Patient Speciﬁc Modelling in Diagnosing Depression

Combining Mixture and Non-linear Mixed Effect Modelling

Johnny T. Ottesen

Dept. of Sciences, System and Models, Roskilde University,

Building 27, Universitetsvej 1, DK-4000 Roskilde, Denmark

Keywords:

Mathematical Modelling, Depression, Parameters Estimation, Patient Speciﬁc.

Abstract:

Depression is a very common disease. Approximately 10% of people in the Western world experience severe

depression during their lifetime and many more experience a mild form of depression. It is commonly believed

that depression is caused by malfunctions in the biological system constituted by the hypothalamus-pituitary-

adrenal (HPA) axis. We pose a novel model capable of showing both circardian as well as ultradian oscillations

of hormone concentrations. We show that these patterns imitate those observed in the corresponding data.

We demonstrate that patient-speciﬁc modelling shows its ability to make diagnoses more precise and to offer

individual treatment plans and drug design. Efﬁcient and reliable methods for parameter estimation are crucial.

Presently we are examine how well the shufﬂed complex evolution algorithm do in estimating parameters. The

next step is to investigate which parameters there are responsible for which pathologies by non-linear mixed

effect modelling and statistical hypothesis testing. Preliminary results are promising. Finally, we plan to

investigate how well the Metropolis-Hastings Algorithm of the Bayesian Markov Chain Monte Carlo method

for estimating the parameters is working and we are about to do the same using iteratively reﬁned principal

differential analysis or the approximated maximum likelihood estimate.

1 INTRODUCTION

Depression is a mental disease normally diagnosed

by psychiatrists. However, it is commonly believed

that it is caused by malfunctions in some coupled

endocrine glands producing hormones. The biolog-

ical system made up by these glands and the hor-

mones they produces are denoted the hypothalamus-

pituitary-adrenal (HPA) axis. The interactions be-

tween these glands are constituted by mainly three

hormones. Corticotropin releasing hormone (CRH)

is secreted in hypothalamus and released into the por-

tal blood vessel of the hypophyseal stalk, where it is

transported to the anterior pituitary and it stimulates

the release of adrenocorticotropic hormone (ACTH)

from the pituitary gland. ACTH moves with the

bloodstream and when it reaches the adrenal glands it

stimulates secretion of cortisol into the blood steam.

Furthermore, cortisol feeds back on hypothalamus

and pituitary inﬂuence the production of CRH and

ACTH, respectively.

The HPA axis plays an important role under

stressed conditions by raising the concentration of the

hormones which leads to energy directed to the organ-

ism (Savic and Jelic, 2006). The return to the basal

hormone levels after a while is an important feature

of the system when it is working properly. Keeping

cortisol concentration within a certain range is im-

portant for various reasons. A maintained, too high

level of cortisol (hypercortisolism) can cause depres-

sion, diabetes, visceral obesity or osteoporosis (Con-

rad et al., 2009). Too low concentration (hypocorti-

solism) is neither desirable since it can result in a dis-

turbed memory formation or life-threatening adrenal

crisis (Conrad et al., 2009) beyond depression. The

regulation of the HPA axis is thus important to stay

healthy.

The cortisol concentration has a circadian pattern.

It is typically low between 8 p.m. and 2 a.m. and rises

to peak in the period 6-10 a.m. (Jelic et al., 2005).

CRH is secreted in a pattern with a frequency of one

to three secretory periods per hour often referred to

as ultradian oscillations (Chrousos, 1998). Through-

out the literature circadian as well as ultradian oscilla-

tions in the hormone concentration of ACTH and cor-

tisol is seen (Grifﬁn and Ojeda, 2004; Carroll et al.,

2007). Thus circadian as well as ultradian oscillations

are characteristics of the system. It is generally be-

lieved that circadian pattern is caused by exogenous

factors (daylight, temperature, psychological as well

658

T. Ottesen J..

Patient Speciﬁc Modelling in Diagnosing Depression - Combining Mixture and Non-linear Mixed Effect Modelling.

DOI: 10.5220/0004622606580663

In Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (BIOMED-2013), pages

658-663

ISBN: 978-989-8565-69-3

Copyright

c

2013 SCITEPRESS (Science and Technology Publications, Lda.)

as physical stress, etc.) but that ultradian pattern is

caused by intrinsic dynamics of the HPA axis.

