Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3787_Библиотеки_им_академика_М_И_Перельмана
.pdf
Appendix 3: Myocardial–Vessel Interaction (Algranati et al., 2010) 415
https://t.me/med1917
The nonlinearity of ℜ(t) stems from the vessel elasticity, expressed by the dependence of the diameter D and length L on the MVI-dependent extravascular loading.
Since each bifurcation pressure P
equals either Pinor P
bif
of the vessels forming
out
that bifurcation, Eq. (6.46) can be combined with Eq. (6.44), resulting in a system of
N nonlinear ordinary differential equations (N denotes the number of network
vessels), which is iteratively solved using the MATLAB
®
ode15s solver until
satisfying periodicity condition. The solution process requires the boundary conditions of the network inlet and outlet pressure, as well as MVI-dependent extravascular loading based on the various conditions defined below.
Varying Elasticity (VE, Fig. 6.17a) Contractility is assum ed to affect coronary
flow through activation-dependent changes of myocardial stiffness (Vis et al., 1995).
To analyze this effect alone, the myocardium is modeled as a hyperelastic solid, and
the effects of LV cavity pressure and associated extracellular (interstitial) pressure
are ignored. Hence, the boundary conditions for this mechanism are: (1) vanishing
extravascular pressure (P
¼0), and (2) linear dependence of vessels dynamic
EV
diameter on activation, namely:
P
¼ 0; D ¼ 1 activation½Dpþ activation D
EV
a
ð6:48Þ
where D
and Daare the vessel diameters under passive and fully active myocar-
p
dium, respectively (Appendix “Mechanics of Vessel-in-Myocardium System”). This
analysis extends the two time points (diastole and peak systole) study of Vis,
Sipkema, and Westerhof (1997) to the entire cardiac cycle.
Shortening-Induced Intracellular Pressure (SIP, Fig. 6.17b) Myocytes are
modeled as membrane-contained fluid compartments that surround the vessels.
During shortening, their thickening (lateral expansion of their membranes) is due
to an internal pressure elevation. This intramyocyte pressure is transmitted to the
vessel due to the impingement by the vessels which is based on data and a model
(Rabbany et al., 1994) that showed a linear increase in pressure with contractile
shortening. Hence, P
is proportional to myocytes shortening (expressed by their
EV
stretch ratio, SSR) through a scale factor α, with baseline value of 0.14 mmHg/%
shortening, implying a systolic extravascular pressure of 15 mmHg in the intact
tissue. This pressure is lower than the measured intracellular pressure in isolated
myocytes (Rabbany et al., 1994), but in the range of measured extravascular pressure
in intact papillary muscle (Heslinga, Allaart, Yin, & Westerhof, 1997). The baseline
value of α is varied through a sensitivity analysis (see below). Here, the vessel
response to pressure is unaffected by ac tivation:
P
¼ α 1 SSR tðÞ½; D ¼ D
EV
p
ð6:49Þ
416 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
https://t.me/med1917
Cavity-Induced Extracellular (Interstitial) Pressure (CEP, Fig. 6.17c) This
mechanism relies on the underlying assumption of both the intramyocardial pump
(Spaan et al., 1981) and the vascular waterfall (Downey & Kirk, 1975) models. P
EV
is assumed to stem from the LVP alone and is taken to vary linearly (Heineman &
Grayson, 1985) with transmural position (expressed by myocardial relative depth,
MRD). As in the SIP mechanism, activation is not considered in this interaction
mode, namely:
P
¼ MRD LVP tðÞ; D ¼ D
EV
p
ð6:50Þ
CEP+VE The extracellular pressure and varying elasticity are combined in this
scenario. The material law of surrounding myocardium is activation dependent (as in
VE) and the extracellular pressure is applied to the myocardium cyli nder external
surface, i.e.:
P
¼ MRD LVP tðÞ; D ¼ 1 activation½Dpþ activation Dað6:51Þ
EV
CEP+SIP Here, myocytes are assumed to contract within an LVP-derived pressurized interstitium. Hence, the assi gned P
equals the algebraic sum of extracellular
EV
pressure and of the shortening-dependent intracellular pressure such that:
P
¼ MRD LVP tðÞþα 1 SSR tðÞ½; D ¼ D
EV
p
ð6:52Þ
A combination of both VE and SIP or of all three basic mecha nisms are not
considered since the extravascular myocytes can be considered either as a solid (as in
the VE mechanism) or as fluid (as is with the SIP), but not as both together.
