sibmedСибирский научный медицинский журналСибирский научный медицинский журнал2410-25122410-2520ИЦиГ СО РАН10.18699/SSMJ20240112sibmed-1395Research ArticleСтатьиМатематическая модель зависимости между размерами желудочков головного мозга и капиллярным давлением у лабораторных животныхMathematical model of the dependence between cerebral ventricular size and capillary pressure in laboratory animalshttps://orcid.org/0000-0003-2014-854XЧеревкоА. А.CherevkoA. A.

Черевко Александр Александрович, к.ф.-м.н. 

630090, г. Новосибирск, пр. Академика Лаврентьева, 15

Alexander A. Cherevko, candidate of physical and mathematical sciences 

630090, Novosibirsk, Academika Lavrentieva ave., 15

cherevko@mail.ruhttps://orcid.org/0000-0002-0782-1819ВаловаГ. С.ValovaG. S.

Валова Галина Сергеевна, к.ф.-м.н.  

630090, г. Новосибирск, пр. Академика Лаврентьева, 15

Galina S. Valova, candidate of physical and mathematical sciences 

630090, Novosibirsk, Academika Lavrentieva ave., 15

galinayankova2703@gmail.comhttps://orcid.org/0000-0002-0623-0363ПетровскийД. В.PetrovskyD. V.

Петровский Дмитрий Валерианович, к.б.н.  

630090, г. Новосибирск, пр. Академика Лаврентьева, 15;630090, г. Новосибирск, пр. Академика Лаврентьева, 10

Dmitry V. Petrovsky, candidate of biological sciences 

630090, Novosibirsk, Academika Lavrentieva ave., 15;630090, Novosibirsk, Aсademika Lavrentieva ave., 10

dm_petr@bionet.nsc.ruhttps://orcid.org/0000-0001-8499-5313АкуловА. Е.AkulovA. E.

Акулов Андрей Евгеньевич, к.б.н. 

630090, г. Новосибирск, пр. Академика Лаврентьева, 15;630090, г. Новосибирск, пр. Академика Лаврентьева, 10;630090, г. Новосибирск, ул. Институтская, 3а

Andrey E. Akulov, candidate of biological sciences 

630090, Novosibirsk, Academika Lavrentieva ave., 15;630090, Novosibirsk, Aсademika Lavrentieva ave., 10;630090, Novosibirsk, Institutskаya st., 3a

akulov_ae@ngs.ruИнститут гидродинамики им. М.А. Лаврентьева СО РАНРоссияLavrentyev Institute of Hydrodynamics of SB RASRussian FederationИнститут гидродинамики им. М.А. Лаврентьева СО РАН; ФИЦ Институт цитологии и генетики СО РАНРоссияLavrentyev Institute of Hydrodynamics of SB RAS; Federal Research Center Institute of Cytology and Genetics of SB RASRussian FederationИнститут гидродинамики им. М.А. Лаврентьева СО РАН; ФИЦ Институт цитологии и генетики СО РАН; Институт «Международный томографический центр» СО РАНРоссияLavrentyev Institute of Hydrodynamics of SB RAS; Federal Research Center Institute of Cytology and Genetics of SB RAS; International Tomography Center of SB RASRussian Federation202407032024441116123Copyright © Черевко А.А., Валова Г.С., Петровский Д.В., Акулов А.Е., 20242024Черевко А.А., Валова Г.С., Петровский Д.В., Акулов А.Е.Cherevko A.A., Valova G.S., Petrovsky D.V., Akulov A.E.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://sibmed.elpub.ru/jour/article/view/1395

