Mathematical Biosciences 257 (2014) 23–32

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Mathematical Biosciences

journal homepage: www.elsevier .com/locate /mbs

Patient-specific modeling of the neuroendocrine HPA-axis and itsrelation to depression: Ultradian and circadian oscillations

http://dx.doi.org/10.1016/j.mbs.2014.07.0130025-5564/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail address: johnny@ruc.dk (J.T. Ottesen).

Johanne Gudmand-Hoeyer a, Stine Timmermann b, Johnny T. Ottesen a,⇑a Department of Science, Systems and Models, Roskilde University, Building 27.1, 4000 Roskilde, Denmarkb Department of Quantitative Pharmacology, H. Lundbeck A/S, 2500 Valby, Denmark

a r t i c l e i n f o

Article history:Available online 3 September 2014

Keywords:DepressionHPA-axisPatient specificNon-linear mixed effects ODE modelParameter estimationBio-marker

a b s t r a c t

In the Western world approximately 10% of the population experience severe depression at least once intheir lifetime and many more experience a mild form of depression. Depression has been associated withmalfunctions in the hypothalamus–pituitary–adrenal (HPA) axis. We suggest a novel mechanistic non-linear model capable of showing both circadian as well as ultradian oscillations of the hormone concen-trations related to the HPA-axis. The fast ultradian rhythm is assumed to originate from the hippocampuswhereas the slower circadian rhythm is assumed to be caused by the circadian clock. The model is able todescribe the oscillatory patterns in hormone concentration data from 29 patients and healthy controls.Using non-linear mixed effects modeling with statistical hypothesis testing, three of the model parame-ters are identified to be related to depression. These parameters represent underlying physiologicalmechanisms controlling the average levels as well as the ultradian frequencies and amplitudes of the hor-mones ACTH and cortisol. The results are promising since they point toward an exact etiology for depres-sion. As a consequence new biomarkers and pharmaceutical targets may be identified.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Depression covers a broad spectrum of mental diseases, such asbipolar and major depressive disorder (MDD). It is diagnosed bypsychiatrists by help of scales, i.e. questionnaires. However,depression is a highly complex disease, which still lacks an exactetiology [1]. In both MDD and other types of depression disorderscomorbidity is high [1–3] and as a result initial misdiagnosis iscommon [2,3]. Depression significantly affects the social life, sleep-ing and eating habits and general health of the depressed personsand has a severe impact on relatives and friends as well [4].Depression is a major cause of morbidity and is very costly for soci-ety [5]. In the Western world, it is estimated that about 10% of thepopulation experience severe depression at least once in their life-time [6]. Obviously, reliable biomarkers are desirable and crucialfor improvement of diagnoses and treatments. However, a morecomplete etiology is a prerequisite.

Depression has been associated with malfunctions in thehypothalamus–pituitary–adrenal (HPA) axis, an endocrine systemof glands and their synthesized hormones [7–9]. The HPA-axisplays an important role during stress where it increases the

concentration of the involved stress hormones leading to directionof energy to the brain and muscles [10]. The return to the basalhormone levels is an important feature of the system when it isworking properly (usually referred to as homeostasis). Since sev-eral feedback mechanisms are working simultaneously in theHPA-axis, causes and effects can be hard to distinguish. A mathe-matical model may enable understanding and can be an importanttool for pinpointing where and how malfunctioning in the HPA-axis may occur.

In this paper we present and analyze a novel mechanism-basednon-linear model able to describe the characteristics of concentra-tion data from the HPA-axis. By using a multi-scale modelingapproach, we attain a model capable of showing both circadianas well as ultradian oscillations. The model is developed from theparadigm of the ultradian oscillations as being driven by cortico-tropin releasing hormone (CRH) originating from the hypothalamicsystem and the assumption that the circadian rhythm is caused bythe circadian clock. The data supporting the modeling processoriginates from a clinical trial consisting of 29 subjects [11]. A pre-viously introduced index [12], the O-index, is used to group thesubjects into three groups termed the nondepressed subjects andlow and high cortisol depressed subjects, respectively. Usingnon-linear mixed effects (NLME) analysis we identify parametersdiffering significantly between the three groups. This result canbe used for hypothesis development concerning increased/

24 J. Gudmand-Hoeyer et al. / Mathematical Biosciences 257 (2014) 23–32

decreased cortisol levels and malfunctioning in the HPA-axis. TheNLME analysis is also seen as a tool for model validation. Thesuccess of our integrated methodology is as important a result asthe specific application to depression.

2. HPA-axis

2.1. Physiology of the HPA-axis

Mainly three hormones are involved in the HPA-axis: CRH issecreted in the hypothalamus and released into the portal bloodvessel of the hypophyseal stalk, where it is transported to the ante-rior pituitary. Here CRH stimulates the synthesis and release ofadrenocorticotropic hormone (ACTH) from the pituitary gland tothe systemic circulation. In the cortex of the adrenal glands ACTHstimulates the synthesis and secretion of the stress hormonecortisol and causes a release of cortisol to the systemic circulation.Cortisol has an impact on the whole body and especially feeds backon the hypothalamus and the pituitary affecting the synthesis andrelease of CRH and ACTH from the respective glands.

Fig. 1 illustrates the anatomy of the HPA-axis and a zoom in onthe endocrine glands constituting it with an indication of how thehormones affect the system.

Keeping cortisol concentration within a certain range is impor-tant for various reasons: A maintained, too high level of cortisol(hypercortisolism) is related to depression, diabetes, visceral obes-ity and osteoporosis [13]. Low concentration (hypocortisolism) canresult in disturbed memory formation or life-threatening adrenalcrisis beside depression [13].

