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Development of a Mathematical Model of the Human Circulatory System

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A mathematical lumped parameter model of the human circulatory system (HCS) has been developed to complement in vitro testing of ventricular assist devices. Components included in this model represent the major parts of the systemic HCS loop, with all component parameters based on physiological data available in the literature. Two model configurations are presented in this paper, the first featuring elements with purely linear constitutive relations, and the second featuring nonlinear constitutive relations for the larger vessels. Three different aortic compliance functions are presented, and a pressure-dependent venous flow resistance is used to simulate venous collapse. The mathematical model produces reasonable systemic pressure and flow behaviour, and graphs of this data are included.

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Abbreviations

μ:

viscosity, kg cm−1 s−1

Φ C :

vessel compliance function

Φ I :

vessel inertance function

Φ R :

vessel resistance function

ρ:

density, kg cm−3

A :

cross-sectional area, cm2

C :

compliance, cm5 N−1

D :

diameter, cm

F :

pumping force, N

I :

fluid inertance, N s2 cm−5

I ht :

heart pumping plate mass, N s2 cm−1

L :

length, cm

P :

pressure, N cm−2

p :

fluid momentum, N s cm−2

p ht :

heart pumping plate momentum, N s

Q :

volume flow rate, cm3 s−1

R :

radius, cm or flow resistance, N s cm−5

R ht :

friction in the pump, N s cm−1

r :

radial position coordinate, cm

s ht :

heart pumping plate position, cm

u :

velocity, cm s−1

V :

volume, cm3

x :

linear position coordinate, cm

□ a :

subscript refers to artery (SCM)

□ao :

subscript refers to aorta (DCM)

□at :

subscript refers to arteries (DCM)

□ht :

subscript refers to heart (pump)

□incn :

subscript refers to inlet cannula

□inv :

subscript refers to inlet valve

□otcn :

subscript refers to outlet cannula

□otv :

subscript refers to outlet valve

□pe :

subscript refers to peripheral system

□ v :

subscript refers to venous system (SCM)

□vc :

subscript refers to vena cava (DCM)

□vn :

subscript refers to venous system (DCM)

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ACKNOWLEDGMENTS

The authors wish to acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada as well as support from the Medical Devices Centre located at the University of Ottawa Heart Institute. We also acknowledge the contributions of Sean Tan and Vinay Menon who contributed to the development of this model during their graduate studies.

Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Martin J. Conlon & Donald L. Russell

  2. University of Ottawa Heart Institute, 40 Ruskin Street, Ottawa, ON, Canada, K1Y 4W7

    Tofy Mussivand

  3. Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, ON, Canada, K1S 5B6

    Donald L. Russell

Authors
  1. Martin J. Conlon
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  2. Donald L. Russell
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  3. Tofy Mussivand
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Corresponding author

Correspondence to Donald L. Russell.

APPENDIX

APPENDIX

A. STATE EQUATIONS FOR THE MODEL

The following differential equations describe the behaviour of the model with the linear arterial and linear venous segments inserted:

$$\displaylines{ \makeatletter\renewcommand\theequation{A.\arabic{equation}}\setcounter{equation}{0}\makeatother \dot{s}_{\rm ht}(t) = \frac{p_{ht}(t)}{I_{ht}}\cr \dot{p}_{ht}(t) = F(t)-\frac{p_{ht}(t)\cdot R_{ht}}{I_{ht}}-A_{ht}\cdot P_{ht}(V_{ht}(t)) \cr \dot{V}_{ht}(t) = \frac{p_{ht}(t)\cdot A_{ht}}{I_{ht}} \cr +\,R_{\rm inv}(P_{\rm incn}(V_{\rm incn}(t))-P_{ht}(V_{ht}(t)))\cr -\,R_{\rm otv}(P_{ht}(V_{ht}(t))-P_{\rm otcn}(V_{\rm otcn}(t))) \cr \dot{V}_{\rm otcn}(t) = R_{\rm otv}(P_{ht}(V_{ht}(t))-P_{\rm otcn}(V_{\rm otcn}(t)))\cr -\,\frac{p_{\rm otcn}(t)}{I_{\rm otcn}}\cr \dot{p}_{\rm otcn}(t) = P_{\rm otcn}(V_{\rm otcn}(t))-\frac{p_{\rm otcn}(t)\cdot R_{\rm otcn}}{I_{\rm otcn}}\cr -\,P_{a}(V_{a}(t))\cr \dot{V}_{a}(t) = \frac{p_{\rm otcn}(t)}{I_{\rm otcn}}-\frac{p_{\rm pe}(t)}{I_{\rm pe}}\cr \dot{p}_{\rm pe}(t) = P_{a}(V_{a}(t))-\frac{R_{\rm pe}\cdot p_{\rm pe}(t)}{I_{\rm pe}}\cr \dot{V}_{v}(t) = \frac{p_{\rm pe}(t)}{I_{\rm pe}}-\frac{p_{\rm incn}(t)}{I_{\rm incn}}\cr \dot{p}_{\rm incn}(t) = P_{v}(V_{v}(t))-\frac{p_{\rm incn}(t)\cdot R_{\rm incn}}{I_{\rm incn}}\cr -\,P_{\rm incn}(V_{\rm incn}(t))\cr \dot{V}_{\rm incn}(t) = \frac{p_{\rm incn}(t)}{I_{\rm incn}} -R_{\rm inv}(P_{\rm incn}(V_{\rm incn}(t))\cr -\,P_{ht}(V_{ht}(t))) }$$

