Identification of Markers from a Set of Spectral Courses · ures. Pentagrams “⋆” : discrete courses of the mea-sured (renal cell carcinoma) spectrum; pentagrams “⋆”: discrete

Identification of Markers from a Set of Spectral Courses

JIRI KNIZEKCharles University in Prague

Faculty of MedicineSimkova 870

500 38 Hradec KraloveCZECH [email protected]

LADISLAV BERANEKUniversity of South Bohemia

Institute of Applied InformaticsBranisovska 31a

370 05 Ceske BudejoviceCZECH [email protected]

PAVEL BOUCHALMasaryk University in Brno

Faculty of ScienceKamenice 753/5

625 00 BrnoCZECH REPUBLUC

[email protected]

BORIVOJ VOJTESEKMasaryk Memorial Cancer

InstituteZluty kopec 543/7


[email protected]

RUDOLF NENUTILMasaryk Memorial Cancer

InstituteZluty kopec 543/7


[email protected]

PAVEL TOMSIKCharles University in Prague

Faculty of MedicineSimkova 870

500 38 Hradec KraloveCZECH [email protected]

Abstract: A brief introduction of algorithms for the statistical identification of markers from a set of spectralcourses is the topic of our paper. Partial results, demonstrated by pictures, are very promising. The proposedalgorithm is generally applicable for an arbitrary problem of marker identification by tests in a set of quantifyingdependences.

Key–Words:Marker, biomarker, regression, tests of hypotheses, software

1 IntroductionThe rapid development of genomic and proteomicmethods led to an enormous increase in experimen-tal data. To be able to extract answers to importantquestions from these data, it is necessary to find aneffective bio-statistical method for their processing.Application of advanced methodologies is necessaryto give us more detailed, structured information.

A (dependence) biomarker, or (dependence) bi-ological marker, is a (dependence) indicator of a bi-ological state. It is a characteristic that is objec-tively measured and evaluated as a (dependence) indi-cator of normal biological processes, pathogenic pro-cesses, or pharmacologic (dependence) responses toa therapeutic intervention. It is used in many scien-tific fields. The presence and concentration of certainbiomarker molecules is then identified and measured.That is why our proposed algorithm based exclusivelyon classical statistical decision making with the helphypotheses testing may facilitate prediction of certainclinical aspects of diseased patients.

Medical and biological research often deals withmiscellaneous dependences. A particular experimen-tal problem in which dependences are dealt with is

then, in terms of the previous paragraph, described bythe means of regression functionsη 1(x), η 2(x), . . . ,ηM (x) [13], [14].

Very often, the problem is specified in such a waythat the first group of experimental dependences mod-els the data measured inthe group of diseased patientsand the second group of experimental dependencesmodels the data measured inthe group of healthy pa-tients. Spectral methodsrepresent a large class ofphysical methods which are based ontwo dimensionaldependences. We denote the group of spectral de-pendences which model the data of the type“groupof diseased patients“by the regression functionsdiseasedη 1(x), diseasedη 2(x), . . . , diseasedηMdiseased

(x)and the group of spectral dependences which modelthe data of the type“group of healthy patients“by theregression functionshealthyη 1(x), healhyη 2(x), . . . ,healthyηMhealthy

(x). The quantityx is a real indepen-dent variable that may represent time, effective massin the case of MS1, etc. Then the total number of de-

1mass spectroscopy; especially by the use of spectral methods,is a monitored process that can take place in some relatively verynarrow spectral region (in relation to the whole possible magni-tude)

WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE

Jiri Knizek, Ladislav Beranek, Pavel Bouchal, Borivoj Vojtesek, Rudolf Nenutil, Pavel Tomsik

E-ISSN: 2224-2902 35 Issue 1, Volume 10, January 2013

pendences or the total number of regression functionsis

M = Mdiseased +Mhealthy.