The cortisol concentration has a circadian as well as a fasterultradian pattern. Cortisol concentration is typically low between8 p.m. and 2 a.m. and peaks in the period 6–10 a.m. [14]. CRH issecreted one to three times per hour similar to the frequency ofthe ultradian oscillations [15] leading to the hypothesis that theultradian oscillations are driven by the CRH secretion [16]. How-ever, when CRH sensitive receptors at the pituitary gland are

Hypothalamus

Pituitary

Adrenal

Hy

Pi

Ad

Fig. 1. Left: Sketch of the anatomy of the HPA-axis showing the relative location of h(different magnification) highlighting the suprachiasmatic nucleus (SCN) controlling circgland (upper green arrow). The pituitary synthesizes ACTH, which affects the adrenal g(inhibitory) feedback on the hypothalamus and the pituitary (red dash-terminated linereferred to the web version of this article.)

blocked, ultradian oscillations have been seen in ACTH and cortisolconcentrations [17]. The paradigm that oscillations are generatedin the isolated pituitary–adrenal sub-system became popular withthe modeling work done in [18]. However, despite CRH being thepredominant ACTH secretagogue in humans as well as in rats[19], its ability to stimulate ACTH secretion can be potentiated byother hypothalamic neuropeptides, such as vasopressin [20].Hence, there exists two competing paradigms; one stating thatultradian oscillations originate from the hypothalamus, via CRH,and the other stating that the isolated pituitary–adrenal subsystemgenerates the ultradian oscillations.

The circadian pattern, on the other hand, is generally believedto be caused by the circadian clocks [21–23] which are biologicaloscillators made by networks of so-called CLOCK genes. These arefound centrally in the suprachiasmatic nucleus as well as peripher-ally in different organs [24]. The circadian clocks are stronglyinfluenced by daylight, but are also coupled to body temperatureand hunger and exogenous factors including psychological andphysical stress [25–27].

Cortisol inhibits the secretion of CRH through glucocorticoidreceptors (GRs) situated in the hypothalamus [28]. In addition cor-tisol also performs a negative feedback on the secretion of ACTHthrough GRs situated in the pituitary [29].

2.2. Models of the HPA-axis

In the so-called ‘minimal model of the HPA-axis’, the HPA-axisis modeled by a system of coupled non-linear ordinary differentialequations with the inhibitory feedback from cortisol on the syn-thesis of CRH and ACTH (see Fig. 2) modeled by sigmoidal decreas-ing functions [30,31]. The models presented in [10,14,32–36] mayall be considered as realizations of the minimal model. The split-ting of the circadian and ultradian rhythms of the HPA-axis isassumed with the ultradian rhythm considered as an inherentbehavior of the HPA-axis whereas the circadian rhythm is thoughtof as an external input to the axis. The minimal model has beeninvestigated thoroughly in [30,31] where extension to the ‘general

pothalamus

tuitary

renal

PVN

CORT

ACTH

CRH

SCN

ypothalamus, pituitary and adrenal glands. Right: The hormone producing organsadian rhythms, paraventricular nucleus (PVN) secreting CRH affecting the pituitary

lands (lower green arrow) stimulating the synthesis of cortisol, which has negatives). (For interpretation of the references to color in this figure legend, the reader is

Fig. 2. Compartmental diagram showing the minimal model of the HPA-axis (blackpart of the figure) with the involved hormones, CRH, ACTH and cortisol (CORT), andits extension to the general model of the HPA-axis (black and gray part of thefigure). The suprachiasmatic nucleus (SCN) is assumed to influence the synthesisand secretion of CRH from hypothalamus. The feedback mediated by cortisol isinhibitory (dashed lines) as well as stimulating (dashed arrow). In the generalmodel parts of the feedback go through hippocampus. Different kinds of receptorsare involved in the feedback; mineralocorticoid receptors (MRs) and glucocorticoidreceptors (GRs) situated at several locations having different action.

J. Gudmand-Hoeyer et al. / Mathematical Biosciences 257 (2014) 23–32 25

model’, which include feedback through the hippocampus, is dis-cussed and analyzed. Despite different approaches the commonaim of the work cited is to show that a stable solution becomesunstable and oscillating for certain parameter values.

It has also been suggested that the ultradian oscillations mayarise from the introduction of a time delay [30,37]. However, it ismathematically well-known that one may easily generate oscilla-tions in solutions to systems of ordinary differential equations byintroducing sufficiently large time delays. Investigations show thatfairly large time delays (i.e. at least 18 min) in the feedback mech-anisms are needed for generating oscillations [30]. The classicdescription of glucocorticoid action involves activation or repres-sion of de novo synthesis of mRNA and proteins. Large time delaysare expected due to the multiple steps in the genomic processes,including glucocorticoid binding to cytosolic GRs and translocationof the GR-complexes into the nucleus [38,39]. However, fast effectGRs do exist at both the pituitary [40] and hypothalamic levels[41,42]. While one may speculate about the affinity of the differenttypes of receptors there seems to be a lack of biological support forsuch speculations. Hence, the role of time-delays in these systemsare still unresolved.

In [10,30,31,34] the strategy was to look for Hopf-bifurcationsof stable fixed points. However, [30] showed that the ’minimalmodel of the HPA-axis’ is not capable of reproducing the character-istic pattern seen in data when using physiologically reasonableparameter values. This implies that the ultradian oscillations mustarise from other mechanisms. Bursting phenomena have been sug-gested previously as a possible explanation [43–45], and attemptshave been made to include competing positive and negative feed-back mechanisms within the hippocampus into the minimal modelas illustrated in Fig. 2 [31,46,47]. However, in order to generateultradian oscillations the feedback rules introduced in the modelwould have to contain a degree of complexity not supported byphysiological knowledge.

