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УДК 611.1.001.576: 51-76
PATIENT-SPECIFIC 1D MODEL OF THE HUMAN CARDIOVASCULAR SYSTEM
Ilya Nikolaevich KISELEV1,2, Fedor Anatol'evich KOLPAKOV1,2, Elina Arnoldovna BIBERDORF3'4, Viktor Il'ich BARANOV5, Tamara Grigor'evna KOMLYAGINA5, Irina Yurevna SUVOROVA5, Vladimir Nikolaevich MELNIKOV5, Sergey Georgievich KRIVOSHCHEKOV5
1 Institute of Systems Biology Ltd. 630112, Novosibirsk, Krasin str., 54
2 Design Technological Institute of Digital Techniques of SB RAS 630090, Novosibirsk, Academician Rzhanov str., 6
3 Sobolev Institute of Mathematics of SB RAS 630090, Novosibirsk, Academician Koptyug av., 4
4 Novosibirsk State University 630090, Novosibirsk, Pirogov str., 2
5 Institute of Physiology and Basic Medicine 630117, Novosibirsk, Timakov str., 4
The technique of parameter personalization of 1D hemodynamic model is discussed and validated against physiological data of 1546 examined patients. Different parameter combinations were used, with the quality of prediction of the systolic and diastolic pressures used as the principle validation criterion. It is shown that with an appropriate personalization the model can provide adequate predictions (correlations near 0.9), where the decisive role is played by the total peripheral resistance parameter. Meanwhile parameters of the largest arteries do not play a significant role in the prediction.
Keywords: cardiovascular system, 1D arterial tree model, mathematical modeling, parameters personalization, validation, experimental data.
This work is concerned with the problem of personalization and validation of a one-dimensional
model of the human cardiovascular system including the arterial tree. This kind of models is widely known and used in many studies [6, 10, 16, 21, 22]. One-dimensional models are signficiantly simpler than three-dimensional models, which usually describe three-dimensional flow with Navier-Stokes
equations in domains with moving boundaries (e.g. [1]). However, they are more complicated than models with lumped parameters (e.g. [15]) and are able to describe particular features of blood flow in separate vessels. Such models are best applicable for global description of arterial systems in order to study propagation of pulse wave or emulating blood loss or zero gravity. The model which we use in the
Kiselev I.N. - junior researcher, e-mail: axec@developmentontheedge.com
Kolpakov F.A. - candidate of biological sciences, director, head of laboratory for bioinformatics
Biberdorf E.A. - candidate of physical mathematical sciences, docent, senior researcher, e-mail: biberdorj@ngs.ru
Baranov V.I. - candidate of biological sciences, senior researcher of laboratory for organism functional reserves,
e-mail: v.i.baranov@physiol.ru
Komlyagina T.G. - researcher of laboratory for organism functional reserves, e-mail: KomlyaginaTG@physiol.ru Suvorova I. Yu. - junior researcher of laboratory for organism functional reserves, e-mail: suvorovaiu@physiol.ru Melnikov V.N. - doctor of biological sciences, leading researcher of laboratory for organism functional reserves, e-mail: mvn@physiol.ru
Krivishchekov S.G. - candidate of medical sciences, professor, head of laboratory for organism functional reserves, e-mail: krivosch@physiol.ru
current work was developed by us earlier in the BioUML platform [9].
Model personalization implies calculation of model parameters according to experimental data for a given patient. Wherein it is necessary to validate personalized model against real data and estimate quality of personalization procedure. Most of the studies considering cardiovascular system personalization use models with lumped parameters. Moreover, they usually are restricted to theoretical description of personalization procedure [15] or validate it against small groups (less than 10 subjects) [20] including animals [7]. The work by Kay-vanpour et al. [8] should be separately noted as it describes personalization of a three-dimensional heart model including electrophysiological, biological and biomechanical blocks conducted on a group of 46 individuals.
To our best knowledge, there are very few works on personalization of one-dimensional arterial tree models and there is no published validation on significant amount of experimental data. In [14] a smaller model with 24 arteries is validated on data measured using MRT on a single individual. The measured parameters include thickness and length of the most of 24 vessels and input blood flow. In [18] a 1d model with coronary vessels is compared to data collected from a small group of volunteers (6-8 depending on type of data) without personalization. In [17] personalization is conducted but the results are validated on a single subject.
In this work we use results of a clinical study [12] comprising more than 1500 subjects. We restricted ourselves to predicting of systolic (maximal) and diastolic (minimal) pressures.
