Mathematical circulatory system model

• US 20060166176A1
• Filed: 03/23/2006
• Published: 07/27/2006
• Est. Priority Date: 09/10/2002
• Status: Active Grant

• Current Assignee: University of Vermont & State Agricultural College
• Original Assignee: University of Vermont & State Agricultural College
• Inventors: Lakin, William D., Stevens, Scott A.
• Granted Patent: US 7,794,230 B2
• Time in Patent Office: Days
• US Class Current: 434/262
• CPC Class Codes: G09B 23/12 of liquids or gases
• CPC Class Codes: G09B 23/30 Anatomical models G09B23/281 - G09B23/288 take precedence
• CPC Class Codes: G09B 23/32 with moving parts
• CPC Class Codes: G16H 50/50 for simulation or modelling of medical disorders

Abstract

A system and method of modeling a circulatory system including a regulatory mechanism parameter. In one embodiment, a regulatory mechanism parameter in a lumped parameter model is represented as a logistic function. In another embodiment, the circulatory system model includes a compliant vessel, the model having a parameter representing a change in pressure due to contraction of smooth muscles of a wall of the vessel.

Claims

1. A method of modeling a circulatory system, the method comprising:

(a) providing a circulatory system model including one or more time-dependent pressure functions, each of said one or more time-dependent pressure functions representing a pressure in a portion of the circulatory system;

(b) using a logistic function to represent a regulatory mechanism parameter, said regulatory mechanism parameter representing a regulatory mechanism having an impact on circulatory system function; and

(c) solving one or more equations for said one or more time-dependent pressure functions and said logistic function to determine a circulatory system dynamic.

2. A method according to claim 1, further comprising:

(a) dividing the circulatory system into a plurality of compartments and a heart pump, said plurality of compartments representing a portion of the circulatory system, said heart pump interacting with at least one of said plurality of compartments; and

(b) assigning one of said one or more time-dependent pressure functions to each of said plurality of compartments.

3. A method according to claim 2, further comprising using a smooth muscle contraction parameter, said smooth muscle contraction parameter representing an autonomic nervous system effect on smooth muscle contraction in a wall of one of said plurality of compartments.

4. A method according to claim 3, wherein said smooth muscle contraction parameter is represented according to the following equation:

{dot over (V)}=C·

({dot over (φ

)}+{dot over (P)}*))

wherein {dot over (V)} is a time derivative of a volume of said one of said plurality of compartments, {dot over (P)} is a time derivative of the pressure inside said one of said plurality of compartments, {dot over (φ

)} is a time derivative of said smooth muscle contraction parameter, {dot over (P)}* is a time derivative of the pressure outside of said one of said plurality of compartments, and C is an active compliance defined according to the following equation;

$C=\frac{{\mathrm{rV}}^{\mathrm{Max}}<br><br>{\mathrm{e<br><br>}}^{-r<br><br>\left(P-\left(\mathrm{\varphi <br><br>}+P^{*}\right)\right)}}{{\left(1+{\mathrm{e<br><br>}}^{-r<br><br>\left(P-\left(\mathrm{\varphi <br><br>}+P^{*}\right)\right)}\right)}^2}$

wherein r>

0, V^Max is a maximum volume of said one of said plurality of compartments, P is the pressure inside said one of said plurality of compartments, P* is the pressure outside said one of said plurality of compartments, and φ

is said smooth muscle contraction parameter.

5. A method according to claim 2, wherein said dividing step (a) divides said circulatory system into a set of compartments consisting of an arterial compartment and a venous compartment, and a heart pump.

6. A method according to claim 2, wherein said dividing step (a) divides said circulatory system into a set of compartments comprising of an arterial compartment and a venous compartment, and a heart pump.

7. A method according to claim 6, further comprising using a logistic function to represent a cerebral blood flow parameter.