Two recent publications are relevant to mention as they bothtake steps, which are elaborated further in our approach. [48] tooka phenomenological and linear approach similar to the earliestHPA-axis models with the inclusion of a very steep switch-likefeedback from cortisol on the CRH release (corresponding to an

unphysiologically large Hill power of approximately 150). Theirmain contribution was to add a simple sleep-wake cycle modula-tion of CRH. Both circadian and ultradian patterns appeared in themodeling results qualitatively resembling observations, but leavingquantitative comparisons for later work. [49] took a pseudo-popu-lation approach to modeling post-traumatic stress disorder inhumans based on a phenomenological model of the HPA-axis usingMichaelis–Menten kinetic descriptions. In their approach theylump subjects in target groups together by averaging over groupdata, with the aim to study variances between the groups.

In the next section we construct the novel non-linear modelshowing patient specific ultradian and circadian oscillations anduse quantitative data to estimate model parameters. Using NLMEmodeling technique, which takes into account both intra-individ-ual and inter-individual variation, contrary to the technique pre-sented in [49], we are able to identify parameters that differbetween groups.

3. CRH-driven model of the HPA-axis: Splitting the time-scales

The starting point of our model development is the minimalmodel. Firstly, we adopt the paradigm promoted in [16], whichargues that the ultradian oscillations are driven by a CRH pulsegenerator. Secondly, we incorporate the effect of the circadianclock [24,50] into time-varying coefficients of the differential equa-tions. By doing so, the parameters become slowly varying overtime compared to the fast ultradian time-scale. The result will bea circadian rhythm.

The approach of letting the parameters vary slowly is inspiredby the fact that in all living organisms nothing is constant overlarge time scales. The stringent differentiation between parametersand variables known from mathematics may be too crude. In biol-ogy one may prefer to define parameters as slowly varying quanti-ties compared to variables. Hence, some biological systems withtwo time-scales may be approached by having incorporated theslow time-scale into the system’s ‘parameters’, a method wedenote splitting time-scales.

In [16] the existence of a CRH-oscillator in the isolated hypo-thalamus was shown experimentally. The very regular oscillationswere concluded to originate from hypothalamus. Hence, the moreirregular observed pattern seen in ACTH and cortisol concentrationdata (typical data are shown in Fig. 7) is either noise or due to thepituitary–adrenal subsystem. With the minimal model as a point ofdeparture, we first introduce a CRH pulse generator responsible forthe ultradian oscillations as suggested by [16]. As indicated inFig. 3 this is done by lumping the pathway regulating the synthesisof CRH into a single regulatory substance (REG), which interactswith CRH in the hypothalamus. REG (the first compartment) inhib-its the self-stimulated synthesis (k2) of CRH (the second compart-ment) whereas CRH stimulates that of REG (k1). At the same timeREG stimulates the degradation of CRH. Both REG and CRH undergoelimination, termed w1 and w2, respectively. According to thephysiology of the HPA-axis outlined in Section 2.1 and illustratedin Fig. 2, CRH feeds into the anterior pituitary (third compartment)and by this into the rest of the system. The coupling is mediatedthrough the CRH-stimulated synthesis of ACTH (a1). In additionACTH has a baseline synthesis (a7). ACTH stimulates the synthesisof cortisol (CORT) with a rate (a5), but CORT (the fourth compart-ment) may also have a baseline synthesis (a4). Elimination of CORT(a6) and ACTH (a3) take place, the latter being stimulated by CORT.There is an inhibitory feedback from CORT onto the CRH-stimu-lated synthesis of ACTH, but we ignore the feedback from cortisolon the hypothalamus subsystem. More specifically, as the first stepof describing the adjusted minimal model, we suggest the follow-ing equations:

Fig. 3. Compartmental diagram of the CRH driven model. The two upper compart-ments describe hypothalamus where a regulatory pathway is lumped into oneregulatory substance (REG), which interacts with CRH. REG inhibits the self-stimulation of CRH synthesis (k2) whereas CRH stimulates that of REG (k1). At thesame time REG stimulates the degradation of CRH. Both REG and CRH undergoelimination (w1 and w2, respectively). This subsystem feeds into the rest of thesystem through CRH stimulating the synthesis of ACTH (a1), which also has abaseline synthesis (a7) at the anterior pituitary. ACTH stimulates the synthesis ofcortisol (CORT) (a5), which also has a baseline synthesis (a4) at the adrenal cortex.There are CORT-stimulated elimination of ACTH (a3) and CORT (a6). There is afeedback from CORT onto the CRH stimulation of ACTH. Four rates, a1; a3; a5, and a6,are affected by the circadian clock.

26 J. Gudmand-Hoeyer et al. / Mathematical Biosciences 257 (2014) 23–32

dx1

dt¼ k1

x2

1þ x2x1 �w1x1; ð1Þ

dx2

dt¼ k2

1aþ x1

x2 �w2x1

aþ x1x2; ð2Þ

dx3

dt¼ a7 þ a1

a2

a2 þ x4x2 � a3x3x4; ð3Þ

dx4

dt¼ a4 þ a5x3 � a6x4; ð4Þ

where x1; x2; x3, and x4 denote the concentration of REG, CRH, ACTH,and CORT, respectively. The model is illustrated in Fig. 3.