ARTERIAL TREE MODEL
A detailed mathematical description can be found in [4]. The Equations for the model consisting of one artery are as follows
d1 A + dQQ = 0 'd zQ + ad z ^ Q2 ] + A dzp + KrQ = 0,
where A(t, z) - cross-sectional area, Q(t, z) - blood flow through this area, p(t, z) - blood pressure in the artery, a - Coriolis coefficient, p - constant blood density, Kr = 8nv - friction coefficient, v - constant blood viscosity. For closing of system (1) we use the equation of state based on the Hooke's law
p(A, A0, P) = P(VA -VA0)/ A0, (1)
where A0 - cross-sectional area of artery in relaxed state (external pressure is equal to internal),
P = \fnh0E - elasticity parameter of the artery, E -
Young modulus, h0 - constant artery wall thickness.
Arteries in the model are organized into binary tree. The model comprising n arteries is described by PDE system
dtUt + B (Ut )d U = S (U, ), (2)
where Ui = (Ai, Qi)T. Further in the text when it is clear from the context that we speak about a single artery, index i will be omitted.
Boundary conditions
Interface conditions at bifurcation points of the arterial tree as well as conditions at the entrance of aorta and ends of terminal arteries (which do not further bifurcate) are formulated as boundary conditions for system (2).
For the aorta entrance it is natural to define input flow Q. Generally speaking, this condition may include aortal and ventricular blood pressures but in the current work we use the explicit form Q = =f (t, SV, HR), where t - time, SV- stroke volume, HR - heart rate, f- certain function. At the ends of terminal vessels we use the filtration condition
Q = Kf (p -pv), (3)
where Kf - filtration coefficient, Q - output flow, p - pressure, pv - pressure in the venous system.
Model parameters
Mathematical modeling of the arterial tree implies numerical solving of initial value boundary problem for hyperbolic system. Model personalization is done by setting individual vessel parameters A0, h0 and E, and boundary conditions according to the experimental data for a given individual. To set the boundary condition one should define function f, values of parameters SV, HR, filtration coefficient Kf and pressure pv.
The data on vessel parameters initially collected by professor A. Noordergraaf [13] have been used for a number of arterial tree models and been modified and adjusted several times. In the work [22] they were extended by data for additional vessels and it was shown that Young modulus is likely to rise along the length of the arterial tree. In [19] it is suggested that the cross-sectional area decreases linearly along each single artery. In [21] cross-sectional areas are adjusted so that reflection coefficients in bifurcation points are minimized. In [10] elastic properties of arteries are also changed. This data is used in the current work and can be found in the supplement materials.
EXPERIMENTAL DATA
As the source of experimental data we use the database [12] including data on common and regional hemodynamics and elasticity of the large arteries of the lower and upper limbs. The Database contains records about 1546 persons examined in a general therapeutic hospital - females and males of different ages, 59 of which are healthy and 1487 have different chronic diseases. Observations were obtained using oscillovasmetry and venous occlusion plethysmography [11].
In total the database contains 127 parameters, including individual and environmental - 17; physiological - 73; clinical - 37. The physiological block contains data on 4 specific arteries in the left and right arms and lower parts of the left and right shins. Particularly the provided data corresponds to brachial arteries in the arms and «generalized» vessels resembling three arteries in the shins [11].
The observations were recorded by the same observer at Scientific Research Institute of Physiology of SB RAMS (currently - Scientific Research Institute of Physiology and Basic Medicine) in the time period from 1993 to 2004. The majority of the examined patients were patients of Clinics of the Institute of Clinical & Experimental Medicine (Novosibirsk). All parameters from the database that were used in the current study are listed in Table 1.
PARAMETERS PERSONALIZATION
The first step in model personalization is creating correspondence between experimental data and model parameters. Particularly for each patient, the
personalized model should define properties of the arteries and boundary conditions.
Parameters of limb vessels
In this paragraph we will deal with one artery, so we will omit the artery index. In order to set the modeled vessel properties let's use the equation of state (2). For each model artery we need to define elastic parameter p and cross- sectional area while artery is relaxed A0.