According to (Chrousos, 1998) the frequency of

the ultradian oscillations is rather insensitive to stress

whereas the amplitude increases. Examples of data

for normal and depressed subjects (diagnosed by psy-

chiatrists) showing circadian and ultradian oscilla-

tions are illustrated in ﬁgure 1 and ﬁgure 2.

0 120 240 360 480 600 720 840 960 1080 1200 1320 1440

0

10

20

30

40

50

60

Time [min]

ACTH concentration[pg/ml]

Hypercortisol depressive

Normal

Lowcortisol depressive

Figure 1: Example of ﬁltered ACTH data of three individ-

uals from the hypercortisol depressed group, the low corti-

sol depressed group and a normal person. Time t=0 corre-

sponds to midnight. Data was sampled every tenth minutes

through 24 hours.

0 120 240 360 480 600 720 840 960 1080 1200 1320 1440

0

2

4

6

8

10

12

14

16

18

20

Time [min]

Cortisol concentration[µg/dl]

Hypercortisol depressive

Normal

Lowcortisol depressive

Figure 2: Example of ﬁltered cortisol data corresponding

to the individuals represented in ﬁgure 1. Time t=0 corre-

sponds to midnight. Data was sampled every tenth minutes

through 24 hours.

Understanding the interplay between the various

elements of the HPA axis is interesting and important.

Since several feedback mechanisms are working si-

multaneously in the HPA axis cause and effect may be

hard to distinguish. A mathematical model may help

understand this and can be an important tool for point-

ing out different ways in which malfunctioning may

occur. More speciﬁcally, if we are able to estimate

parameters based on a correct model of the HPA axis

and individual data, i.e. concentrations of the hor-

mones ACTH and cortisol in blood plasma samples,

and some of these parameters varies signiﬁcantly be-

tween groups of depressed subjects and normal sub-

jects, then these parameters characterise the state of

the disease and at the same time pinpoint the mecha-

nisms which are malfunctioning.

2 MODELS OF THE HPA AXIS

It is commonly known that cortisol inhibits the secre-

tion of CRH through glucocorticoid receptors (GR)

situated in the hypothalamus (Wilson and Foster,

1992). In addition cortisol also performs a negative

feedback on the secretion of ACTH through GR situ-

ated in pituitary (Tortora and Derrickson, 2006). This

description is called ’the minimal model of the HPA-

axis’ and has been thoroughly investigated in (Vinther

et al., 2011). In (Vinther et al., 2011) as well as

in (Kyrylov et al., 2004; Kyrylov et al., 2005; Jelic

et al., 2005; Savic and Jelic, 2005; Savic and Jelic,

2006; Liu et al., 1999; Bingzhenga et al., 1990) the

splitting of the circadian and ultradian rhythm is as-

sumed. This is in such a way that the ultradian rhythm

is considered an inherent behaviour of the HPA axis

whereas the circadian rhythm is thought of as an ex-

ternal input to the axis. The models presented in

(Vinther et al., 2011; Kyrylov et al., 2004; Kyrylov

et al., 2005; Jelic et al., 2005; Savic and Jelic, 2005;

Savic and Jelic, 2006; Liu et al., 1999; Bingzhenga

et al., 1990) all consist of a system coupled non-linear

ordinary differential equations. Despite different ap-

proaches the common aim of these publications is

to have unstable and oscillating solution curves. In

(Vinther et al., 2011; Savic and Jelic, 2005; Savic

and Jelic, 2006) this was done by looking for a Hopf-

bifurcation of a stable ﬁxed point, thus guaranteeing

oscillating solutions. However previous results docu-

mented in (Vinther et al., 2011) was that this ’minimal

model of the HPA axis’ was not capable of reproduc-

ing the characteristics seen in data using reasonable

parameter values. This suggests that the ultradian os-

cillations arise from other mechanisms like bursting

which has been investigated successfully by (Veld-

huis et al., 1989; Keenan et al., 2001; Keenan and

Veldhuis, 2003). It has also been suggested that the

ultradian oscillations may arise from the introduction

of a time delay (Vinther et al., 2011; Bairagi et al.,

2008). However, investigations show that rather large

time-delays (i.e. at least 18 minutes) in the feedback

mechanisms are needed (Vinther et al., 2011). Fur-

thermore, one might suggest that the ultradian oscil-

lations are not an inherent behaviour of the HPA-axis

but imposed from outside. A last possibility is that

something is missing in the minimal description of

the HPA-axis. In the latter case, we have suggested

the inclusion of mechanisms from hippocampus, see

(Andersen et al., 2010; Andersen et al., 2013).