In the second step we infer the effects of the circadian clock. Thecircadian clock causes diurnal rhythms of the body temperatureresulting in altered biochemical reaction rates. Hence, we allowthe four rates a1; a3; a5, and a6 to depend on the circadian clock(the baseline synthesis parameters a4 and a7 do not need to dependon the circadian clock in order for the model to be able to predictthe desired oscillatory pattern). To meet the parsimonious princi-ple we take the circadian rhythm to be sinus-shaped,

XðtÞ ¼ 1þ sinðxct þ dÞ ð5Þ

where t denotes time, xc ¼ 2p1440 [min�1] is the circadian clock fre-

quency, while d denotes the circadian phase. The four circadiandependent parameters are chosen as simple functions of XðtÞ,

a1 ¼ p1 � ð2�XðtÞÞ; ð6Þ

a3 ¼ p3 þ p4 �XðtÞ; ð7Þ

a5 ¼ p6 þ p7 � ð2�XðtÞÞ; ð8Þ

a6 ¼ p8 � p9 � ð2�XðtÞÞ; ð9Þ

where we have introduced new parameters p1; p3;p4;p6; p7;p8, andp9 to substitute a1; a3; a5, and a6. Note that these equations consistof two terms; a circadian independent part and a circadian

dependent part. The circadian dependent parts involve the factor2�XðtÞ ¼ Xðt � p

xcÞP 0, which merely represent phase-shifted

circadian dependencies. In addition we simplify the model bychoosing w � w1 ¼ w2 and setting a4 ¼ 0.

Theorem 1. The hypothalamus submodel in Eqs. 1 and 2 has a uniqueneutral steady state at ðx1;ss; x2;ssÞ ¼ ðk2

w2; w1

k1�w1Þ in the positive

quadrant for k1 > w1. Solutions starting in the positive quadrant staythere for all later times. The small amplitude frequency is

xh ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� w1

k1

1� w2k2

w1w2

vuut :

All solutions have closed periodic trajectories in the anti-clockwisedirection around the steady state.

Proof. The proof is straight forward except for the part showingthat the trajectories are closed for all initial conditions in the posi-tive quadrant. To show that, we adopt the idea from the similarproof for the Lokta–Volterra model [51]. Eqs. (1) and (2) and usingseparation of variables yields

f ðx1Þgðx2Þ ¼ C0 � f ðx1ð0ÞÞgðx2ð0ÞÞ; ð10Þ

where C0 is a constant and

gðx2Þ ¼xw1

2

ðx2 þ 1Þk1

and

f ðx1Þ ¼xk2=a

1

ðx1 þ aÞk2=aþw2:

Both functions are C1ðRþÞ, are zero at zero, tend to zero at infinityand have unique maxima Mg and Mf , respectively. Hence, forC0 ¼ M � Mf Mg exactly one solution exists, the steady state solu-tion. If C0 < M and x1 ¼ x1;ss, Eq. (10) has two solutionsx2 ¼ x2;m < x2;ss and x2 ¼ x2;M > x2;ss. For x2 R ½x2;m; x2;M�, Eq. (10)has no solution. For x2 ¼ x2;m or x2 ¼ x2;M , exactly one solution existsfor Eq. (10). Finally, for x2 2 ðx2;m; x2;MÞ, Eq. (10) has exactly twosolutions. At x2 ¼ x2;m and x2 ¼ x2;M , where dx2=dt ¼ 0, the twosolutions are glued together resulting in a periodic trajectorycorresponding to closed periodic solutions. h

The hypothalamus submodel in Eqs. 1 and 2 is feeding into thepituitary–adrenal submodel given by Eqs. 3 and 4. Using the resultfrom Theorem 1 we can analyze the pituitary–adrenal submodel:The solutions of the hypothalamus submodel oscillates periodi-cally, CRH stimulates the synthesis of ACTH with a time-dependentrate, which is furthermore inhibited by CORT. The elimination rateof ACTH is time-dependent and stimulated by CORT. ACTH stimu-lates the synthesis of CORT with a time-dependent rate and theelimination of CORT is likewise time-dependent. Thus, the circa-dian rhythm modulates the ultradian oscillations that propagatefrom the hypothalamus subsystem.

Theorem 2. The pituitary–adrenal submodel in Eqs. 3 and 4 has anattracting trapping region in the positive quadrant.

Proof. Note that CRH is bound from below and above by, lets say,Hm > 0 and HM > Hm, respectively. Thus,

a1 2 ½0;4p1� � ½0; a1M�; ð11Þ

a3 2 ½p3;p3 þ 2p4� � ½a3m; a3M�; ð12Þ

a5 2 ½p6;p6 þ 2p7� � ½a5m; a5M�; ð13Þ

J. Gudmand-Hoeyer et al. / Mathematical Biosciences 257 (2014) 23–32 27

a6 2 ½p8 � 2p9; p8� � ½a6m; a6M �; ð14Þ

with a6m > 0, i.e. 0 < p9 < p8=2. As illustrated in Fig. 4 we define foreach r P 1,

‘1 : x4 ¼1r

a4

a6M; ð15Þ

‘2 : x4 ¼a5M

a6mx3 þ r

a4

a6m; ð16Þ

h1 : x4 ¼1r

a7

a3M

1x3

ð17Þ

h2 : x4 ¼ ra7

a3mþ a1MHM

a3m

� �1x3: ð18Þ

The straight line ‘2 lies above the horizontal line ‘1 and the hyper-bola h2 lies above the hyperbola h1 in the positive quadrant. LetQ1 ¼ ðA1;C1Þ be the unique intersection between ‘1 and h2 and letm1 denote the vertical line x3 ¼ A1 through Q1. The line m1 has aunique intersection with ‘2;Q2 ¼ ðA2; C2Þ ¼ ðA1;C2Þ. Let m2 denotethe horizontal line x4 ¼ C2 through Q2. The line m2 intersects h1