Equations for elastic resistance Kin and characteristic impedance ZW are taken from [11]. Writing the equation of state separately for systole and diastole we obtain algebraic system with 4 equations and 3 variables (AS, A0, P)
p _ p VAa - VA0 p _ P \[Ad
/ c A , D c ' (4)
K _ c Ps - PD Z _ [jpL _L
^ m~lAs - AD ' W"V 2jAsAs '
where Vs - volume of the artery in systole (maximal), VD - volume of the artery in diastole (minimal), l - artery length, p - blood density, coefficient c ~ 133.32 serves to translate units from mm hg to dyn/cm2. From the experimental data we know values of Ad, Ps, PD, Kin, ZW. Variables to be found are As, A0, p System (4) is overdetermined. All its equations are only approximations and the known parameters also contain errors. Therefore magnitude of discrepancy for this system may be considered as measure for compatibility between the database and equation of state used in our model. Let's use the first three equations to calculate unknown variables
Table 1
Parameters from the database used in this study (in total - 31 parameters)
Notation Description Units
Individualization parameters
Age Age years
HR Heart rate beats/min
H Height cm
W Weight kg
Physiological parameters (for each of 4 limbs)
Pd Blood diastolic pressure mmHg
Ps Blood systolic pressure mmHg
Pp Pulse pressure mmHg
rD Arterial effective diastolic radius cm
Kin Arterial elastic resistance dyn/cm5
ZW Characteristic impedance (dyn x s)/cm5
Classification is given according to [12]
and then compare simulated characteristic impedance with given by the fourth equation. Explicit equations for A0, p can be derived from (4):
A =
P - P
1 S 1 D
,P = cA
p - p
1 S 1 D
4a -^jad
(5)
These values should be necessarily set to arteries in order to obtain real correspondence between pressures and elastic resistance Kin from the database.
Characteristic impedance can then be calculated from the last equation of (4) and compared to this from the database. It appeared that value Z,Ratio = = ZWim/ ZW divides all subjects from the database into two clusters which persist for all 4 vessels. These clusters are not correlated with physiological parameters but are defined by date when observations were taken (Fig. 1). The mean square error of impedance calculation is 0.098 before May 1999 and 0.02 - after. It is reasonable to expect that the overall accuracy of personalization will not exceed accuracy of system (4) solution. We have divided the database into two subgroups according to date when observation were taken. The first group (before 01.05.1999) will be used as a control set. The second group which demonstrates relatively low discrepancy for system (4) will be used as the main dataset for personalization and validation.
Tree structure qualification
Data provided by [10] includes the subclavian artery (a. subclavia), which branches into radial (a. radialis) and ulnar (a. ulnaris) arteries. This indicates that the brachial artery, which is missing, is actually included a. subclavia II which is one of the longest vessels in the model (42,2 cm) and its relaxed cross-sectional area is 0,51 mm2 which significantly exceeds typical value for a. brachialis in the database (average for left hand is 0.135 mm2). Probably this is due to a relatively complex structure of the arterial system in the arm, which includes two anastomoses that cannot be correctly modeled with a binary tree, so this structure is described as one vessel, wherein parameters were taken from the main branch and are constant through the vessel.
To make the model and the database compatible, the model was modified as follows. The last 15 centimeters of a. subclavia II were extracted to form a separate artery with an additional branch, which models a. profunda brachii and ensures the binary structure of the tree. The new arterial tree includes 59 arteries. Parameters for the new vessels for each individual from the database were calculated using formulas (5). The elasticity parameter of both added arteries was considered equal. The cross-sectional
area of a. profunda brachii was set to 30 % from this of a. brachialis (based on [2]).
Unlike in the upper limbs, in the lower limbs area of measurements covers three arteries with close diameter values (a. tibialis posterior, a. tibialis anterior, a. peronea) while the database records -due to features of measurement techniques - are given for the «generalized» vessel whose volume is equal to the sum of the three arteries. In the model we have only two of these arteries. A. peronea, which in reality is a branch of a. tibialis posterior, is not included. We will consider the model vessel a. tibialis posterior as a «generalized» vessel which resembles real arteries a. tibialis posterior and a. peronea. Elasticity parameters are considered to be equal for both vessels while cross-sectional areas are split in proportion 1 : 2, which agrees with our tabular data and data given in [2].
Reference patient
In this study we consider tabular data from the literature as representative of «average» parameters. Due to the tree extension we need to extend this table as well. We will do it by choosing one patient from the database and using his parameters to extend tabular data. The «reference» patient should satisfy the following criteria.
1. His parameters should be close to mean values of the population. It is checked using the next formula:
Dist = ^ jx, - x ^ / xt,
i
where xi - mean value of i-th parameter. The next parameters were used: age, height, weight, heart rate, diastolic and systolic pressures.