2.1 Inclusion of Hippocampal Dynamics

It has been suggested that hippocampus is also play-

PatientSpecificModellinginDiagnosingDepression-CombiningMixtureandNon-linearMixedEffectModelling

659

ing a role for the dynamics of the HPA axis in such

a way that hippocampus stimulates hypothalamus to

produce CRH (Jelic et al., 2005; de Kloet et al., 1998;

Jacobson and Sapolsky, 1991; Oitzl et al., 1995). Fur-

thermore cortisol should be able to exert a feedback

on mineralcorticoid receptors (MR) and GR situated

in hippocampus. More speciﬁc cortisol should exert a

positive feedback through GR in hippocampus and a

negative feedback through MR in hippocampus (Jelic

et al., 2005; Oitzl et al., 1995).

To our knowledge there is no known hormone se-

creted from hippocampus to stimulate secretion of

CRH from hypothalamus neither does data for con-

centration of CRH exist for humans. Therefore one

is faced with the challenge of how to model the hip-

pocampal dynamics. The amount of cortisol binding

to MR compared to the amount binding to GR deter-

mines whether the inclusion of hippocampal mecha-

nisms corresponds to a positive or negative feedback.

However this may depend on the concentration of cor-

tisol thus given a positive feedback for some concen-

trations and a negative for others.

The inclusion of a competing positive and neg-

ative feedback mechanism widens the possible be-

haviour of the model compared to a purely negative

feedback model. The idea is that the inclusion of hip-

pocampal mechanisms would give an unstable ﬁxed

point that could explain the ultradian oscillations seen

in data.

The known features the model reﬂects are feed-

back from cortisol on the secretion of CRH and

ACTH and the combined feedback effect from hip-

pocampus. Since no known hormone is secreted from

hippocampus the known features from hippocampus

is directly included in the differential equation gov-

erning CRH. This approach leads to the model given

in equations (1) - (3) and is denoted the general

model,

dx

1

dt

= f

1

(x

3

) − w

1

x

1

, (1)

dx

2

dt

= f

2

(x

3

)x

1

− w

2

x

2

, (2)

dx

3

dt

= k

2

x

2

− w

3

x

3

. (3)

The overall feedback from cortisol (x

3

) in hippocam-

pus and on CRH (x

1

) is modelled through the func-

tion f

1

(x

3

). The negative feedback from corti-

sol on ACTH (x

2

) is modelled through the func-

tion f

2

(x

3

). w

1

, w

2

, w

3

are elimination constants and

w

1

, w

2

, w

3

> 0, f

1

, f

2

: R

+

∪{0} 7→ R

+

∪{0}, f

1

, f

2

∈

C

1

, sup( f

1

) = M

1

∈ R

+

, sup( f

2

) = M

2

∈ R

+

,f

1

(0) >

0 and f

2

(0) > 0. This means f

1

and f

2

are bounded

functions mapping non negative real numbers into

non negative real numbers. f

1

and f

2

have non neg-

ative domains since the cortisol concentration is non

negative. The ranges of f

1

and f

2

are non negative

since the positive stimulation of the hormones must

not turn negative. The criteria that f

1

and f

2

are

bounded reﬂects the saturation of receptors through

which the feedbacks are realized. When no cortisol is

present the feedbacks must not completely shut down

the stimulation of hormone production. This justiﬁes

f

1

(0) > 0 and f

2

(0) > 0. It is further assumed that the

feedback on ACTH is negative meaning f

′

2

(x

3

) < 0,

∀x

3

∈ R

+

∪{0}. An approach similar to this has been

investigated by (Conrad et al., 2009). However, in

(Conrad et al., 2009) the ACTH and CRH compart-

ment are pooled together.

The major conclusions (see (Vinther et al., 2011;

Andersen et al., 2010; Andersen et al., 2013)) con-

cerns the possibility of oscillating solutions:

• For the general model there exist a trapping re-

gion. This means that all solutions are guaranteed

not to become negative or tend to inﬁnity.

• The characteristic ultradian oscillations seen in

data has been suggested to be an inherent be-

haviour of the HPA-axis. However the general

model is not capable of giving oscillating solu-

tions within physiologically reasonable parameter

values. This suggests that origin of the ultradian

rhythm should be found in different mechanisms

than the ones included in the general model.

• Using the default parameters the system is glob-

ally stable. This means that the solution curves

will converge to a unique ﬁxed point for all ini-

tial conditions. This can be interpreted as healthy

people returns to normal cortisol levels after peri-

ods of stress.