at the unique point Q3 ¼ ðA3;C3Þ ¼ ðA3; C2Þ. Let m3 denote the verti-cal line x3 ¼ A3 through Q3. This line intersects ‘1 at the uniquepoint Q4 ¼ ðA4; C4Þ ¼ ðA3;C1Þ. The rectangle Q1Q2Q3Q4 will serveas an attracting trapping region for r > 1. From Eqs. 3 and 4 it fol-lows that dx4=dt > 0 on and below the line ‘1;dx4=dt < 0 on andabove the line ‘2;dx3=dt > 0 on and below the hyperbola h1, anddx3=dt < 0 on and above the hyperbola h2. Hence, the vector fieldpoints into the box Q1Q2Q3Q4, since Q1Q2 is above h2;Q2Q3 isabove ‘2;Q3Q4 is below h1, and Q4Q1 is on ‘1 by construction, seeFig. 4. For r ¼ 1 some derivatives may vanish causing the rectangleto be a trapping region in this case. We emphasize that on the upperboarder of the trapping region and between the two hyperbolas thevector field is downward, but its horizontal component may vary insize and sign. Similar on the right boarder of the trapping region andbetween the two hyperbolas the vector field is leftward, but its ver-tical component may vary in size and sign. The rectangle Q1Q2Q3Q4

is then an attracting trapping region for all r > 1 and it is a trappingregion for r ¼ 1. h

Q4 Q1

Q3Q2

h2

h1

x4=CORT

x3=ACTH

l2

l1

Fig. 4. Illustrating the attracting trapping region Q1Q2Q3Q4. The green arrowsindicate the overall direction of the flow along the boundaries of the trappingregion. Crossing arrows indicate that either one of the directions may occurdepending on time. (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

It follows trivially that there exists a trapping region for the fullmodel given by Eqs. (1)–(4) in the positive octahedron of R4. Thussolutions to the full model are uniformly persistent and bounded.

4. Data

The estimation of the model parameters will be carried outusing data from a clinical study of 29 depressed patients andhealthy controls published in [11]. During 24-h, plasma ACTHand serum cortisol concentrations were measured every ten min-ute in each subject. The subgroup consisting of 12 depressed adultswas found by a screening at the inpatient psychiatry services ofJohn Umstead Hospital and Duke University Hospital, North Caro-lina US. The control group consisted of 17 normal healthy adults,matched on age, sex, and race. All 12 depressed subjects were diag-nosed with primary major depressive disorder according to theResearch Diagnostic Criteria. Following the DSM-IV criteria nineof the depressed subjects were diagnosed with major depressivedisorder and three were diagnosed with bipolar disorder in thedepressive phase. None of the depressed subjects were absolutetreatment resistant, while all of them responded positively totreatment after the study protocol was carried out. Both thedepressed patients and the controls were carefully screened forany additional health or behavioral issues that could influencethe HPA-axis (further study details may be found in [11]).

Both the ACTH and cortisol concentration data from depressedpatients as well as the healthy controls (diagnosed by psychia-trists) show circadian and ultradian oscillations as illustrated forthree typical subjects in Fig. 5, where noise has been filtered outusing discrete Fourier analysis (sampling every ten minute leadsto a smallest observable ultradian period of 20 min). Frequencieslarger than 1/3 h�1 is considered as noise.

In order to identify a quantitative relationship between depres-sion and the HPA-axis, [11] divided the subgroup consisting of 12depressed patients into two groups, the hypercortisolemic andnon-hypercortisolemic depressed groups, using the 24 h meanplasma cortisol level derived from the normal controls as abiomarker.

As an alternative to this classification, [12] introduced a novelindex, the O-index, which was used to classify the same 29 sub-jects. The O-index describes the sum of the normalized averagedeviations from normal values in cortisol as well as ACTH. Usingthis index, the 29 subjects could be classified into three distinctgroups termed non-depressed (12), hypercortisolemic depressed(7) and hypocortisolemic depressed subjects (10), see Fig. 6. Theclassification showed good performance compared to the classicalmethods such as Carroll’s Depression Scale (CDS) [52]. In [12], datawere analyzed independently with mixture effects modeling [53],cluster analysis [54] as well as a two-dimensional hidden Markovchain modeling approach of the low frequency signal [55,56], allresulting in the same categorization of data. Hence, the index sug-gests a scientifically based measure for depression and moves theetiology of depression from considering cortisol to also includingACTH.

5. Non-linear mixed effects modeling

One have to face some limitations in the validation process ofmodels when working with complex illnesses. First of all thereare ethical, practical and economical issues in obtaining sufficientdata material: Large clinical trials are extremely expensive, e.g.the cost for developing new drugs by pharmaceutical companiesis estimated to around 5 billions (5 � 109) dollars per success [57].In addition many investigations are not possible in humans orare considered unethical to perform, since these may harm the test

0 120 240 360 480 600 720 840 960 1080 1200 1320 14400

10

20

30

40

50

60

70

Time [min]

AC

TH c

once

ntra

tion

[pg/

ml]

Hypercortisol depressiveNormalLowcortisol depressive

0 120 240 360 480 600 720 840 960 1080 1200 1320 14400

5

10

15

20

25

Time [min]

Cor

tisol

con

cent

ratio

n [µ

g/d

l]

Hypercortisol depressiveNormalLowcortisol depressive

Fig. 5. Example of filtered ACTH data (left) and cortisol data (right) of three individuals; one from the hypercortisolemic (high cortisol) depressed group (blue curves), onefrom the hypocortisolemic (low cortisol) depressed group (green curves) and one from the normal (non-depressed) group (red curves). Data show circadian as well asultradian oscillations. Time t = 0 corresponds to midnight. Data were collected every ten minute during 24 h. Left: The hypercortisolemic subject (blue curve) clearly deviatesfrom the others. Right: The hypocortisolemic subject (green curve) deviates noticeably from the others. The unfiltered data were provided by [11]. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

28 J. Gudmand-Hoeyer et al. / Mathematical Biosciences 257 (2014) 23–32

persons. Consequently, one is often left with data material limitedin the number of subjects, but sometimes rich in sampling.