Цель исследования – выполнить адаптацию математической модели, описывающей взаимодействие жидких сред и вещества головного мозга, с целью описания зависимости между размерами желудочков головного мозга и капиллярным давлением у лабораторных животных двух генотипов: BALB/c и C57BL/6. Материал и методы. В качестве объекта исследования использовано по четыре самца мышей инбредных линий C57BL/6 и BALB/c в возрасте 12 недель. Изображения головного мозга и ликворной системы получены с помощью горизонтального МР-томографа 11,7 Тл. В качестве геометрии для математического моделирования выбран аксиальный срез на уровне –0,5 мм от брегмы. Для описания полученных данных проведена адаптация математической модели путем выбора масштабного коэффициента на основе известных значений скорости образования цереброспинальной жидкости для человека и мышей. Результаты и их обсуждение. Для всех рассмотренных животных наблюдается одинаковая качественная картина взаимосвязи капиллярного давления и среднего смещения стенок желудочков. Несмотря на то что выбранные генетические линии мышей BALB/c и C57Bl существенно отличаются с точки зрения размеров мозговых желудочков, данные различия в генотипе животных не сказались на характере этой взаимосвязи. Изменение параметров взаимодействия жидких сред в области сжатия или умеренного расширения желудочков почти не приводит к выходу из области физиологически допустимого значения капиллярного давления. При этом размер желудочков существенно меняется. В области большого расширения желудочков, напротив, размер желудочков меняется незначительно, что сопровождается резким повышением капиллярного давления, выходящего далеко за физиологические пределы. Таким образом, изменение размеров желудочков – адаптивный процесс, связанный с колебаниями давления, вызванного изменением перетоков внутричерепной жидкости. Сам факт выхода части значений в зону физиологически недопустимых, сопряженных, по сути, со смертью, при условии практического отсутствия изменения размеров желудочков указывает на то, что данная ситуация редко реализуется и возможна при нарушении внутричерепных перетоков жидких сред, связанных с тем, что увеличение размеров желудочков лимитирует адаптивные возможности. Заключение. Представленная модель позволит улучшить понимание установленной нами закономерности и перейти к попыткам прогнозирования. 

Aim: To adapt a mathematical model describing the interaction between fluid media and brain matter for the purpose of definition of the dependence between brain ventricle size and capillary pressure in laboratory animals of two genotypes, BALB/c and C57BL/6. Material and methods. The study included 4 male mice of each inbred strain C57BL/6 and BALB/c at the age of 12 weeks. The brain and cerebrospinal fluid system images were obtained using an 11.7 T horizontal MR scanner (Bruker, BioSpec 117/16 USR, Germany). An axial section at the level of -0.5 mm from bregma was chosen as the geometry for mathematical modelling. To describe the data obtained, the mathematical model was adapted by selecting a scale factor based on the known values of the cerebrospinal fluid formation rate for humans and mice. Results and discussion. The same qualitative pattern of relationship between capillary pressure and mean ventricular wall displacement was observed for all animals considered. Although the selected genetic strains of BALB/c and C57Bl mice differ significantly in terms of cerebral ventricle size, these differences in animal genotype did not affect the nature of this relationship. Changing the parameters of the fluid media interaction in the area of compression or moderate ventricular dilation almost does not lead to an exit from the physiologically acceptable capillary pressure value. In this case, the size of the ventricles changes significantly. In the area of large ventricular dilation, in contrast, there is little change in ventricular size, and this is accompanied by a dramatic increase in capillary pressure far beyond physiologic limits. Thus, the change in ventricular size is an adaptive process associated with pressure fluctuations caused by changes in intracranial fluid flow. The mere fact that some of the values reach the zone of physiologically unacceptable pressures associated, in fact, with death, provided that there is practically no change in ventricular size indicates that such a situation is rarely realized and is possible in case of violation of intracranial fluid media flows associated with the fact that the increase in ventricular size limits adaptive capabilities. Conclusions. The presented animal model will further increase the understanding of the pattern we have established and allow us to move on to attempts at prediction.