Effects of Dynamic Axial Stretch on the Flow Under all tested mechanisms, the
instantaneous diameter and length of each vessel are taken to be affected by the
dynamic axial stretch as outlined in Appendix “Mechan ics of Vessel-in-Myocardium System”.
Table 6.2 Comparison of various mechanisms (VEM-varying elasticity model; IPM-intramyocyte
pressure model; CPM-cavity pressure model) with experimental observations
Mechanism
Observation
Transmural distribution of perfusion
under contraction is close to
homogeneous
Elevated contractility attenuates total
perfusion under similar left ventricle
cavity pressure (LVP) values
Elevated heart rate attenuates both total
perfusion and endocardial/epicardial perfusion ratio.
VEM IPM CPM CPM + VEM CPM + IPM
––++ +
++–– +
––++ +
(continued)
Appendix 4: Coronary Flow Regulation 417
https://t.me/med1917
Mechanism
Observation
Reduced inlet pressure attenuates both
total perfusion and endocardial/epicardial
perfusion ratio
Epicardial arteriolar waveforms follow
aortic pressure
Predicted systolic/diastolic diameters
change follows measured data
Predicted velocity waveforms follow
measured data
Reproduced with permission from Algranati et al. (2010)
VEM IPM CPM CPM + VEM CPM + IPM
––++ +
– ++ – +
––++ +
––+ – +
Appendix 4: Coronary Flow Regulation
The Network Structure
A morphological reconstruction of left circumflex (LCx) arterial tree (see Chap. 2)is
used for a comprehensive flow analysis. A sub-endocardial subtree consisting of
400 vessels (orders 6 to 0) is pruned from the LCx coronary tree. The subtree
contained 195 bifurcations, 3 trifurcations, and 79 terminal order 1 vessels. Diameters are randomly assigned to the tree vessel segments based on morphological
statistical data of the measured diameters (Chap. 2) at 80 mmHg. Preliminary flow
analysis revealed that this method of diameter assignment resulted in excessive
heterogeneity in both perfusion and wall shear stress. An iterative diameter
re-assignment is implemented to reduce flow heterogeneity to be in agreement
with published data. Re-assignment of diameters is based on the assumption that
the flow in each upstream network vessel should be proportional to the number of
terminal vessels it perfuses. The iteration is terminated when perfusion heterogeneity
(coefficient of variation, CV ¼ SD/mean) is reduced to within the range of
published data.
To study the effects of transmural location on the network flow, the same
sub-endocardial subtree is “implanted” in the sub-epicardium layer. The network
flow analysis is subjected to the sub-epicardium boundary conditions (i.e., inlet and
outlet pressures and extravascular loading by the myocardium–vessel interaction,
MVI).
418 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
https://t.me/med1917
Fig. 6.18 Flow models in a
single vessel and vessel
bifurcation. (a) Scheme of a
single uniform cylindrical
coronary vessel and its
pressure boundary
conditions: P
pressure; P
pressure; P
surrounding tissue pressure.
(b) Lumped three-element
Windkessel model of a
single vessel segment
consisting of two nonlinear
resistors connected in series
and a parallel nonlinear
capacitor. (c) Lumped
model of a single bifurcation
of a parent vessel into two
daughters. Reproduced from
Namani et al. (2018)by
permission
, the inlet
in
, the outlet
out
, the
T
Network Flow Analysis
The flow in each elastic vessel is modeled by a three-element Windkessel made of
two nonlinear resistors in series and one parallel capacitor as shown in Figs. 6.18a, b.
Flow resistance is governed by Poiseuille’s equation and vessel capacitance by the
vessel pressure–diameter relationship (PDR). Network flow is solved by imposing
flow continuity at each vessel midpoint and at each network junction, i.e., the net
flow at each junction v anishes to conserve mass. This formulation resulted in a
system of ordinary differential equation s which are numerically solved as outlined
below. The network dynamic flow is analyzed subject to boundary conditions of
inlet and outlet pressures, and the extravascular loading by myocardial contractions.