2. Pressures of the «reference» individual should be close to those produced by the model without changing vessel parameters personalization. We estimate this property by comparison of simulated and expected pressures in the left brachial artery
Error = PS - PS
/ Ps
p _ ps"
\TD rD
/ Pn
1086420
15.06.1994 01.05.1999 01.09.2002
Fig. 1. Squared error of calculated characteristic impedance
where Ps, Pd - real pressure values, PSSim, PDS™ - simulated with personalized values of HR, SV and TPR.
3. It should be symmetric in the sense that it minimizes the Asym value of
1 A0,lh + 1 Ая + 1 ß0,lh + 1 ß0,ll
A0,rh ß0, rh ß0,rl
where Ih, rh, ll, rl - indices of the left and right upper and lower arteries given in the database.
Parameters of the the «reference» patient selected according to these criteria are given in Table 2. We assume that vessel parameters of this patient agree well with the tabular data obtained from the literature.
Extrapolation to the tree
In order to personalize parameters of other arteries we will make the next assumptions. For two subjects, the ratio between cross-sectional areas AO remains close to constant for all the arteries. Particularly if a. brachialis sinistra of one subject is twice as thick as this artery of another subject then the same can be said about all other arteries of upper part of the body. Analogically, a. tibialis posterior sinistra serves as reference for the lower part (division into two parts was done at the branching point between IV and V segments of a. abdominalis, see the supplement). Thus if we know the parameters of the entire arterial tree of one person and parameters of certain arteries of another then we can restore parameters for all vessels of a second person.
We introduce parameters AF,up and AFdown into the model, which relate to the upper and lower parts of the body. Before simulation starts, all parameters of all vessels are multiplied by the corresponding factor. For the «reference» patient we set AFup = = AF,down = 1 (i.e. his parameters are default values presented in the supplement). For arbitrary patient
(6)
AF, up = Ao, h / Д), h, AF, down = Ao, l / Д), l
where * means parameters of «reference» patient, index h means index of a. brachialis sinistra and l -a. tibialis posterior sinistra.
For elastic parameter p analogical factors are introduced into the model. In the current work we will not distinguish between left and right parts of tree,
Table 2
Parameters of «referenced» subject
Value Min Max Median Mean
Dist 0.37 0.06 1.84 0.64 0.68
Asym 0.405 0.235 4.17 0,87 1.1
Error 0.124 0.048 0.486 0.159 0.183
i.e. parameters of right part of the tree will be set the same as for the left.
Parameters of boundary conditions
Natural boundary conditions are blood flow from the heart at the aorta entrance and filtration through porous medium (capillaries) at the terminal arteries ends.
For further calculations we will assume that the systole duration is proportional to the heart cycle duration. Function to model blood flow:
Qm (t) = SV (1 + sin( x(t))/ Ts,
— - 0.5, s(t) < a, a
s(t) - a
x(t) =
+ 0.5, s(t) < Ts, 1.5, otherwise,
ts - a
where SV - stroke volume, TS - systole duration, TC -heart cycle duration, TC* - «reference» heart cycle duration, TS* - «reference» systole duration, t - time, % - modulo division, s(t) maps intervals [NTC, (N + 1)Tc] to [0, T*], where N = 0, 1, ... Assuming that the systole duration is proportional to the heart cycle duration, the simulated blood volume for one heart beat will be equal to SV. The heart cycle duration is given by formula TC = 60/HR. Parameters a « 0.091 and TS « 0,331 are fitted to match the desired blood flow profile [3]. The stroke volume can be obtained from the cardiac output and heart rate
SV = CO/HR.
(7)
To calculate the cardiac output we use the al-lometric equations from [5]. One particular equation which produces the best results was used (see supplement materials for more details)
CO (ml/min) = 2499 H (m)
1,16
(8)
To define the output boundary conditions (3) venous pressure Pveins and filtration coefficient Kf should be specified. As simulation results show, the total resistance of arterial tree comprising 59 largest vessels is less than 1 % of the normal value of the total peripheral resistance (TPR) which is ~ 1.1 mmHg/(ml/s). Thus, to compensate for the lack of resistance we should set Kf = 1/TPR, which can be calculated from database values:
TPR = 60 (Ps/3+2Pd/3)/CO.
(8)
RESULTS
Numerical simulations were conducted using a specially written Java program using the BioUML
ID Common parameters R. upper limb L. upper limb R. lower limb L. lower limb
О
□
Branching point Observation point
artery | g | heart
model and database relation
Fig. 2. Ratio between simulated and expected systolic, diastolic and pulse pressures
API for model construction and numerical calculations (www.biouml.org).