In this paper,a new regression algorithm for sta-tistical identification of markers from a set of spec-tral coursesis described. There aredefinition mat-ters presented in section2. There isa model “setof multiple linear regressions”presented in section2.1 which serves as an initial basis of whole algo-rithm. Section2.2 deals with a more narrow modelfor our purpose - model“a set of orthogonal poly-nomial regressions”, i.e., practically with model“aset of spectral courses”. Section2.3 deals with thenewly well-established statistical tool“the definitionmatrix” which simplifies the definition of various sta-tistical tests of dependences. For the solution of theproblem, key principals are presented in section3.Some numerical-mathematical aspects of used algo-rithms (and their solution with the help of“highly ef-fective algorithm for orthogonalization”) are referredto in section3.1. Section3.2, named“identificationof markers by simultaneous tests in a set of quantify-ing dependences”, deals with compliance to the fun-damentalbiophysical principlesat the algorithm ap-plication. There areexperimental resultspresented insection4. Description of“biomarker pictorial exem-plifications on real data”is presented in section4.1.Section4.2 deals with very promising results ofiden-tifying biomarker areas in SELDI-TOF mass spectraof data which has been obtained from 10 patients suf-fering from renal cell carcinoma.

2 Def nitions

2.1 The model “A set of multiple linear re-gressions”

The beginning of the algorithmic study described be-low is based on following test criterion:

λF =(r −Rβ)′(RCR′)−1(r −Rβ)/J

(y −Xβ)′(Σ−1⊗ I) (y −Xβ)/ (MT −K)

∼F(J, MT−K),

C = [X ′(Σ−1⊗ I)X]−1 ,

(1)for the standard statistical model called the“Disturbance-Related Sets of Regression Equa-tions”

y1

y2...

yM

=

X1

X2

. . .XM

β1

β2...

βM

+

e1e2...

eM

(2)

(or brieflyy = Xβ + e) for the null hypothesis

H0 : Rβ = r, (3)

where the form of theR(J×K)matrix of constants andthe form of ther(J×1) vector of constants in rela-tion (3) concretizethe null hypothesisH0. Dimen-sion K of regression vectorβ is given as a sum ofsingle regression vectorsβ1, β2, . . . ,βM , i.e. K =ΣMi=1(Ki + 1). The covariance matrixΩ of the joint

disturbance vectore is given byΩ = Σ ⊗ I and soΩ

−1 = Σ−1

⊗ I [6], [12].It is important that it is possible to test arbitrary

linear mutual relations among particular multiple lin-ear regressions in (2) with the help of the test criterion(1).

2.2 The model known as “A set of orthogonalpolynomial regressions”

It is necessary to approach model (2) more narrowlyfor our purpose (namely)“the statistical identificationof markers from a set of spectral courses”. Every mul-tiple linear regression in model (2) is interpreted asan orthogonal polynomial regression describing oneappropriate spectral course. Thus, we can test (withthe help of (3) appropriately modified) arbitrary lin-ear mutual relations among particular spectral courses[13], [14].

2.3 The def nition matrix

When we summarize the values of regression func-tions (polynomial regressions) into the vector

η (x)=(η 1(x), η 2(x), . . . , ηM (x))′, (4)

we can formally transcribe a null hypothesis (3) intothe form

H0 : kη (x) = r (x) ,

where an abscissax is the arbitrary value of used spec-tral independent variable and

k =

k1, 1 k1, 2 . . . k1, M

k2, 1 k2, 2 . . . k2, M...

.... ..

...kJ, 1 kJ, 2 . . . kJ, M

is the so calleddefinition matrix[14]. Definition ma-trix k expresses generally all conceivable linear mu-tual relations among regression functions (4).




3 Main idea

3.1 A highly effective algorithm for orthogo-nalization

Computational practice showed that the currently usedGram-Schmidt’s polynomials [5], [19] are not able toprovide satisfactory measure of orthogonality. For ourpurpose – the polynomial approximation of a set ofspectral courses – we had to use special, outstandinglyefficient algorithms [1], [5], [7] - [10], [16], [18].

3.2 Identif cation of markers using simulta-neous tests in a set of quantifying depen-dences

As substantial limitation while using the“Test of theHypothesis That One Group of Dependences is Con-sistent with Another Group of Dependences”[14] isthat the null hypothesisH0 : k(J×M)η(M×1)(x) =

r(J×1)(x) can be rejected in favour of the double-sided alternative that at leastone of the J linear re-lationsk(J×M)η(M×1)(x) = r(J×1)(x) is not valid.However, the biophysical principlesof the problemforce the experimenter to assume that changes inthe concentration of a given biomarker arenatu-ral, i.e., complete. It means that the experimenterwould need to reject the null hypothesisH0 :k (J×M)η (M×1)(x) = r(J×1)(x) in favour of thedouble-sided alternative that allJ linear relationsk (J×M)η (M×1)(x) = r(J×1)(x) together are notvalid. Resulting from these necessities is the fact thatmutual conformity is availableonly and onlyin thecases where the number of tested linear relations isJ = 1.