NLME modeling embeds deterministic models of individuals(given by solutions to the differential equations) into a statisticalframework whereby inference for repeated measurements from apopulation, extraction of knowledge and assumptions on variationin outcome within and across individuals become formalized.Repeated measurements may include time-varying concentrationmeasurements of a specific substance. NLME modeling goes backto [58] who studied correlations of trait values between relatives.Since then the area has developed strongly and along with thedevelopment of high speed computer technology in the early1990s it has become possible to perform MLME modeling of largestudies [59]: It is frequently assumed that one has a continuousresponse evolving over time (often denoted profiles) for the singlesubjects of the population. In fact, it is assumed that one has a

0 5 10 15 20 25 30−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Subjects ordered after groups

O−i

ndex

Hypocortesolemic

Normo−cortisolemic

Hypercortesolemic

Fig. 6. The O-index for the 29 subjects and the most likely grouping. Three statesemerge; a hypocortisolemic depressed state (yellow circles), a non-depressed state(green circles) and a hypercortisolemic depressed state (red circles). The subjectsare renumbered according to the ascending order of the value of the O-index. Thethree groups are separated by threshold values marked by (yellow and red) dottedlines. The hypocortisolemic depressed subjects are located below the (yellow)dotted line and the hypercortisolemic depressed subject are located above theupper (red) dotted line. Between the two dashed lines the non-depressed subjectsare located. After [12]. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

correct (ODE-) model for the individual profiles. Finally, statisticalinference is introduced focusing on the parameters that underliethe individual profiles and on how these vary across thepopulation.

Let data consist of an independent sample of n subjects and letthe ith subject have mi measurements at times ti;1; ti;2; . . . ; ti;mi

. Weconcatenate all these measurements in a single vector

y ¼ ðy1;1; y1;2; . . . ; y1;m1; . . . ; yn;1; yn;2; . . . ; yn;mn

Þ:

NLME modeling is most easily presented by separating themodeling process into two stages; stage 1 dealing with intra-individual (or within-subject) variation and stage 2 dealing withinter-individual (or between-subject) variation.

At the intra-individual level (stage 1) we assume that data canbe described by a mean response profile changing over time by areal function f and a simple additional error vector � describingrandom intra-individual variation (residual variability),

y ¼ f ðt; bÞ þ �

or, as in our case, a proportional error vector

y ¼ f ðt; bÞ 1þ �ð Þ;

where b denotes the vector of parameters to be estimated. In ourspecific application the non-linear function f will be the solutionto the ODE-model with initial conditions included into the param-eter vector. The residuals � are assumed to be independent, oftennormally distributed with zero mean and constant variance r2

� . Col-lectively the set of all residual variances is referred to as the residualvariance matrix R�.

At the inter-individual level (stage 2) the components bk of theparameter vector b are either assumed to be log-normally distrib-uted with one common distribution for the whole population or tovary across the population. Sometimes the distribution is assumedGaussian, but in the present case we wish to exclude non-positiveparameter values justifying the assumption of a log-normal distri-bution. In the case of varying the bk across the population we spe-cifically have three groups defined by the O-index consisting of thehypocortisolemic depressed, the non-depressed, and the hypercor-tisolemic depressed subjects, respectively, and bk may varybetween these groups. We use the symbol g to denote either ofthese three groups. The k0th component of the i0th subject isdenoted bk;i and is assumed log-normally distributed, i.e.logðbk;iÞ � N ðlogðlk;gÞ;r2

k;gÞ with mean logðlk;gÞ and variance r2k;g

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Fig. 7. Data (circles) and model prediction (solid line) for ACTH (upper panel in blueof (a)–(c)) and cortisol (CORT) (lower panel in orange of (a)–(c)) for a hypocorti-solemic depressed subject (a), a non-depressed subject (b) and a hypercortisolemicdepressed subject (c). For fitting the novel model presented in these palletsindividually to Carroll’s data the shuffled complex evolution (SCE) algorithm wasused to minimize the least square deviation. The SCE algorithm is a step-wise globalmethod where a number of complexes in each step make use of the simplexalgorithm and transition between steps evolving according to a random procedure.(For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

J. Gudmand-Hoeyer et al. / Mathematical Biosciences 257 (2014) 23–32 29

where subject i belongs to group g; i 2 g. These inter-individuallyvarying parameters bk are not fixed across the population, butare allowed to depend on the subject-specific categorical covariateg. Hence, we write

log bk;i

� �¼ log lk;g

� �þ gk;i;

where gk;i is normally distributed with mean zero and variance r2k;g ,

which may be assumed g-independent for simplicity. For thoseparameters, which have a common population distribution, wehave a similar description, but with both lk;g ¼ lk and rk;g ¼ rk

being group independent. Generally, it is assumed that the �s andthe gs are independent. Collectively, the r2

k;gs constitute the diago-nal of the variance–covariance matrix Rg. The ls are denoted thefixed effects whereas gs are denoted random effects.

6. Results

Initially, the parameters in the proposed model were estimatedusing the shuffled complex evolution (SCE) algorithm to minimizethe least square deviation between data and model prediction. Asatisfactory fit of each of the 29 subjects was obtained. This is illus-trated in Fig. 7, where profiles of ACTH and cortisol for a single sub-ject from each group are shown.

A numerical sensitivity analysis of the parameters of the modelwas performed. The parameter w was the most sensitive parame-ter influencing both ultradian amplitude and frequency, followedby k2 affecting mainly the ultradian amplitude and d affectingthe circadian phase. Next, p1; p3; p6; p8; k1, and a had approximatelyequal sensitivity mostly affecting the ultradian amplitude, fre-quency and phase, but also the mean values of ACTH and cortisolconcentrations. The rest of the parameters showed minorsensitivity.