Before simulation starts, case deletion was performed, reducing the number of patients from 1546 to 1207. To avoid outliers affecting the results, we censored patients by parameters A0 and P: bottom and upper 5 % were excluded from the groups. In total, there were 556 patients retained in the training group (recorded after 01.05.1999) and 420 in the control group (before 01.05.1999).
For each patient, personal parameters were set to the model and a simulation was performed for 12 seconds of the model time. The last 2 seconds were used to calculate the maximum (systolic) and minimum (diastolic) pressures. As experience has shown, 12 seconds is enough for the model to reach a quasi steady state (see supplement for details).
We have conducted several simulations using different combinations of parameters to personalize. If a parameter is not included to the selected combination, then its default value is used. The total list of personalized parameters is presented below.
Resistance to blood flow from the terminal vessels is set to TPR (8).
Heart rate HR is taken directly from the database. The default value is 72 beats/min.
Stroke volume SV is calculated according to (7). Default value is 75 ml.
Parameters A0 and P of a. brachialis sinistra, a. brachialis dextra and generalized arteries of lower limbs are calculated with formulas (6). There is no default values, these parameters are personalized in all simulations.
Parameters A0 and P of all other arteries are obtained multiplying default values by AFlh and pFlh for upper part of the tree and by AFU and pFll for lower part of the tree. Default values are given in the supplement.
Best simulation results are presented on Fig. 2. Scheme of relations between database and model are given on Fig. 3. Simulated systolic and diastolic pressures were compared to real values utilizing Pearson correlation coefficient and mean relative error. The results for different personalized parameters combinations for a. brachialis dextra are given in Table 3. The first column indicates which parameters were personalized. Additional results can be found in the supplement.
DISCUSSION
The mathematical model helps in revealing of artifacts in experimental data such as unexpected
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Simulation results for left upper limb
Table 3
Parameter Correlations Mean relative error
Ps Pd Pp Ps Pd Pp
TPR A0 ß SV HR 0.7841 0.8641 0.011 0.055 0.208 0.372
A0 ß SV HR 0.007 -0.1621 0.046 0.171 0.126 0.342
TPR SV HR 0.8971 0.9251 -0.070 0.041 0.164 0.283
TPR HR 0.4841 0.5561 -0.3182 0.169 0.391 0.214
TPR SV 0.4221 0.5301 -0.075 0.119 0.249 0.283
TPR A0 ß 0.6931 0.7811 0.006 0.158 0.472 0.328
TPR A0 0.5691 0.8381 -0.1252 0.209 0.385 0.285
TPR ß 0.8061 0.7591 0.0852 0.123 0.486 0.400
TPR 0.8101 0.8381 -0.4011 0.164 0.425 0.210
Index 1 denotes P-value< 0.001, index 2 denotes P-value< 0.05.
data clustering, which may be hard or nearly impossible to find by analysis of the database alone (Fig. 1).
The simulation results show that the single parameter - TPR is able to explain up to 70 % of systolic (r ~ 0,838) and diastolic (r ~ 0,81) pressures dispersion while other parameters cannot account for more than 10 % of dispersion. The best result is obtained when personalization of the heart rate and stroke volume is added to TPR r ~ 0,897 and r ~ ~ 0,925. In comparison to using TPR only it demonstrates a better accuracy with the mean relative error decreasing nearly fourfold. Thus, the structure of the arterial tree and parameters distribution are unlikely to play a significant role in prediction of systolic and diastolic pressures.
It should be kept in mind that the database contains records of people who (in the majority) had suffered from cardiovascular diseases and received medicament treatment for certain periods of time. Their artery properties and relationships between them could have changed as a result.
It was also shown that the model using only TPR can predict pressures for different age groups (these results can be found in the supplement).
Meanwhile none of the used personalization parameters could explain the variability of pulse pressure. In the simulated data the pulse pressure is negatively correlated (r = -0,41) with diastolic pressure while this is not the case in reality (r = 0,13). This can be a result of inaccurate equation of state (1) with constant parameters A0 and p. It is possible to improve the model by making the parameters dependent on the pressure.
The error rate in modeling is around 10 %, resulting from a combination of the following factors: observation record error (e.g. heart rate was measured for 10 seconds and multiplied by 6), connection between database and model - equations in system (5) are approximations, the «reference» patient is modeled with ~10% error itself (see Table 2.)