Under the myogenic and shear regulation, network flow is iteratively solved by
adjusting each vessel diameter according to the local pressure and flow. For cases of
metabolic regulation set to achieve a certain level of perfusion to match the metabolic demand, coronary flow is solved by optimizing the distribution of metabolic
activation which produced terminal flows similar to the desired perfusion level.
Flow in Single Vessel Flow in a single vessel (Fig. 6.18a) is governed by Poiseuille
relation as:
4
QtðÞ¼
πRtðÞ
8μ RðÞ
PintðÞP
L
tðÞðÞ
out
ð6:53Þ
Appendix 4: Coronary Flow Regulation 419
https://t.me/med1917
where Q(t) is the flow rate, R(t) is the vessel radius, (Pin(t)–P
(t)) is the longitudinal
out
input/output pressure drop, μ(R) is the dynamic viscosity which is a function of
vessel radius, and L is the length of the vessel (Jacobs et al., 2008). The shear stress,
τ, is given by:
τ ¼
PintðÞP
2L
out
tðÞðÞRtðÞ
ð6:54Þ
Flow in each vessel is simulated by a validated lumped three-element Windkessel
model (Jacobs et al., 2008). Two nonlinear resistors (R
, R2) in series are connected
1
in parallel to a capacitor C (Fig. 6.18b). The capacitive element represents the
pressure-induced volume change in each elastic vessel. The resistance R (and
conductance G), and the capacitance C of each vessel are:
ℜ tðÞ¼2ℜ
tðÞ¼2ℜ2tðÞ¼
1
CtðÞ¼
dV
dΔP
8μ tðÞL
4
πRtðÞ
¼ 2πLR tðÞ
, GtðÞ¼1=ℜ tðÞ ð6:55Þ
dR
dΔP
ð6:56Þ
The junction of the three elements in the single vessel (Fig. 6.18b) is the geometric
center of the vessel with an unknown pressure, P
. The bifurcation node between
mid
vessels is the junction of three resistors, each belonging to a different vessel
(Fig. 6.18c). The nodal pressure at the junction of the three vessels is P
unknowns P
mid
and P
are solved based on conservation of mass .
node
node
. The
The model is based on the following assumptions: (a) each vessel is of uniform
cross-section and wall thickness, (b) flow in the vessels is laminar, (c) blood
viscosity in each vessel is diameter dependent following measured data of the
apparent viscosity in microvascular beds (Pries et al., 1994), (d) active and passive
vessel properties are homogeneous in each vessel but vary between vessels,
(e) dynamic extravascular pressure P
is constant along each vessel but varies
T
between vessels depending on their transmural location, and (f) active vessel
response depends on the time-averaged pressure, flow, and metabolic signal.
Network Flow Analysis Flow in the multiple vessel network is deter mined via the
iterative solution of a system of ordinary differential equations (ODEs) based on the
conditions of conservation of mass which requires that the net flow in each node be
zero, and hence for a vessel midpoint as:
i
out
dV
þ
dt
i
Q
þ Q
in
i
i
P
in
¼
hi
d
π R
dt
i
tðÞP
ℜ
i
tðÞ
mid
i
=2
2
þ
i
LiP
mid
P
P
i
out
i
T
i
tðÞP
mid
i
ℜ
=2
¼ 0; i ¼ 1,2,3 ...
tðÞ
ð6:57Þ
420 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
https://t.me/med1917
where P
i
is the given input signal of the extravascul ar pressure which depends on the
T
myocardial transmural wall location. The mass conservation at the midpoint in each
vessel is given by:
i
dt
P
i
T
ð6:58Þ
where P
i
P
in
(t) is the vessel input pressure signal and P
in
tðÞP
i
ℜ
1
=2
i
mid
tðÞ
i
P
out
þ
tðÞP
i
ℜ
=2
2
i
mid
tðÞ
¼ C
dP
mid
i
(t) is the vessel output
out
pressure.