• A perturbation of the parameters may lead to bi-

furcations where the system undergo a transition

from having a unique stable ﬁxed point to hav-

ing three ﬁxed points where one is unstable and

the two others are stable. All solution curves con-

verge to one of the two stable ﬁxed points depend-

ing on the initial conditions. This is accordance

with the fact that depressed individuals is either

hypercortisolemic depressive or hypocortisolemic

depressive.

In the following we will pose a novel model ex-

tending the general model capable of showing both

circadian as well as ultradian oscillations.

2.2 A Novel Model of the HPA Axis

The fundamental extension from earlier, e.g. equa-

tions (1) - (3), is that we allow a positive (super-short)

feedback in hypothalamus, i.e. from CRH on its own

SIMULTECH2013-3rdInternationalConferenceonSimulationandModelingMethodologies,Technologiesand

Applications

660

production. However, this is not based on pure specu-

lation but on experimental evidence, see (Motta et al.,

1970; Louis et al., 1987; Zanisi et al., 1987). In com-

parison with most speciﬁc models in the literature an-

other novelty is the inclusion of non-linear terms in

the feedbacks and the feedforwards (i.e. the terms x

2

3

and x

2

2

) thus large concentration of ACTH and cortisol

affects the production of cortisol and CRH, respec-

tively, more pronounced than linear terms.

Hence we propose the improved novel model,

dx

1

dt

= C(t)

a

1

+ a

2

x

1

1+ a

3

x

3

+ a

4

x

2

3

− w

1

x

1

, (4)

dx

2

dt

=

a

5

x

1

1+ a

6

x

3

− w

2

x

2

, (5)

dx

3

dt

= a

7

x

2

+ a

8

x

2

2

− w

3

x

3

. (6)

Here the circadian rhythm is generated by the endoge-

nous input

C(t) = t

3

m

exp(−

t

m

120

), (7)

where t

m

denotes time t modulo 24 hours (the time

units used is minutes).

3 METHODS

Models should be developed so they incorporate the

responsible mechanisms for the modelled phenom-

ena, i.e. they should be mechanisms-based and they

should be based on ﬁrst principles (conservationlaws,

etc.) whenever possible. Thus mechanisms-based

models may be rather detailed models. Somehow in

oppose to this demand but in order to identify and es-

timate patient-speciﬁc parameters in an effective and

reliable way, the number of parameters has to be kept

as low as possible, which means that any unimpor-

tant factors and elements should be excluded. Hence

a compromise between these conﬂicting demands of-

ten result in models based on elements resembling the

underlying mechanisms as well as lumped elements.

In any case, all parameters should have physiological

interpretations. Following the principle of parsimony

a model should be as simple as possible fulﬁlling the

purpose of the modelling task without contradicting

existing knowledge. This has been the guidance in

deriving the novel model presented in section 2.2.

Patient-speciﬁc models are (preferable

mechanisms-based) models with physiologically

interpretable parameters related to different patholo-

gies and healthy states in which the values of the

parameters can be individually estimated. Thus,

patient-speciﬁc models are models that can be

adjusted to speciﬁc individuals. Hence, in patient-

speciﬁc models, pathologies are clariﬁed by the

values of certain parameters. The parameters are

estimated from measurements in combination with

the model, thus giving rise to more precise clinical

diagnoses and more reliable suggestions for treat-

ments than are known based on today’s practices. In

addition, existing classes of diagnosed cases may be

reﬁned into subclasses of pathologies corresponding

to the actual defect of the physiological system

by use of such patient-speciﬁc models. Moreover,

knowing the actual defect(s) makes the develop-

ment of target-speciﬁc drugs and other treatments

possible. Development of this kind can guide the

pharmaceutical industry in its search for new and

improved drugs. In addition, a huge reduction in the

cost of developing new drugs may be expected not

only due to a more beneﬁcial process when searching

for drug candidates but also because models may be

used to substitute some costly animal and human

experiments in future pre-clinical and clinical trials,

respectively.

The parameters have to be estimated by statisti-

cally founded algorithms (for example, the extended

Kalman ﬁlter, the Nelder-Mead algorithm combined

with simplex methods, multidirectional search, par-

ticle ﬁlter/sequential Monte Carlo methods, genetic

algorithms, Bayesian methods etc.) or by functional

analysis, i.e. optimal control, functional differential

analysis, collocation methods, etc. Not all the pa-

rameters will necessarily be identiﬁable due to lim-

itations concerning available data and/or an over-

parameterization of the model. Thus, the estimation

process has to be an iterative procedure coupled with

sensitivity analyses or generalized sensitivity analy-

ses combined with subset selection strategies, for in-

stance.