It emerges from these reasons that instead of test-ing one null hypothesisH0 : k (J×M)η (M×1)(x) =

r(J×1)(x), we must testκ simultaneous null hypothe-

sesHj0 : kj

(1×M)η (M×1)(x) = rj

(1×1)(x) = rj (x),

where the index for thejth simultaneous null hypoth-esis isj = 1, 2, . . . , κ. The size of the numberκ, theconcrete form ofdefinition row vectorskj

(1×M) and

elementsr j (x) is then dependent on whether our dataare paired, unpaired or combined. This means that(for a given abscissax) the appropriatesimultaneousnull hypotheses are rejected when un-equalities

pj(x) < α/κ, j = 1, 2, . . . , κ,p(x) = pj′(x) = maxj=1,2,...,κ pj(x) < α/κ,

(5)are simultaneously valid. Along with this condition,

the appropriatepower analysis-un-equalities

1− βj(x) ≥ convention limit , j = 1, 2, . . . , κ,1− β(x) = 1− βj′(x),

(6)must be fulfilled.

The requested power of the test (in other wordsthe convention limit) depends on the test significancelevelα: 1− βreq(α = 0.05) = 0.8 and1− βreq(α ≤

0.01) = 0.95 [3], [4].For test significance levelsα greater thanα =

0.05, the requested powers of the test are1−βreq(α =0.1) = 0.6125, possibly1−βreq(α = 0.2) = 0.2375.

4 Experiments

4.1 Pictorial exemplif cations of real data,Figures 1-9

The potential biomarker areaswere obtained by theproposed data-treatment of the mass spectral data,measured with the aim ofidentifying renal cell carci-noma biomarkers. Two experimental groups (diseasedand healthy, i.e. red and blue) are demonstrated in fig-ures. Pentagrams “⋆” : discrete courses of the mea-sured (renal cell carcinoma)spectrum; pentagrams“⋆”: discrete courses of the measured (not from renalcell carcinoma) spectrum; solidlines: statistical esti-mations of the courses offunction dependencesbasedon the experimental courses of “⋆” and “⋆”.

Conventional decisionmaking conditions (5) and(6) are satisfied in the whole measurement range at allthe 1st-9th figures. The physical unit of the indepen-dent variable (effective mass) in all pictures is Dal-ton. The physical unit of the dependent variable (in-tensity of mass-spectrum) in all pictures is as a %. Ap-propriate potential biomarker areas are then locatedaround thex-ordinates of appropriate dependent vari-able maximums.

1.745 1.75 1.755 1.76 1.765 1.77 1.775 1.78 1.785 1.79

x 104

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figure027: EXPERIMENTAL DEPENDENCES

independent variable

depe

nden

t var

iabl

e

Figure 1. See very detail comments in the section “Picto-rial exemplificationsof real data, Figures 1-9”.




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13figure107: EXPERIMENTAL DEPENDENCES


depe

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iabl

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4.2 Identifying biomarker areas in SELDI-TOF mass spectra

A large set of (normalized) mass spectral data, mea-sured with the aim ofidentifying renal cell carcinomabiomarkers, was subjected to the algorithm describedabove. A group of data was obtained from 10 patientssuffering from renal cell carcinoma. One group ofdata was obtained from renal cell carcinoma tissue,the second group of data was obtained from the samepatients but from healthy (i.e. not renal cell carci-noma) tissue. Naturally,the paired versionof the pro-posed algorithm was used here. Spectra were dividedinto segments containing 200 points. The findingsof the alreadydiscovered biomarker“αB-crystallin”2

[11] by the proposed algorithmwas confirmed. Theproposed algorithm isvery sensitive, because addi-tional potential biomarker areashave been found. Itmanaged to find at least 12 cases of otherbiomarker

2Ciphergen-software [2]

areas. See figures 1-93.

5 Discussion and conclusions

There is no doubt at present that computerized tech-nologies in medicine and biological research, e.g. pro-teomics and genomics, need new approaches. This pa-per deals with“The Regression Algorithm for Statis-tical Identification of Markers From a Set of SpectralCourses”in cases where data error disturbances havea normal distribution.

The proposed algorithm works in practice verywell. At first sight, this property of the algorithmcould appear rather unexpected, considering the veryrigorous necessary requirements for the simultaneoustesting (1) of the appropriatep(x)-values.