8. A method according to claim 7, wherein said using of a logistic function to represent said cerebral blood flow parameter comprises representing said cerebral blood flow parameter according to the following logistic function:

Q₁=L_inc(P_AV−

{overscore (P)}_AV,0.15,1.0001,0)·

{overscore (Q)}₁

wherein Q₁ is said cerebral blood flow parameter, P_AV is a pressure difference P_A−

P_V, wherein P_A is said time-dependent pressure function for said arterial compartment and P_V is said time-dependent pressure function for said venous compartment, {overscore (P)}_AV is a mean systemic indicative pressure, {overscore (Q)}₁ is a mean cerebral blood flow, and L_inc is an increasing logistic function represented by the following equation;

$L_{\mathrm{inc}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and min<

L_inc.

9. A method according to claim 6, further comprising using a logistic function to represent an output versus pressure parameter, said output versus pressure parameter representing an effect of venous pressure on cardiac uptake.

10. A method according to claim 9, wherein said using of a logistic function to represent an output versus pressure parameter comprises representing said output versus pressure parameter according to the following logistic function:

OVP(P_V)=L_inc(P_V−

{overscore (P)}_V,0.5,2.5,0).

wherein OVP is said output versus pressure parameter, P_V is said time-dependent pressure function for said venous compartment, {overscore (P)}_V is the mean systemic venous pressure, and L_inc is an increasing logistic function represented by the following equation;

$L_{\mathrm{inc}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and min<

L_inc.

11. A method according to claim 6, wherein said regulatory mechanism parameter represents an autonomic nervous system effect on cardiac output according to the following equation:

ANSo=L_dec(P_A−

{overscore (P)}_A,0.1,2,0)

wherein ANS_o is said regulatory mechanism parameter representing said autonomic nervous system effect on cardiac output, P_A is said time-dependent pressure function for said arterial compartment, {overscore (P)}_A is an initial mean arterial pressure, and L_dec is a decreasing logistic function represented by the following equation;

$L_{\mathrm{dec}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{-\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and L_dec<

max.

12. A method according to claim 6, wherein said regulatory mechanism parameter represents an autonomic nervous system effect on non-cerebral blood flow according to the following equation:

ANSz=L_inc(P_A−

{overscore (P)}_A,0.3,1.1,0.7)

wherein ANSz is said regulatory mechanism parameter representing said autonomic nervous system effect on non-cerebral blood flow, P_A is said time-dependent pressure function for said arterial compartment, {overscore (P)}_A is an initial mean arterial pressure, and L_inc is an increasing logistic function represented by the following equation;

$L_{\mathrm{inc}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

o<

x<

∞

, and min<

L_inc.

13. A method according to claim 6, wherein said regulatory mechanism parameter represents a central nervous system effect on cardiac output according to the following equation:

CNSo=L_dec(Q₁−

{overscore (Q)}₁,0.01,5,0.9)

wherein Q₁ is cerebral blood flow, {overscore (Q)}₁ is mean cerebral blood flow, and L_dec is a decreasing logistic function represented by the following equation;

$L_{\mathrm{dec}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{-\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and L_dec<

max.

14. A method according to claim 6, wherein said regulatory mechanism parameter represents a central nervous system effects on non-cerebral blood flow according to the following equation:

CNSz=L_inc(Q_l−

{overscore (Q)}₁,0.01,1.01,0.1)

wherein Q₁ is cerebral blood flow, {overscore (Q)}₁ is mean cerebral blood flow, and L_inc is an increasing logistic function represented by the following equation;

$L_{\mathrm{inc}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and min<

L_inc.

15. A medical device for monitoring a circulatory system, the medical device employing a method according to claim 1.

16. A method of monitoring a circulatory system, the method comprising:

(a) measuring a desired circulatory system value; and

(b) comparing said desired circulatory system value to a corresponding circulatory system value calculated using a method according to claim 1.

17. A computer readable medium containing computer executable instructions implementing a method of modeling a circulatory system, the instructions comprising:

(a) a first set of instructions for providing a circulatory system model including one or more time-dependent pressure functions, each of said one or more time-dependent pressure functions representing a pressure of a portion of the circulatory system;

(b) a second set of instructions for using a logistic function to represent a regulatory mechanism parameter, said regulatory mechanism parameter representing a regulatory mechanism having an impact on circulatory system function; and

(c) a third set of instructions for solving one or more equations for said one or more time-dependent pressure functions and said logistic function to determine a circulatory system dynamic.

