Current Directions in Biomedical Engineering 2019;5(1):533-536 Lorena Krames*, Rosa Daschner, Yannick Lutz, Axel Loewe, Giorgio Cattaneo, and Olaf Dössel Modeling of the Human Cerebral Collateral Circulation: Evaluation of the Impact on the Cerebral Perfusion in Case of Ischemic Stroke https://doi.org/10.1515/cdbme-2019-0134 1 Introduction Abstract: Stroke is the third-most cause of death in devel- Therapeutic hypothermia (TH) can possibly lead to an im- oped countries. A new promising treatment method in case proved outcome for ischemic stroke patients [1]. Currently, of an ischemic stroke is selective intracarotid blood cool- several techniques which induce TH of the whole body are ing combined with mechanical artery recanalization. However, used in case of ischemia. However, these can cause several side the control of the treatment requires invasive or MRI-assisted effects like pneumonia. In order to avoid these disadvantages, measurement of cerebral temperature. An auspicious alterna- targeted temperature management (selective TH) can be used. tive is the use of computational modeling. In this work, we This approach requires a detailed information of the cerebral extended an existing 1D hemodynamics model including the circulation and a precise resolution of the brain temperature. In characteristics of the anterior, middle and posterior cerebral this work, we used a computational 1D hemodynamics model, artery. Furthermore, seven ipsilateral anastomoses were addi- which includes a detailed cerebral arterial and collateral struc- tionally integrated for each hemisphere. A potential stenosis ture to predict the cerebral blood flow distribution. was placed into the M1 segment of the middle cerebral artery, due to the highest risk of occlusion there. The extended model 2 Methods was evaluated for various degrees of collateralization (“poor”, 2.1 Transmission Line Approach “partial” and “good”) and degrees of stenosis (0%, 50%, 75% One main advantage of a 1D transmission-line approach is the and 99.9%). Moreover, cerebral autoregulation was considered real time capability for extensive fluid systems. For use in our in the model. The higher the degree of collateralization and the case, the blood is regarded as an incompressible Newtonian degree of stenosis, the higher was the blood flow through the fluid. Making use of a simplified form of the Navier-Stokes collaterals. Hence, a patient with a good collateralization could equation, two relations between pressure p and flow q can be compensate a higher degree of occlusion and potentially has a established: better outcome after an ischemic stroke. For a 99.9% stenosis, @p 휌 @q 8휂 an increased summed mean blood flow through the collater- − = 2 · + 4 · q; (1) @z 휋r0 @t 휋r0 als of +97.7% was predicted in case of good collateralization. @q 3휋r3 @p Consequently, the blood supply via the terminal branches of − = 0 · : (2) the middle cerebral artery could be compensated up to 44.4% @z 2Ed @t to the physiological blood flow. In combination with a temper- Due to analogies between the hydrodynamical and electrical ature model, our model of the cerebral collateral circulation systems, the coefficients of the differential equations per unit can be used for tailored temperature prediction for patients to length Δz can be defined as follows: be treated with selective therapeutic hypothermia. 8휂Δz 휌Δz 3휋r3Δz R = ;L = ;C = 0 : (3) 4 2 2Ed Keywords: ischemic stroke, cerebral circulation, ipsilateral 휋r0 휋r0 collaterals, M1 stenosis, hemodynamics model While the electric current corresponds to the flow, voltage rep- resents the analog of the pressure. The coefficients can be de- rived from characteristics of the vessel (radius r0, Young’s modulus E and wall thickness d) and from parameters of the *Corresponding author: Lorena Krames, Institute of Biomedical −3 kg fluid (dynamic blood viscosity 휂 = 2:3 · 10 ms and density Engineering, Karlsruhe Institute of Technology (KIT), Karlsruhe, kg 휌 = 1020 m3 ). Moreover, the electrical resistance corresponds Germany, [email protected] to the flow resistance, the inductance represents the inertia of Rosa Daschner, Yannick Lutz, Axel Loewe, Olaf Dössel, Institute of Biomedical Engineering, Karlsruhe Institute of the fluid and the capacity mirrors the compliance, a quantity Technology (KIT), Karlsruhe, Germany for the elasticity of the vessel. For a discretization, the follow- Giorgio Cattaneo, Adceris GmbH & Co KG, Pforzheim, Germany ing differential relations for one short arterial segment can be Open Access. © 2019 Lorena K rames et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. 