Presently we are examine how well the shufﬂed

complex evolution (SCE) algorithm do in estimat-

ing parameters and are about to do the same using

the Metropolis-Hastings Algorithm of the Bayesian

Markov Chain Monte Carlo (MCMC) method and the

iteratively reﬁned pricipal differentialanalysis (iPDA)

also denoted approximated maximum likelihood esti-

mate (AMLE).

When well-validated models with patient-speciﬁc

estimated parameters exist, the identiﬁcation of po-

tential biomarkers becomes achievable. Potentially

parameter estimation by patient-speciﬁc models may

identify windows for parameter values deﬁning dif-

ferent states for patients, e.g diseased or healthy. This

would be a big step forward for healthcare compared

with empirical developed biomarkers, since the for-

mer also pinpoint the pathological part of the system

PatientSpecificModellinginDiagnosingDepression-CombiningMixtureandNon-linearMixedEffectModelling

661

0 200 400 600 800 1000 1200 1400

0

2

4

6

8

10

12

14

16

18

20

Time t [min]

CRH [pg/ml]

Figure 3: Example of simulated CRH concentration for an

individual.

0 200 400 600 800 1000 1200 1400

0

10

20

30

40

50

60

Time t [min]

ACTH [pg/ml]

Figure 4: Example of simulated ACTH concentration for an

individual.

0 200 400 600 800 1000 1200 1400

0

2

4

6

8

10

12

14

16

18

20

Time t [min]

Cortisol [pg/ml]

Figure 5: Example of simulated cortisol concentration for

an individual.

for diseased patients. When such potential biomark-

ers are identiﬁed, different groups of patients, i.e.

pathological subjects versus non-pathological sub-

jects, can be examined. Notice that, some of the pa-

rameters between two different groups have to vary.

To determine whether there is a real difference be-

tween the values of the parameters (i.e. the biomark-

ers) within two groups or whether suggested biomark-

ers can identify variant causes (i.e. pathologies) of the

illness (diagnosed by symptoms), statistical tests have

to be performed. The biomarkers will deﬁnitely give

rise to a classiﬁcation of variants of the illness be-

cause they have inherent features that are naturally in

accordance with data from clinical diagnoses.

4 RESULTS

The values of the 11 parameters of the model pre-

sented in section 2.2 are initially guessed to be a

1

=

4.6, a

2

= 7.6, a

3

= 2.0, a

4

= 0.5, w

1

= 0.34, a

5

=

0.126, a

6

= 3.0, w

2

= 0.011, a

7

= 0.064, a

8

= 0.0165

and w

3

= 0.057. In the near future the parameters will

be estimated by a Bayesian MCMCM method as de-

scribed in section 3 using these parameter values as

initial guesses. The outcome is shown on ﬁgures 3, 4

and 5

5 CONCLUSIONS

Depression is a very common disease. Approximately

10% of people in the Western world experience se-

vere depression during their lifetime and many more

experience a mild form of depression. Endocrine

pathologies are believed to be responsible for depres-

sion as well as for stress. Patient-speciﬁc modelling

has shown its ability to make diagnoses more precise

and to offer individual treatment plans and drug de-

sign. Efﬁcient and reliable methods for parameter es-

timation are crucial. We have proposed a novel model

capable of showing both circardian as well as ultra-

dian oscillations. These patterns imitate those ob-

served in the correspondingdata. Presently we are ex-

amine how well the shufﬂed complex evolution (SCE)

algorithm do in estimating parameters. The SCE al-

gorithm is a stepwise global method where a num-

ber of complexes in each step make use of the sim-

plex algorithm and transition between steps evolve

according to a random procedure. The next step is

to investigate which parameters there are responsi-

ble for which pathologies by non-linear mixed effect

(NLME) modelling and statistical hypothesis testing

(ANOVA). Preliminary results are promising. Fi-

nally, we plan to investigate how well the Metropolis-

Hastings Algorithm of the Bayesian Markov Chain

Monte Carlo (MCMC) method for estimating the pa-

rameters is working and we are about to do the same

using iteratively reﬁned principal differential analysis

(iPDA) or the approximated maximum likelihood es-

timate (AMLE).

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