18. A computer readable medium according to claim 17, further comprising:

(a) a fourth set of instructions for dividing the circulatory system into a plurality of compartments and a heart pump, said plurality of compartments representing a portion of the circulatory system, said heart pump interacting with at least one of said plurality of compartments; and

(b) a fifth set of instructions for assigning a time-dependent pressure function to each of said plurality of compartments.

19. A computer readable medium according to claim 18, further comprising a sixth set of instructions for using a smooth muscle contraction parameter, said smooth muscle contraction parameter representing an autonomic nervous system effect on smooth muscle contraction in a wall of one of said plurality of compartments.

20. A computer readable medium according to claim 19, wherein said smooth muscle contraction parameter is represented according to the following equation:

{dot over (V)}=C·

({dot over (P)}−

({dot over (φ

)}+P*))

wherein {dot over (V)} is a time derivative of a volume of said one of said plurality of compartments, {dot over (P)} is a time derivative of the pressure inside said one of said plurality of compartments, {dot over (φ

)} is a time derivative of said smooth muscle contraction parameter, {dot over (P)}* is a time derivative of the pressure outside of said one of said plurality of compartments, and C is an active compliance defined according to the following equation;

$C=\frac{{\mathrm{rV}}^{\mathrm{Max}}<br><br>{\mathrm{e<br><br>}}^{-r<br><br>\left(P-\left(\mathrm{\varphi <br><br>}+P^{*}\right)\right)}}{{\left(1+{\mathrm{e<br><br>}}^{-r<br><br>\left(P-\left(\mathrm{\varphi <br><br>}+P^{*}\right)\right)}\right)}^2}$

wherein r>

0, V^Max is a maximum volume of said one of said plurality of compartments, P is the pressure inside said one of said plurality of compartments, P* is the pressure outside said one of said plurality of compartments, and φ

is said smooth muscle contraction parameter.

21. A computer readable medium according to claim 19, wherein said fourth set of instructions comprises a seventh set of instructions for dividing the circulatory system into an arterial compartment and a venous compartment.

22. A computer readable medium according to claim 21, further comprising a eighth set of instructions for using a logistic function to represent a cerebral blood flow parameter.

23. A computer readable medium according to claim 22, wherein said eighth set of instructions comprises a ninth set of instructions for representing said cerebral blood flow parameter according to the following logistic function:

Q₁=L_inc(P_AV−

{overscore (P)}_AV,0.15,1.0001,0)·

{overscore (Q)}₁

wherein Q₁ is said cerebral blood flow parameter, P_AV is a pressure difference P_A−

P_V, wherein P_A is said time-dependent pressure function for said arterial compartment and P_V is said time-dependent pressure function for said venous compartment, {overscore (P)}_AV is a mean systemic indicative pressure, {overscore (Q)}₁ is a mean cerebral blood flow, and L_inc is an increasing logistic function represented by the following equation;

$L_{\mathrm{inc}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and min<

L_inc.

24. A computer readable medium according to claim 21, further comprising a tenth set of instructions for using a logistic function to represent an output versus pressure parameter, said output versus pressure parameter representing an effect of venous pressure on cardiac uptake.

25. A computer readable medium according to claim 24, wherein said tenth set of instructions comprises a eleventh set of instructions for representing said output versus pressure parameter according to the following logistic function:

OVP(P_V)=L_inc(P_V−

{overscore (P)}_V,0.5,2.5,0).

wherein OVP is said output versus pressure parameter, P_V is said time-dependent pressure function for said venous compartment, {overscore (P)}_V is the mean systemic venous pressure, and L_inc is an increasing logistic function represented by the following equation;

$L_{\mathrm{inc}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and min<

L_inc.

26. A computer readable medium according to claim 22, wherein said second set of instructions comprises a twelfth set of instructions for representing said regulatory mechanism parameter as an autonomic nervous system effect on cardiac output according to the following equation:

ANSo=L_dec(P_A−

{overscore (P)}_A,0.1,2,0)

wherein ANSo is said regulatory mechanism parameter representing said autonomic nervous system effect on cardiac output, P_A is said time-dependent pressure function for said arterial compartment, {overscore (P)}_A is an initial mean arterial pressure, and L_dec is a decreasing logistic function represented by the following equation;

$L_{\mathrm{dec}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{-\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and L_dec<

max.