534 Krames et al., Modeling of the Human Cerebral Collateral Circulation ACA MCA Anterior communicating a. communicatingPosterior a. Ophthalmic Circle of artery Willis Internal carotid a. Basilar a. PCA Vertebral a. Arch of Ascending aorta aorta Fig. 2: Extended model of the cerebral circulation with ipsilateral collaterals, bifurcating MCA, and typical callosomarginal artery. 2.3 Terminal Resistances The total periphery resistance is mainly governed by the re- sistance of the capillary network. The resistance of this fine branched network is defined by so-called terminal resistances. The values were calculated depending on the tissue volume and type of the supplied cerebral area of the respective artery Fig. 1: Avolio’s model with 128 segments [3]. Every segment is using: marked with a number. The highlighted segments are the part shown in Fig. 2. qterm = vGM · Vterm;GM + vWM · Vterm;W M (6) used: with vGM and vWM denoting the perfusion rates of grey (GM) dpout 1 = · (qin − qout); (4) and white matter (WM). Vterm = Vterm;GM + Vterm;W M dt C represents the volume of the supply area of each terminal seg- dqin R 1 = − + · (pin − pout): (5) ment, which consists of an individual composition of WM and dt L L GM. With series and parallel connections, in accordance to Kirch- 2.4 Autoregulation hoff’s first law, it is possible to build a complete arterial net- Cerebral blood flow is regulated by the active change ofthe work. vessel lumen, caused by dilatation and constriction. Thus, a 2.2 Model of the Cerebral Arteries constant supply of the organs is ensured for changing blood As a basic model, a multi-branched model by Avolio (see Fig. flow conditions. In order to integrate this autoregulation, ev- 1) was used [2]. In this model, the vessels of the human body ery terminal resistance has its own controller. The controller were implemented in 128 segments. Later, Schwarz extended modifies the value of the terminal resistance by controlling the the model by taking an arterial ring structure into account, actual flow. In this process the flow averaged over time iscom- the Circle of Willis [3]. However, a more detailed model is pared to a reference flow qterm (eq. 6). The mean value of the required to investigate the impact of ipsilateral collaterals on temporal flow is calculated by a third-order low-pass Butter- the cerebral circulation. Therefore, we refined the strucutre of worth filter (fc = 0:1 Hz) and afterwards compared to a refer- the three major cerebral arteries (anterior, middle and poste- ence value. The block diagram of the autoregulation is shown rior cerebral artery (ACA, MCA, PCA)). Characteristics of in Fig. 3. the vessels were determined from literature. Since, the cere- 2.5 Stenosis Model bral anatomy varies widely between individuals, we focused Considering the impact of ischemic stroke on the cerebral on the development of four standardized models depending blood flow, we included an M1-stenosis. The stenosed artery on characteristics of the ACA and MCA. The most common is subdivided into three parts: the proximal part (in front of the structure for humans is a bifurcation of the MCA and an ACA stenosis), the stenosis part itself, and the distal part (behind with a typical callosomarginal artery [4]. The complete model the stenosis). The stenosis part is implemented as a single re- is shown in Fig. 2. All simulations were performed in MAT- sistance. Moreover, the degree S is variable and defined by LAB/SIMULINK (2019a, The MathWorks, Natick, MA, USA). S the stenosis radius rsten = r0(1 − 100 ), with r0 equal to the initial M1-artery radius. In this work, four different degrees of Krames et al., Modeling of the Human Cerebral Collateral Circulation 535 Collaterals 2 f=1 f=1.5 f=2 1.5 1 flow in ml/s 0.5 0 Fig. 3: Autoregulation of the terminal segments. The block on 58 58.5 59 59.5 60 ml the left symbolizes the heart with a cardiac output of 85 s . The time in s blood flow splits into lower flow rates depending on the arterial tree structure. Every terminal segment is concluded by a terminal Fig. 4: Summed flow through the 3 ACA-MCA and 3 MCA-PCA resistance Rterm with a flow qterm. The autoregulation itself con- collaterals in case of a 99.9% stenosis. sists of a low-pass filter and a controller, which needs a reference Collaterals S=0% value of the flow, and modifies the terminal resistance. S=50% 0.8 S=75% stenosis were considered (S=0% (physiological case), S=50%, S=99.9% 0.6 S=75% and S=99.9%) 2.6 Collateralization 0.4 flow in ml/s In case of an M1-stenosis, the blood supply through ipsilat- 0.2 eral collaterals is particularly important. In a study by Van- 0 der Eecken and Adams, information about the occurrence and 58 58.5 59 59.5 60 the size of the collaterals is given [5]. We included seven sec- time in s ondary collaterals between the main cerebral arteries in both Fig. 5: Summed flow through the 3 ACA-MCA and 3 MCA-PCA hemispheres (3 ACA-MCA, 3 MCA-PCA, 1 ACA-PCA). To collaterals with r = 1:5 · rinitial.