The net flow at a bifurcation or trifurcation between a mother vessel and daughter
vessels at a designated “network node” is zero. Application of mass conservation at
each node yields additional equations for the nodal pressures. For an ith vessel,
which is neither source nor sink, mass balance at the vessel inlet (Fig. 6.18a) yields:
i, n
P
mid
i
1
P
G
in
i, n
1
þ P
i
mid
P
i
þ P
i, n
mid
i
2
P
i,n
2
G
in
¼ 0 ð6:59Þ
i
G
in
Hence, the inlet nodal pressure as a function of neighboring vessel pressures and
conductance is given by:
i, n
i
P
i
P
tðÞ¼
in
Giþ P
mid
Giþ G
mid
i, n
1
1
G
i,n
1
þ G
þ P
i, n
2
i, n
mid
i, n
2
2
G
ð6:60Þ
Similarly, applying mass balance at the outlet node of the ith vessel gives:
i, n
i
P
i
P
tðÞ¼
out
For a source vessel, the inlet pressure P
out
P
(t) are prescribed boundary conditions. The governing equations are assembled
Giþ P
mid
Giþ G
in
i, n
1
G
mid
i, n
1
(t) and for a sink vessel, the outlet pressure,
1
þ P
þ G
i,n
2
i, n
mid
i, n
2
2
G
ð6:61Þ
for the network into a system of ordinary differential equations (ODEs) which is
written in matrix form as:
dP
mid
¼ AP
dt
þ B ð6:62Þ
mid
Expressions for the A and B matrices are given below. The trifurcation node is the
junction of four resistors, each belonging to a different vessel. Similar to a bifurcation, the nodal pressure (P
) at the junction of the four elements is an unknown.
node
The network flow is solved based on mass conservation at each vessel midpoint and
at each vessel nodal junction.
The network structure matrix in Eq. (6.62) for a subtree is used to build the
coefficient matrices A (n n) and B (n 1). The indices of the non-zero elements
at an ith row in A correspond to the ith vessel properties and its neighbors. A vessel
Appendix 4: Coronary Flow Regulation 421
https://t.me/med1917
connected to bifurcating vessels at its origin and end has indices (i, n-1), (i, n-2), (i, n1),
and (i, n
conductances are: G
has an additional daughter with index (i, n
trifurcating vessel at its origin has an additional sister with index (i, n
) for the mother, sister, and two daughters respectively. The corresponding
2
n
n
n
1
2
, G
i
, G
i
n
1
2
, G
i
. A vessel with a trifurcating branch at its end
i
) with conductance G
3
n
3
:A vessel with a
i
).
-21
For a vessel, i, connected to bifurcations at both ends, the elements of matrices A
are as follows:
mid
dP
i
dt
A
i, n
A
where P
i, n
¼ A
A
2
2
mid
mid
P
þ A
, n
i
1
2G
i
¼
C
Giþ G
i
2G
i
¼
C
Giþ G
i
i
Giþ G
i
i
Giþ G
i
i, n
¼
¼
i,n
1
A
i, i
1
2G
C
2G
C
mid
P
i, n
2
, n
i
2
G
i
n
1
þ G
i
n
1
G
i
n
1
þ G
i
n
2
G
i
n
i
G
i
n
i
1
n
1
þ G
2
þ G
n
i
n
2
i
þ A
2
i, i
n
2
i
n
2
i
, A
P
mid
þ
Giþ G
i, n
þ A
1
mid
P
i, n
2G
¼
þ A
1
, n
i
1
G
i
n
1
þ G
i
i
C
Giþ G
i
mid
P
2
, n
i
n
1
i
n
1
þ G
i
þ Bið6:63Þ
2
,
,
n
2
i
i, n
2
n
2
i
G
ð6:64Þ
, ð6:65Þ
is the press ure in the vessel midpoint, and C is its capacity (Eq. 6.56).
