. However, although intrinsic to the cells and tissuesinvolved in the various subprocesses of Ca control, sto-chasticity is not a defining characteristic of overall plasmaCa regulation. Therefore, taken together, group averagesof the different Ca regulatory subprocesses provide a rea-sonable representation of overall plasma Ca control. Assuch, we have demonstrated how the control engineeringframework provides an efficient means for organizing thevarious physiological subprocesses constituting calciumregulation and facilitates a fundamental understanding ofthis complex physiological process. The resulting mathe-matical model adequately predicts the short- and long-term dynamic characteristics of the Ca regulatory systemin both healthy and pathological states. The model pro-vides novel insights into Ca regulation by generating re-sults about physiological variables that are currently un-measurable. Furthermore, the framework facilitates thetesting of hypotheses about mechanisms underlying theemergence of known pathologies. Currently, we are usingthe model to: 1) explore the differential diagnosis of Ca-related pathologies with similar pathophysiologies; and 2)identify potential sites for therapeutic intervention in dif-ferent Ca-related pathologies.

Acknowledgments

Address all correspondence and requests for reprints to:Babatunde A. Ogunnaike, College of Engineering, 102 DuPontHall, University of Delaware, Newark, DE 19716. E-mail:ogunnaike@udel.edu.

Themanuscriptwas fundedbytheNationalScienceFoundationand the Department of Chemical Engineering, Virginia Tech.

Disclosure Summary: The authors have nothing to disclose.

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A control engineering model of calcium regulation

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