The NLME analysis of the model and data was conducted usingthe NONMEM 7 version 2.0 software program with the ADVAN13subroutine [60]. The Iterative Two-Stage method (an expecta-tion–maximization method) was used for parameter estimation.In order for the method to converge, it was necessary to fix someof the parameters. The less sensitive parameters identified in thesensitivity analysis were fixed to their mean values obtained usingthe SCE algorithm and the random effect associated with theparameter was fixed to zero.

The first step in the NLME modeling process was to estimate allnon-fixed parameters by treating all the subjects as one group.

Next, the parameter with the largest variance (p1) was allowedto vary between the three groups. A v2-test showed that the modelthen improved significantly (Fð99;2Þ and p-value 1:6 � 10�22). Thesame procedure was used for the parameter with the next largestvariance (p3), and the model improved significantly by letting p3

vary between the groups (Fð10;2Þ and p-value 3:4 � 10�3) while fix-ing all the other parameters including p1. The parameter with thirdhighest variance was w and again the model improved significantlyby letting this parameter vary between the three groups (Fð138;2Þand p-value 5:4 � 10�31) while keeping all the others fixed. Contin-uing this process no other parameter resulted in significantimprovements when allowed to vary between the three groups.

Next step was to allow the two parameters, which improved themodel the most, i.e. w and p1, to vary between the groups whilekeeping the rest fixed. A significant improvement of the modelwas obtained (Fð17;2Þ and p-value 1:0 � 10�4). Hence, we tested allthree identified parameters by allowing them to vary between thegroups simultaneously while keeping the rest fixed. A significantimprovement was obtained (Fð52;2Þ and p-value 2:6 � 10�12).Finally, we continued by adding one parameter in turn to seewhether the model would improve significantly. The result wasnegative. In all cases the estimates were produced by

maximum-likelihood estimation, i.e. by maximizing the usual like-lihood function (sometimes denoted the objective function), bywhich all parameters including those from the probability distribu-tions were determined. The approach taken is denoted ‘forwardinclusion’. In contrast one may start with including covariates onall non-fixed parameters and then excluding those, which are notsignificant. That procedure, denoted ’backward exclusion’, is morecomputational demanding.

Hence, we conclude that the best model is where the threeparameters, x; p1 and p3, are allowed to vary between the groupswhile the rest are kept fixed for the population as a whole, i.e. hav-ing a common population distribution with one mean value andone variance for each parameter, except for the parameters thatoriginally were fixed (the parameters with minor sensitivity). The

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Fig. 8. Boxplot for p1 (left), p3 (middle) and w (right) for the three groups; hypocortisolemic depressed (hypocort), non-depressed and hypercortisolemic depressed(hypercort). In each box the central mark is the median, the edges of the box are the 25th and 75th percentiles, the whiskers extend to the most extreme data points notconsidered outliers, and outliers are plotted individually. The population parameter value (i.e. mean value for each group) for p1 is 2.89, 13.59 and 99.35 min�1, for p3 it is0.025, 0.062 and 0.244 dl � (lg �min)�1 and for x it is 0.046, 0.041 and 0.036 min�1, for the hypocortisolemic depressed group, the non-depressed group and thehypercortisolemic depressed group, respectively.

30 J. Gudmand-Hoeyer et al. / Mathematical Biosciences 257 (2014) 23–32

findings are confirmed by box plots as illustrated in Fig. 8 for theresulting best model.

Thus, we have identified three parameters p1; p3 and w, whichdescribe the deviation between the three groups. p1 is the circa-dian-dependent rate for how well CRH stimulated the synthesisof ACTH in the low cortisol limit, p3 is the circadian-independentcomponent of the elimination of ACTH per cortisol concentration,and w is the common elimination rate for CRH and REG. Thus,these parameters describe underlying physiological mechanismsand thereby describe possible malfunctioning for the depressedgroups. The specific values describe the strength of malfunctioning

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Fig. 9. (a) ACTH concentration profiles for all subjects of the three groups, hypocortisohypercortisolemic depressed (the last 7 subjects) are shown. After each group a windowgroup average (green curve). Each window shows data as red circles and the model predicThe figure gives an impression of how well the model describes the individual data as wevariability and the intra-variability, in both data and model. (For interpretation of the refearticle.)

in each of these groups. We emphasize that these three parametersare among the sensitive ones, but without being the three mostsensitive. This is because the NLME analysis takes the inter-indi-vidual variation seen in data into account in contrast to a simplerdata-less sensitivity analysis.

In the trellis plot in Fig. 9, the individual profiles of all subjectswithin the three groups, hypocortisolemic depressed (the first 10subjects), non-depressed (the next 12 subjects) and hypercortiso-lemic depressed (the last 7 subjects) are shown separated by spa-ghetti plots for each group (i.e. panel Nos. 11, 24 and 32). Thesespaghetti plots show the individual profiles within the group (blue

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lemic depressed (the 10 first subjects), non-depressed (the next 12 subjects) and(panel Nos. 11, 24 and 32) shows the preceding group profiles (blue curves) and thetions as blue lines. (b) the similar are shown for cortisol (CORT) data and predictions.ll as the population (both within the groups and the whole population), i.e. the inter-rences to color in this figure legend, the reader is referred to the web version of this

J. Gudmand-Hoeyer et al. / Mathematical Biosciences 257 (2014) 23–32 31

curves) and the group average (green curve). Each panel showsdata as red circles and model predictions as blue curves. Fig. 9(a)shows the profiles of ACTH concentrations and Fig. 9(b) showsthe profiles of the cortisol concentrations. The trellis plot givesan impression of how well the model describes data, and showthe inter- and intra-variability in both data and model.

7. Discussion and conclusion

Endocrine pathologies are believed to be related to stress aswell as to depression. A novel model capable of showing both cir-cadian as well as ultradian oscillations of the hormones involved inthe HPA-axis is proposed. These patterns imitate those observed indata from 29 subjects diagnosed as either depressed or normal.