For a vessel, i, connected to a bifurcation at its origin and a trifurcation at its end,
the elements of matrices A are given by:
mid
dP
i
¼ A
dt
2G
A
i,i
i, n
2G
¼
1
A
i
¼
C
Giþ G
i
i
C
Giþ G
i
i
n
G
i
n
1
þ G
i
þA
G
n
1
1
i, n
i
i
n
1
i,n
þ G
2
þ G
mid
P
þ A
, n
i
1
mid
P
þ A
1
, n
i
1
þ
n
2
Giþ G
i
, A
n
3
i
i, n
i,n
mid
P
2
, n
i
2
mid
P
þ A
2
, n
i
2
G
n
1
þ G
i
2G
¼
i, n
2
C
mid
þ A
P
i, i
mid
P
i, n
i
i
i
i
þ B
3
i
n
2
þ G
Giþ G
i
, n
3
2
n
3
i
G
i
n
1
þ G
i
ð6:66Þ
ð6:67Þ
n
2
n
n
2
3
þ G
i
i
ð6:68Þ
For a vessel, i, connected to a trifurcation at its origin and a bifurcation at its end, the
elements of matrices A are given by:
2G
A
i,i
i
¼
C
Giþ G
i
G
i
n
i
n
1
þ G
2
þ G
i
þ
n
21
i
Giþ G
G
i
n
1
i
þ G
2
n
2
i
ð6:69Þ
422 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
https://t.me/med1917
2G
A
i, n
1
A
i, n
2
i
¼
C
Giþ G
i
2G
i
¼
C
Giþ G
i
n
1
G
i
n
i
n
i
n
1
1
þ G
G
i
þ G
2
þ G
i
n
2
n
2
þ G
i
,
n
21
i
n
21
i
If an ith vessel is terminating, the elements at each row in A are:
2G
i
¼
A
i, i
C
Giþ G
i
G
i
n
1
þ G
i
A
i, n
2
2
, A
i, n
n
2
i
2G
i
¼
C
Giþ G
i
1
n
2
G
i
n
1
i
2G
i
¼
C
Giþ G
i
n
2
þ G
i
For the source vessel, the elements of A are as follows:
2G
1
¼
A
1,1
C
G1þ G
1
2G
1
¼
A
1, n
2
C
G1þ G
1
G
i
n
1
1
n
2
G
1
n
1
þ G
1
þ G
2
, A
1, n
n
2
1
n
2
1
1
2G
1
¼
C
G1þ G
1
For all interior vessels, the elements of B are given by:
ð6:70Þ
n
1
G
i
n
i
n
1
þ G
2
i
ð6:71Þ
ð6:72Þ
n
1
G
i
n
1
þ G
1
,
n
2
1
ð6:73Þ
t
dP
i
¼
B
i
dt
ð6:74Þ
For a terminal vessel, i, the elements at each row in B are:
t
dP
2G
i
B
i
þ
¼
i
dt
out
P
C
i
ð6:75Þ
For the source vessel, the elements of B are as follows:
t
dP
2G
1
1
þ
¼
1
dt
P
in
and P
B
out
are the network input and output pressures, respective ly.
Terminal Arteriole Flow The flow in the pre-capillary arterioles, q
in
P
C
1
term
ð6:76Þ
, represents
the cardiac perfusion. The flow must satisfy the myocyte demand for oxygen as
represented by the target flow q
the heart metabolic demand. Several levels of q
. This is an input to the model which depends on
target
are used to repres ent a range of
target
physiological activity levels.