многофазная пороупругостьМРТлабораторные животныепатологии церебральной циркуляциичисленное моделированиеmulticomponent poroelasticityMRIlaboratory animalspathologies of cerebral circulationnumerical modelingИсследование выполнено за счет гранта Российского научного фонда №22-11-00264.The research was supported by the grants the Russian Science Foundation, 22-11-00264, https://rscf.ru/ project/22-11-00264/.ReferencesWorld Health Organization. World health statistics 2022: monitoring health for the SDGs, sustainable development goals. Available at: https://www.who.int/publications/i/item/9789240051157World Health Organization. World health statistics 2022: monitoring health for the SDGs, sustainable development goals. Available at: https://www.who.int/publications/i/item/9789240051157Zhou X., Xia J. Application of Evans index in normal pressure hydrocephalus patients: a mini review. Front. Aging. Neurosci. 2022;13:783092. doi: 10.3389/fnagi.2021.783092Zhou X., Xia J. Application of Evans index in normal pressure hydrocephalus patients: a mini review. Front. Aging. Neurosci. 2022;13:783092. doi: 10.3389/fnagi.2021.783092He W., Fang X., Wang X., Gao P., Gao X., Zhou X., Mao R., Hu J., Hua Y., Xia J. A new index for assessing cerebral ventricular volume in idiopathic normal-pressure hydrocephalus: a comparison with Evans’ index. Neuroradiology. 2020;62(6):661–667. doi: 10.1007/s00234-020-02361-8He W., Fang X., Wang X., Gao P., Gao X., Zhou X., Mao R., Hu J., Hua Y., Xia J. A new index for assessing cerebral ventricular volume in idiopathic normal-pressure hydrocephalus: a comparison with Evans’ index. Neuroradiology. 2020;62(6):661–667. doi: 10.1007/s00234-020-02361-8Linninger A.A., Xenos M., Sweetman B., Ponkshe S., Guo X., Penn R. A mathematical model of blood, cerebrospinal fluid and brain dynamics. J. Math. Biol. 2009;59(6):729–759. doi: 10.1007/s00285-009-0250-2Linninger A.A., Xenos M., Sweetman B., Ponkshe S., Guo X., Penn R. A mathematical model of blood, cerebrospinal fluid and brain dynamics. J. Math. Biol. 2009;59(6):729–759. doi: 10.1007/s00285-009-0250-2Ficola A., Fravolini M.L., Anile C. A physical model of the intracranial system for the study of the mechanisms of the cerebral blood flow autoregulation. IEEE Access. 2018;6:67166–67175.Ficola A., Fravolini M.L., Anile C. A physical model of the intracranial system for the study of the mechanisms of the cerebral blood flow autoregulation. IEEE Access. 2018;6:67166–67175.Doron O., Zadka Y., Barnea O., Rosenthal G. Interactions of brain, blood, and CSF: a novel mathematical model of cerebral edema. Fluids Barriers CNS. 2021;18(1):1–14. doi: 10.1186/s12987-021-00274-zDoron O., Zadka Y., Barnea O., Rosenthal G. Interactions of brain, blood, and CSF: a novel mathematical model of cerebral edema. Fluids Barriers CNS. 2021;18(1):1–14. doi: 10.1186/s12987-021-00274-zBaghbani R. An electrical model of hydrocephalus shunt incorporating the CSF dynamics. Scientific Reports. 2019;9(1):1–12. doi: 10.1038/s41598-019-46328-zBaghbani R. An electrical model of hydrocephalus shunt incorporating the CSF dynamics. Scientific Reports. 2019;9(1):1–12. doi: 10.1038/s41598-019-46328-zRyu J., Hu X., Shadden S.C. A coupled lumpedparameter and distributed network model for cerebral pulse-wave hemodynamics. J. Biomech. Eng. 2015;137(10):1–13. doi: 10.1115/1.4031331Ryu J., Hu X., Shadden S.C. A coupled lumpedparameter and distributed network model for cerebral pulse-wave hemodynamics. J. Biomech. Eng. 2015;137(10):1–13. doi: 10.1115/1.4031331Clarke E.C., Fletcher D.F., Stoodley M.A., Bilston L.E. Computational fluid dynamics modelling of cerebrospinal fluid pressure in Chiari malformation and syringomyelia. J. Biomech. 2013; 46(11):1801–1809. doi: 10.1016/j.jbiomech.2013.05.013Clarke E.C., Fletcher D.F., Stoodley M.A., Bilston L.E. Computational fluid dynamics modelling of cerebrospinal fluid pressure in Chiari malformation and syringomyelia. J. Biomech. 2013; 46(11):1801–1809. doi: 10.1016/j.jbiomech.2013.05.013Gholampour S., Fatouraee N., Seddighi A.S., Seddighi A. Numerical simulation of cerebrospinal fluid hydrodynamics in the healing process of hydrocephalus patients. Journal of Applied Mechanics and Technical Physics. 