Models should be developed so they incorporate the responsi-ble mechanisms for the modeled phenomena, i.e. they should bemechanism-based and they should be based on first principles(conservation laws, etc.) whenever possible. Thus, mechanism-based models may be rather detailed models. However, in orderto identify and estimate patient-specific parameters in an effectiveand reliable way, the number of parameters has to be kept as lowas possible, which means that non-essential factors and elementshave to be excluded. Hence, a compromise between these conflict-ing demands often results in models based on elements resemblingthe underlying mechanisms as well as lumped elements. In anycase, all parameters should have physiological interpretations. Fol-lowing the principle of parsimony a model should be as simple aspossible, fulfilling the purpose of the modeling task without con-tradicting existing knowledge. This has been the guiding principlein deriving the novel model presented in this paper.

The backbone of the proposed model is the minimal model,which has been extended with a hypothalamus submodel (Eqs. 1and 2). The submodel has a unique neutral steady state in the posi-tive quadrant for k1 > w1. Solutions starting in the positive quad-rant stay there for all later times. The small amplitude frequencyis explicitly found. It is shown that all solutions have closed peri-odic trajectories in the anti-clockwise direction around the steadystate. Moreover, it follows that there exist box-shaped attractingtrapping regions in the positive quadrant of the pituitary–adrenalsub-space, which gives a trapping region in the positive octahe-dron of R4 for the full model (Eqs. (1)–(4)). Thus, solutions are qual-itatively meaningful, e.g. they are uniformly persistent andbounded.

Data from 29 subjects are used for individual as well as forwhole population estimation of parameters. This makes the modelpatient-specific and characterizes at the same time three groupswithin the population. The categorization of the subjects were per-formed using the previously introduced O-index. The three groupsare denoted hypocortisolemic depressed, non-depressed andhypercortisolemic depressed and are in agreement with, but notidentical to the groups originating when using standard subjectivepsychiatric procedures.

NLME modeling was used. It showed that two parameters (p1

and w) influencing ultradian frequency and amplitude deviate sig-nificantly between the three groups. Similar, a parameter (p3)describing the elimination of ACTH deviates significantly betweenthe three groups. This parameter influences the mean values ofACTH and cortisol. No other parameters deviate significantlybetween the three groups.

The identification of p3 as a parameter differentiating betweenthe three groups agrees with the statistical relation providedthrough the O-index, since p3 affects the average levels of ACTHand then of cortisol. However, the identification of the two otherparameters, p1 and w, as parameters differentiating between thethree groups add extra information: They have an impact on

ultradian frequency and amplitude. That is very important newinsight, which has been proposed multiple times without any val-idation until now.

The three parameters characterizing depression in the modelrepresent underlying physiological mechanisms, which controlthe average level as well as the ultradian frequency and amplitudefor the hormones ACTH and cortisol. The results are promisingsince they offer a step towards an exact etiology for depression.

The model presented in this paper is one example of a patient-specific model. Patient-specific models are (preferable mechanism-based) models with physiologically interpretable parametersrelated to different pathologies and healthy states and in whichthe parameter values can be estimated for each individual patient.Hence, in patient-specific models, pathologies can be described bycertain parameter values or by certain ranges of values. In the spe-cific HPA-axis model, the parameters separating the three groupsindicate in which parts of the HPA-axis the malfunctioning maybe located. In general, the parameters are estimated using mea-surements from the individual patient in combination with a spe-cific model, potentially resulting in more precise clinical diagnosesand more reliable suggestions for treatment. Furthermore, usingpatient-specific models existing classes of diseases may be dividedinto subclasses of pathologies corresponding to the actual defect ofthe physiological system. In the specific case, depression mighthave a subclass termed malfunctioning in the HPA-axis with anadditional subclass of ACTH elimination dysfunctions. Knowingthe pharmacological defect(s) makes it possible to identify targetsfor the development of new and perhaps more effective drugs. Inaddition, the cost of drug development may be reduced not onlydue to a more effective process when searching for new drug can-didates, but also because modeling and simulation may substitutesome of the expensive and time-consuming animal and humanexperiments in pre-clinical and clinical trials.

When well-validated models with patient-specific parametersexist, the identification of potential biomarkers becomes achiev-able. Such biomarkers will definitely give rise to a classificationof variants of the disease because they possess inherent featuresthat are naturally in accordance with clinical data and diagnoses.This would be a big step forward for healthcare compared to theexisting empirically developed biomarkers, since the former alsopinpoints the pathological parts. In the specific example, theO-index serves as a classical biomarker in the sense that it is aquantitative measure that allows diagnose and assessment ofresponse to treatment. The three specific parameters varyingbetween the groups (hypocortisolemic depressed, hypercortiso-lemic depressed and non-depressed) may also serve as biomarkers,although not in the classical interpretation. In case of the O-index,we have a biomarker, which is statistically correlated with the dis-eased state and it could potentially be used to assess the effect oftreatment. However, it does not point toward the dysfunctionalmechanisms in the same way that the three identified parametersdo. Thus, we are using the model to interpret the blurred andcomplex information hidden in the quantitative measurements ofACTH and cortisol.

In this paper, we have illustrated how patient-specific modelingcan be used to make diagnoses more precise and offers a potentialtool for target identification leading to the design of new drugs andimproved treatment for patients. This may also contribute toreplace evidence based medicine with model based medicine inthe future.

Acknowledgements

We would like to thank MD Carroll and MD Veldhuis for grant-ing permission to use their clinical data. The work has been

32 J. Gudmand-Hoeyer et al. / Mathematical Biosciences 257 (2014) 23–32

supported by funding from The Danish Society for the Protection ofLaboratory Animals and from Alternativfondet.

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