Appendix 4: Coronary Flow Regulation 423
https://t.me/med1917
Table 6.3 Parameters for the reference case in the model of comprehensive blood flow
Parameter Value Units Reference
n—Number of network
400 – Kaimovitz, Lanir, and Kassab (2005)
vessels
r—Number of network termi-
79 – Kaimovitz et al. (2005)
nal vessels
—Vessel tethering strut
C
str
2.0 N/mm3–
a
stiffness (Eq. 6.55)
C1(Eq. 6.67) 30.6 mm
-1
Halpern, Mulvany, and Warshaw
(1978)
(Eq. 6.67) 4.85 kPa Halpern et al. (1978)
C
0
H
—Hematocrit 0.45 – Pries et al. (1994)
D
η-Blood viscosity Diameter dependent Pries et al. (1994)
L
—Length scale of meta-
0
bolic decay (Eq. 6.79)
A
, ϕp, Cp(Eq. 6.77) As in Table 6.4A Liao and Kuo (1997)
p
ρ
, ϕm, Cm(Eq. 6.95) As in Table 6.4B Liao and Kuo (1997)
m
F
, Kτ(Eq. 6.98) As in Table 6.4C Liao and Kuo (1997)
τmax
q
target
P
in
P
out
Reproduced from Namani et al. (2018) by permission
a
The value of C
is selected such that the vessel dynamic stiffness (Eq. 6.87) yielded smooth
str
1.0 mm Hald, Jensen, Sorensen, HolsteinRathlou, and Jacobsen (2012)
1.5 103mm3/s Tillmanns et al. (1974)
100 mmHg Algranati et al. (2010)
41.7 mmHg Algranati et al. (2010)
variation of vessel compliance and solution convergence
Vascular Mechanical Properties
The Passive Vessel Properties The passive vessel response consists of the intrinsic
passive mechanics (given by the PDR) and the tethering effect of the myocardial
tissue. The PDR of isolated in vitro passive vessels under positive trans-vascular
pressure showed the radius to be a sigmoidal function of the trans-vascular pressure,
ΔP (Young, Choy, Kassab, & Lanir, 2012)as:
where A
are the asymptotical highest and lowest radii, respectively, ϕpis the
p,Bp
trans-vascular pressure corresponding to the average of radii A
passive response bandwidth. The tethering model is described below and the data on
the PDR constants for orders 1 –6 vessels are listed in Table A.1A.
Coronary vessels are tethered to the surrounding myocardium by a network of
short collagen struts (Borg & Caulfield, 1979; Caulfield & Borg, 1979). These struts
prevent the vessels from collapse under negative trans-vascular pressure which
R
ΔPðÞ¼Bpþ
p
Ap B
π
p
π
þ arctan
2
ΔP ϕ
p
C
p
and Bp, and Cpis the
p
ð6:77Þ
424 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
https://t.me/med1917
occurs in the deeper myocardial layers (Kajiya et al., 2008). The importance of
tethering is clearly seen when considering the wall tension balance under negative
trans-vascular pressure in the absence of tethering. In the case of passive vessels, the
tension balance is expressed by:
ΔP R
reg
¼ T
pas
ð6:78Þ
The compliant untethered vessel wall can only sustain a very small negative
transmural pressure before collapse. In the presence of active regulation, the wall
smooth muscle cells contract thereby adding to the vessel tendency to collapse.
Tethering prevents collapse by adding a positive pressure-like term to the combined
tension balance equation, i.e.:
ΔP R
reg
þ T
teth
¼ T
pas
þ T
act
ð6:79Þ
Two assumptions are adopted in the analysis. First, under negative trans-vascular
pressure (ΔP < 0), the passive wall tension T
vanishes. Second, under positive
pas
trans-vascular pressure, the tension in the tethering struts increases quadratically
with the gap between the zero-pressure radius (R
embeds the vessel and that of the vessel (R
) of the myocardial “tunnel” which
0
). This nonlinearity in the tension–gap
reg
relationship reflects the gradual recruitment of the non-uniformly undulated struts
with stretch. Under positive transvascular pressure (ΔP < 0), the tension in the
tethering struts is assumed to vanish while the vessel adheres to the myocardial
tunnel. The struts tension is thus C
coronary vessels of all orders with the same value of C
struts density on the myocardial tunnel wall is indepe ndent of R
str(R0
– R
)2. This expression cannot be used for
reg
since it assumes that the
str
. It is likely that the
0
struts density is higher for smaller vessels with smaller perimeters and vice versa for
larger vessels. Hence, the strut pressure-like stress per unit myocardial tunnel area
can be expressed as:
The strut density on the vessel wall under negative trans-vascular pressure (i.e.,
when R
< R0) is higher than that on the myocardium by a factor of R0/R
reg
the strut stress on the vessel wall is P
contribution of the strut pressure-like stress to the vessel wall tension (“tethering
tension”) is given by:
myo
P
str
teth
¼ P
vessel
str
T
¼ C
strR0
vessel
str
R
¼ C
reg
2
R
reg
=R
0
¼ C
strR0
R
R
strR0
reg
ð6:80Þ
, so that
2
=R
reg
2
reg
. Finally, the
reg
ð6:81Þ
Соседние файлы в папке Библиотека им академика М.И. Перельмана