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A dynamic nonlinear lumped parameter model for skeletal muscle circulation

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Abstract

A dynamic nonlinear lumped parameter model of the circulation of skeletal muscle for constant vasoactive state is presented. This model consists of four compartments that represent the large arteries, the arterioles, the capillaries and venules, and the veins, respectively. The first compartment consists of a linear compliance (C1) and resistance (R1). The third compartment possesses no compliance and is represented by a linear resistance (R3). The second and fourth compartments each consist of a nonlinear pressure-volume relation, resulting in a pressure dependent compliance (C2, C4, respectively) and nonlinear resistance (R2, R4, respectively). The eleven model parameters were collected in a complementary way: they were partly obtained from a priori knowledge including, information at the microscopic level, and partly determined by means of an estimation algorithm. Estimated values of the compliances (in cm3·kPa−1·100 g−1, 1kPa=7.5 mmHg) and resistances (in kPa·s·cm−3·100 g) at an (arterial) inflow pressure of 10 kPa and a (venous) outflow pressure of 0 kPa were: C1: 0.014; R1: 6.6; C2: 0.565; R2: 84.6; R3: 37.9; C4: 1.044; R4: 24.5. The model (with the nonlinear pressure-volume relations) is able to predict the static and dynamic instantaneous (i.e., for constant vasomotor tone) pressure-flow relation and the instantaneous zero flow pressure intercept. These phenomena are therefore not necessarily the result of the rheological properties of blood. The secondary or delayed dilatation upon a positive inflow pressure step (or negative step in venous pressure) is predicted by the model implying that delayed dilatation is not necessarily related to changes in vasomotor tone. Venous outflow delay, upon a positive inflow pressure step (starting from zero flow), is also predicted by the model.

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Abbreviations

A :

slope ofP-V relation atV 0

C :

compliance (dV/dP)

D :

diameter

F :

flow

F in :

input (arterial) flow, model

R out :

output (venous) flow, model

F p :

input (arterial) flow, measurements

F v :

output (venous) flow, measurements

i :

generation number

K :

8νπL 3

K eff :

effective value ofK

L :

length

N :

number of vessels

P :

pressure

P in :

input (arterial) pressure, model

P out :

output (venous) pressure, model

P 0a :

apparent zero flow pressure intercept

P 0i :

instantaneous zero flow pressure intercept

P 0s :

steady state zero flow pressure intercept

P p :

perfusion (arterial) pressure, measurements

P v :

output (venous) pressure, measurements

R :

resistance

t :

time

V :

volume

V 0 :

half maximal volume of compartment

V 0ef :

half effective maximal volume of compartment

W :

working point

ν:

viscosity

τ:

time constant

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Authors and Affiliations

  1. Laboratory for Physiology, Free University, van der Boechorststraat 7, 1081 BT, Amsterdam, The Netherlands

    Robert Braakman, Pieter Sipkema & Nicolaas Westerhof

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  1. Robert Braakman
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  2. Pieter Sipkema
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  3. Nicolaas Westerhof
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Braakman, R., Sipkema, P. & Westerhof, N. A dynamic nonlinear lumped parameter model for skeletal muscle circulation. Ann Biomed Eng 17, 593–616 (1989). https://doi.org/10.1007/BF02367465

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