It should be noted that model parameters are personalized in the natural way without a fitting procedure and despite the difference between training and control groups (see Fig. 1.), the model pre-
dicts pressures for control group only slightly worse (Table 4).
CONCLUSIONS
We have performed personalization of 1d arterial tree model parameters and consequent validation of this model on a group of 1207 persons (after case deletion from the original 1546). For each subject his own identified parameters were calculated and set to the model and a simulation was performed. The results show good correlations and mean error in the predictions of systolic and dia-stolic pressures and inability of the model (with used personalized parameters).
Although vessels elasticity and relaxed cross-sectional area do not play a significant role in the predictions, they are expected to be important for the pulse wave profile and velocity prediction. They also can be used for modeling of special states, such as vasoconstriction in certain arteries, stress. Along with model modifications/improvements, this is planned for future work.
The model described in this paper is available as a part of the free open-source platform BioUML at www.biouml.org. One can access the model either through the standalone version of BioUML or using its web-interface. The supplementary materials are available at http://wiki.biouml.org/index.php/Patient-specific cardiovascular model.
ACKNOWLEDGEMENTS
The reported study was supported by RFBR, research project № 16-01-00779 A.
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Table 4
Simulation results for main and control groups (depending on observations date)
Date n Correlations Mean Error
Ps Pd Pp Ps Pd Pp
Control 420 0.7331 0.8011 0.1112 0.073 0.199 0.335
Main 556 0.7841 0.8641 0.011 0.055 0.208 0.372
Any 976 0.7541 0.8361 0.098 0.062 0.205 0.356
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ПЕРСОНАЛИЗИРОВАННАЯ ОДНОМЕРНАЯ МОДЕЛЬ СЕРДЕЧНО-СОСУДИСТОЙ СИСТЕМЫ ЧЕЛОВЕКА
Илья Николаевич КИСЕЛЕВ1'2, Федор Анатольевич КОЛПАКОВ1'2, Элина Арнольдовна БИБЕРДОРФ3,4, Виктор Ильич БАРАНОВ5, Тамара Григорьевна КОМЛЯГИНА5, Ирина Юрьевна СУВОРОВА5, Владимир Николаевич МЕЛЬНИКОВ5, Сергей Георгиевич КРИВОЩЕКОВ5
1 ООО «Институт системной биологии» 630112, г. Новосибирск, ул. Красина, д. 54
2 Конструкторско-технологический институт вычислительной техники СО РАН 630090, г. Новосибирск, ул. Академика Ржанова, 6
3 Институт математики им. им. С.Л. Соболева СО РАМН 630090, г. Новосибирск, просп. Академика Коптюга, 4
4 Новосибирский государственный университет 630090, г. Новосибирск, ул. Пирогова, 2
5 НИИ физиологии и фундаментальной медицины 630117, г. Новосибирск, ул. Тимакова, 4
В работе обсуждается методика персонализации параметров одномерной модели гемодинамики и ее валида-ция на основе физиологических данных 1546 пациентов. Использованы различные комбинации параметров, в качестве главного критерия валидации выступало качество прогнозирования систолического и диастолическо-го давления. Показано, что при точной персонализации модель может обеспечить адекватное предсказание давления (коэффициенты корреляции около 0,9), при этом решающую роль играет общее периферического сопротивление, а параметры крупных артерий не играют значительную роль в прогнозировании.
Ключевые слова: сердечно-сосудистая система, одномерная модель артериального дерева, математическое моделирование, персонализация параметров, валидация, экспериментальные данные.
Киселев И.Н. - младший научный сотрудник, e-mail: axec@developmentontheedge.com Колпаков Ф.А. - к.б.н., директор ООО; зав. лабораторией биоинформатики.
Бибердорф Э.А. - к.ф.-м.н., доцент кафедры, старший научный сотрудник, e-mail: biberdorf@ngs.ru Баранов В.И. - к.б.н., старший научный сотрудник лаборатории функциональных резервов организма, e-mail: v.i.baranov@physiol.ru
Комлягина Т.Г. - научный сотрудник лаборатории функциональных резервов организма, e-mail: KomlyaginaTG@physiol.ru
Суворова И.Ю. - младший научный сотрудник лаборатории функциональных резервов организма, e-mail: suvorovaiu@physiol.ru
Мельников В.Н. - д.б.н., ведущий научный сотрудник лаборатории функциональных резервов организма, e-mail: mvn@physiol.ru
Кривощеков С.Г. - д.м.н., проф., зав. лабораторией функциональных резервов организма, e-mail: krivosch@physiol.ru