2017;58(3):386–391. doi: 10.1134/S0021894417030026Gholampour S., Fatouraee N., Seddighi A.S., Seddighi A. Numerical simulation of cerebrospinal fluid hydrodynamics in the healing process of hydrocephalus patients. Journal of Applied Mechanics and Technical Physics. 2017;58(3):386–391. doi: 10.1134/S0021894417030026Khani M., Sass L.R., Xing T., Sharp M.K., Balédent O., Bryn A.M. Anthropomorphic model of intrathecal cerebrospinal fluid dynamics within the spinal subarachnoid space: spinal cord nerve roots increase steady-streaming. J. Biomech. Eng. 2018;140(8):081012. doi: 10.1115/1.4040401Khani M., Sass L.R., Xing T., Sharp M.K., Balédent O., Bryn A.M. Anthropomorphic model of intrathecal cerebrospinal fluid dynamics within the spinal subarachnoid space: spinal cord nerve roots increase steady-streaming. J. Biomech. Eng. 2018;140(8):081012. doi: 10.1115/1.4040401Gholampour S. FSI simulation of CSF hydrodynamic changes in a large population of non-communicating hydrocephalus patients during treatment process with regard to their clinical symptoms. PLoS One. 2018;13(4):e0196216. doi: 10.1371/journal.pone.0196216Gholampour S. FSI simulation of CSF hydrodynamic changes in a large population of non-communicating hydrocephalus patients during treatment process with regard to their clinical symptoms. PLoS One. 2018;13(4):e0196216. doi: 10.1371/journal.pone.0196216Penn R.D., Basati S., Sweetman B., Guo X., Linninger A. Ventricle wall movements and cerebrospinal fluid flow in hydrocephalus. J. Neurosurg. 2011;115(1):159–164. doi: 10.3171/2010.12.JNS10926Penn R.D., Basati S., Sweetman B., Guo X., Linninger A. Ventricle wall movements and cerebrospinal fluid flow in hydrocephalus. J. Neurosurg. 2011;115(1):159–164. doi: 10.3171/2010.12.JNS10926Nagashima T., Tamaki N., Matsumoto S., Horwitz B., Seguchi Y. Biomechanics of hydrocephalus: a new theoretical model. Neurosurgery. 1987;21(6):898–904. doi: 10.1227/00006123-198712000-00019Nagashima T., Tamaki N., Matsumoto S., Horwitz B., Seguchi Y. Biomechanics of hydrocephalus: a new theoretical model. Neurosurgery. 1987;21(6):898–904. doi: 10.1227/00006123-198712000-00019Tully B., Ventikos Y. Cerebral water transport using multiple-network poroelastic theory: application to normal pressure hydrocephalus. J. Fluid Mech. 2011;667:188–215. doi: 10.1017/S0022112010004428Tully B., Ventikos Y. Cerebral water transport using multiple-network poroelastic theory: application to normal pressure hydrocephalus. J. Fluid Mech. 2011;667:188–215. doi: 10.1017/S0022112010004428Vardakis J.C., Tully B.J., Ventikos Y. Multicompartmental poroelasticity as a platform for the integrative modeling of water transport in the brain. In: Computer models in biomechanics. Springer, 2013; 305–316.Vardakis J.C., Tully B.J., Ventikos Y. Multicompartmental poroelasticity as a platform for the integrative modeling of water transport in the brain. In: Computer models in biomechanics. Springer, 2013; 305–316.Chou D., Vardakis J.C., Guo L., Tully B., Ventikos Y. A fully dynamic multi-compartmental poroelastic system: Application to aqueductal stenosis. J. Biomech. 2016;49(11):2306–2312. doi: 10.1016/j.jbiomech.2015.11.025Chou D., Vardakis J.C., Guo L., Tully B., Ventikos Y. A fully dynamic multi-compartmental poroelastic system: Application to aqueductal stenosis. J. Biomech. 2016;49(11):2306–2312. doi: 10.1016/j.jbiomech.2015.11.025Guo L., Vardakis J.C., Lassila T., Mitolo M., Ravikumar N., Chou D., Lange M., Sarrami-Foroushani A., Tully B.J., Taylor Z.A., … Ventikos Y. Subjectspecific multi-poroelastic model for exploring the risk factors associated with the early stages of Alzheimer’s disease. Interface focus. 2018;8(1):20170019. doi: 10.1098/rsfs.2017.0019Guo L., Vardakis J.C., Lassila T., Mitolo M., Ravikumar N., Chou D., Lange M., Sarrami-Foroushani A., Tully B.J., Taylor Z.A., … Ventikos Y. Subjectspecific multi-poroelastic model for exploring the risk factors associated with the early stages of Alzheimer’s disease. Interface focus. 