B. STATE EQUATIONS FOR THE MODEL

The following differential equations describe the behaviour of the model with the nonlinear arterial and nonlinear venous segments inserted:

$$\displaylines{\makeatletter\renewcommand\theequation{B.\arabic{equation}}\setcounter{equation}{0}\makeatother \dot{s}_{ht}(t) = \frac{p_{ht}(t)}{I_{ht}}\cr \dot{p}_{ht}(t) = F(t)-\frac{p_{ht}(t)\cdot R_{ht}}{I_{ht}}-A_{ht}\cdot P_{ht}(V_{ht}(t))\cr \dot{V}_{ht}(t) = \frac{p_{ht}(t)\cdot A_{ht}}{I_{ht}}+R_{\rm inv}(P_{\rm incn}(V_{\rm incn}(t)) \cr -\,P_{ht}(V_{ht}(t)))\quad -R_{\rm otv}(P_{ht}(V_{ht}(t)) \cr -\,P_{\rm otcn}(V_{\rm otcn}(t)))\cr \dot{V}_{\rm otcn}(t) = R_{\rm otv}(P_{ht}(V_{ht}(t))-P_{\rm otcn}(V_{\rm otcn}(t)))\cr \dot{p}_{\rm otcn}(t) = P_{\rm otcn}(V_{\rm otcn}(t))-\frac{p_{\rm otcn}(t)\cdot R_{\rm otcn}}{I_{\rm otcn}}-P_{a}(V_{ao}(t)) \cr \dot{V}_{ao}(t) = \frac{p_{\rm otcn}(t)}{I_{\rm otcn}}-\frac{p_{ao}(t)}{I_{ao}} }$$
figure 10

 

figure 11

 

figure 12

 

$$\displaylines{\makeatletter\renewcommand\theequation{B.\arabic{equation}}\setcounter{equation}{6}\makeatother \dot{p}_{ao}(t) = P_{ao}(V_{ao}(t))-\frac{p_{ao}\cdot R_{ao}}{I_{ao}}-P_{at}(V_{at}(t))\cr \dot{V}_{at}(t) = \frac{p_{ao}(t)}{I_{ao}}-\frac{p_{\rm pe}(t)}{I_{\rm pe}}\cr \dot{p}_{\rm pe}(t) = P_{at}(V_{at}(t))-\frac{R_{\rm pe}\cdot p_{\rm pe}(t)}{I_{\rm pe}}-P_{vn}(V_{vn})\cr \dot{V}_{vn}(t) = \frac{p_{\rm pe}(t)}{I_{\rm pe}}-R_{vn}(P_{vn}(V_{vn}(t))-P_{vc}(V_{vc}))\cr \dot{V}_{vc}(t) = R_{vn}(P_{vn}(V_{vn}(t))-P_{vc}(V_{vc}))-\frac{p_{\rm incn}(t)}{I_{\rm incn}} \cr \dot{p}_{\rm incn}(t) = P_{vc}(V_{vc}(t))-\frac{p_{\rm incn}(t)\cdot R_{\rm incn}}{I_{\rm incn}}-P_{\rm incn}(V_{\rm incn}(t)) \cr \dot{V}_{\rm incn}(t) = \frac{p_{\rm incn}(t)}{I_{\rm incn}}-R_{\rm inv}(P_{\rm incn}(V_{\rm incn}(t))-P_{ht}(V_{ht}(t))) }$$

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Conlon, M.J., Russell, D.L. & Mussivand, T. Development of a Mathematical Model of the Human Circulatory System. Ann Biomed Eng 34, 1400–1413 (2006). https://doi.org/10.1007/s10439-006-9164-y

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