The discovered principles are generally usablein analogical spectroscopy studies, i.e., not only fortreatment of MS for the purpose of biomarker identifi-cation. They are even generally applicable to the arbi-trary problem of marker identification (used in miscel-laneous branches of human activity) by simultaneoustests in a set of quantifying dependences.

With the help of an appropriate mass spectradatabase analysis, the proposed methodological ap-proach will lead to the construction ofa clinic run-ning systemwhich will allow statistical decision mak-ing concerning suspicion of disease in patients[17],[18].

Acknowledgements: This work was supported bythe Czech Science Foundation (grant No. P304/10/0868) and by the European Regional Devel-opment Fund (RECAMO; CZ 1.05/2.1.00/03.0101).This study was supported by the programs PRVOUKP37/01, PRVOUK P37/09 and GACR P303/12/P536..

References:

[1] Arnoldi, W.E., The principle of minimized it-eration in the solution of the matrix eigenvalueproblem.Quart. Appl. Math.,9, 17-29 (1951).

[2] CiphergenR© Biosystems, Inc., ProteinChip soft-ware 3.1. Operation Manual (2002).

3Numbers ofbiomarker areasin particular figures: f.1: 1, f.2:1, f.3: 1, f.4: 2, f.5: 1, f.6: 2, f.7: 2, f.8: 1, f.9: 1. Note:Num-bers of figures inheadings of particular figures (e.g. “figure063”and the like) are order numbers of particular (200 points) originalspectral segments.




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[11] Holcakova, J., Hernychova, L., Bouchal, P.,Brozkova, K., Zaloudik, J., Valik, D., Nenutil,R., Vojtesek, B.,Identification ofαB-crystallin,a biomarker of renal cell carcinoma by SELDI-TOF MS, The International Journal of Biologi-cal Markers, Italy: Wichtig editore, 23, 1, 48-53(2008).

[12] Judge, G.G, Griffiths, W.E, Hill, R.C, Lutkepohl,H., Tsoung-Chao, L.,The Theory and Practiceof Econometrics,J. Wiley, New York (1985).

[13] Knizek, J., Sindelar, J., Beranek, L., Vojte-sek, B., Nenutil, R., Brozkova, K., Drazan, V.,Hubalek, M. & Kubacek, L.,Power function fortests of null hypotheses on mutual linear regres-sion functions’ relations, International Journalof Applied Mathematics & Statistics, Volume2; Number S08; Bull. Stat. Econ., ISSN 0973-7022. pp. 26-33 (2008).

[14] Knizek, J., Sindelar, J., Pulpan, Z., Vojtesek, B.,Nenutil, R., Brozkova, K., Drazan, V., Hubalek,M. & Beranek, L.,Test of the Hypothesis ThatOne Group of Dependences is Consistent withAnother Group of Dependences, InternationalJournal of Applied Mathematics & Statistics,Volume 2; Number A08; Bull. Stat. Econ., ISSN0973-7022. pp. 2-18 (2008).

[15] Knizek, J., Sindelar, J., Vojtesek, B., Bouchal,P., Nenutil, R. & Beranek, L.,Identificationof Markers by Simultaneous Tests in a Set ofQuantifying Dependences, International Journalof Statistics & Economics (formerly known asthe “Bulletin of Statistics & Economics”), A10,Volume 5 [Special], Number A10. pp. 12-20(2010).

[16] Knizek, J., Tichy, P., Beranek, L., Sindelar, J.,Vojtesek, B., Bouchal, P., Nenutil, R. & Dedik,O.,Note on Generating Orthogonal Polynomialsand Their Application in Solving ComplicatedPolynomial Regression Tasks, International Jour-nal of Mathematics and Computation, ISSN0974-570X (Online), ISSN 0974-5718 (Print),Vol. 7; No. J10; June 2010. pp. 48-60 (2010).

[17] Knizek, J., Sindelar, J., Vojtesek, B., Bouchal,P., Nenutil, R., Beranek, L. & Dedik, O.,UsingMarkers to Aid Decision Making in Diagnostics,International Journal of Tomography & Statis-tics, ISSN 0973-7294 (Online), ISSN 0972-9976(Print), W11, Volume 16, Number W11. pp. 41-55 (2011).

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Identification of Markers from a Set of Spectral Courses · ures. Pentagrams “⋆” : discrete courses of the mea-sured (renal cell carcinoma) spectrum; pentagrams “⋆”: discrete