2018;8(1):20170019. doi: 10.1098/rsfs.2017.0019Vardakis J.C., Guo L., Peach T.W., Lassila T., Mitolo M., Chou D., Taylor Z.A., Varma S., Venneri A., Frangi A.F., Ventikos Y. Fluid–structure interaction for highly complex, statistically defined, biological media: Homogenisation and a 3D multicompartmental poroelastic model for brain biomechanics. J. Fluids Struct. 2019;91:102641. doi: 10.1016/j.jfluidstructs.2019.04.008Vardakis J.C., Guo L., Peach T.W., Lassila T., Mitolo M., Chou D., Taylor Z.A., Varma S., Venneri A., Frangi A.F., Ventikos Y. Fluid–structure interaction for highly complex, statistically defined, biological media: Homogenisation and a 3D multicompartmental poroelastic model for brain biomechanics. J. Fluids Struct. 2019;91:102641. doi: 10.1016/j.jfluidstructs.2019.04.008Ye W., Chen Q. Potential applications and perspectives of humanized mouse models. Annu. Rev. Anim. Biosci. 2022;10:395–417. doi: 10.1146/annurevanimal-020420-033029Ye W., Chen Q. Potential applications and perspectives of humanized mouse models. Annu. Rev. Anim. Biosci. 2022;10:395–417. doi: 10.1146/annurevanimal-020420-033029Ma Y., Hof P.R., Grant S.C., Blackband S.J., Bennett R., Slatest L., McGuigan M.D., Benveniste H. A three-dimensional digital atlas database of the adult C57BL/6J mouse brain by magnetic resonance microscopy. Neuroscience. 2005;135(4):1203–1215. doi: 10.1016/j.neuroscience.2005.07.014Ma Y., Hof P.R., Grant S.C., Blackband S.J., Bennett R., Slatest L., McGuigan M.D., Benveniste H. A three-dimensional digital atlas database of the adult C57BL/6J mouse brain by magnetic resonance microscopy. Neuroscience. 2005;135(4):1203–1215. doi: 10.1016/j.neuroscience.2005.07.014Masoumi N., Framanzad F., Zamanian B., Seddighi A.S., Moosavi M.H., Najarian S., Bastani D. 2D computational fluid dynamic modeling of human ventricle system based on fluid-solid interaction and pulsatile flow. Basic and Clinical Neuroscience. 2013;4(1):64–75.Masoumi N., Framanzad F., Zamanian B., Seddighi A.S., Moosavi M.H., Najarian S., Bastani D. 2D computational fluid dynamic modeling of human ventricle system based on fluid-solid interaction and pulsatile flow. Basic and Clinical Neuroscience. 2013;4(1):64–75.Petrov I.B. Solution of deformable solid mechanics dynamical problems with use of mathematical modeling by grid-characteristic method. In: Continuum mechanics, applied mathematics and scientific computing: Godunov’s legacy. Springer, 2020; 299–305.Petrov I.B. Solution of deformable solid mechanics dynamical problems with use of mathematical modeling by grid-characteristic method. In: Continuum mechanics, applied mathematics and scientific computing: Godunov’s legacy. Springer, 2020; 299–305.Cheng S., Jacobson E., Bilston L.E. Models of the pulsatile hydrodynamics of cerebrospinal fluid flow in the normal and abnormal intracranial system. Comput. Methods. Biomech. Biomed. Eng. Imaging. Vis. 2007;10(2):151–157. doi: 10.1080/10255840601124753Cheng S., Jacobson E., Bilston L.E. Models of the pulsatile hydrodynamics of cerebrospinal fluid flow in the normal and abnormal intracranial system. Comput. Methods. Biomech. Biomed. Eng. Imaging. Vis. 2007;10(2):151–157. doi: 10.1080/10255840601124753Valova G., Bogomyakova O., Tulupov A., Cherevko A. Influence of interaction of cerebral fluids on ventricular deformation: A mathematical approach. PLoS One. 2020;17(2):e0264395. doi: 10.1371/journal.pone.0264395Valova G., Bogomyakova O., Tulupov A., Cherevko A. Influence of interaction of cerebral fluids on ventricular deformation: A mathematical approach. PLoS One. 2020;17(2):e0264395. doi: 10.1371/journal.pone.0264395Vinje V., Eklund A., Mardal K.A., Rognes M.E., Støverud K.H. Intracranial pressure elevation alters CSF clearance pathways. Fluids and Barriers of the CNS. 2020;17(1):29. doi: 10.1186/s12987-020-00189-1Vinje V., Eklund A., Mardal K.A., Rognes M.E., Støverud K.H. Intracranial pressure elevation alters CSF clearance pathways. Fluids and Barriers of the CNS. 2020;17(1):29. doi: 10.1186/s12987-020-00189-1

The authors declare that there are no conflicts of interest present.