27. A computer readable medium according to claim 22, wherein said second set of instructions comprises a thirteenth set of instructions for representing said regulatory mechanism parameter as an autonomic nervous system effect on non-cerebral blood flow according to the following equation:

ANSz=L_inc(P_A−

{overscore (P)}_A,0.3,1.1,0.7)

wherein ANSz is said regulatory mechanism parameter representing said autonomic nervous system effect on non-cerebral blood flow, P_A is said time-dependent pressure function for said arterial compartment, {overscore (P)}_A is an initial mean arterial pressure, and L_inc is an increasing logistic function represented by the following equation;

$L_{\mathrm{inc}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and min<

L_inc.

28. A computer readable medium according to claim 22, wherein said second set of instructions comprises a fourteenth set of instructions for representing said regulatory mechanism parameter as a central nervous system effects on cardiac output according to the following equation:

CNSo=L_dec(Q₁−

{overscore (Q)}₁, 0.01,5,0.9)

wherein Q₁ is cerebral blood flow, {overscore (Q)}₁ is mean cerebral blood flow, and L_dec is a decreasing logistic function represented by the following equation;

$L_{\mathrm{dec}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{-\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and L_dec<

max.

29. A computer readable medium according to claim 22, wherein said second set of instructions comprises a fifteenth set of instructions for representing said regulatory mechanism parameter as a central nervous system effects on non-cerebral blood flow according to the following equation:

CNSz=L_inc(Q₁−

{overscore (Q)}₁,0.01,1.01,0.1)

wherein Q₁ is cerebral blood flow, {overscore (Q)}₁ is mean cerebral blood flow, and L_inc is an increasing logistic function represented by the following equation;

$L_{\mathrm{inc}}<br><br>\left(x,r,\max ,\min \right)=\max +\frac{\min -\max }{1+\frac{1-\min }{\max -1}<br><br>{\mathrm{e<br><br>}}^{\mathrm{rx}}}$

wherein max is a maximum value for x, min is a minimum value for x, r>

0, −

∞

<

x<

∞

, and min<

L_inc.

30. A method of modeling a pressure and volume relationship in a compliant vessel, the method comprising:

(a) defining a first parameter as a change in pressure within the vessel, said change in pressure being due to a contraction of smooth muscles of a wall of the vessel; and

(b) defining a second parameter as an active compliance for the vessel, said active compliance varying with internal pressure, external pressure, and said first parameter.

31. A method according to claim 30, further comprising relating said first parameter and said second parameter via a differential equation.

32. A method according to claim 31, wherein said relating step comprises relating said first parameter and said second parameter according to the following equation:

{dot over (V)}=C·

({overscore (P)}+{dot over (P)}*))

wherein {dot over (V)} is a time derivative of a volume of said vessel, {dot over (P)} is a time derivative of the pressure inside said vessel, {dot over (φ

)} is a time derivative of said first parameter, {dot over (P)}* is a time derivative of the pressure outside of said vessel, and C is said second parameter.

33. A method according to claim 32, wherein said second parameter is defined according to the following equation:

$\mathrm{??}=\frac{r<br><br>\text{\hspace{1em}<br><br>}<br><br>V^{\mathrm{Max}}<br><br>{\mathrm{e<br><br>}}^{-r<br><br>\left(P-\left(\mathrm{\varphi <br><br>}+P^{*}\right)\right)}}{{\left(1+{\mathrm{e<br><br>}}^{-r<br><br>\left(P-\left(\mathrm{\varphi <br><br>}+P^{*}\right)\right)}\right)}^2}$

wherein C is said second parameter, r>

0, V^Max is a maximum volume of said vessel, P is the pressure inside said vessel, P* is the pressure outside said vessel, and φ

is said first parameter.

34. A method according to claim 30, further comprising using said first parameter and said second parameter in modeling a circulatory system.

35. A computer readable medium containing computer executable instructions implementing a method of modeling a pressure and volume relationship in a compliant vessel, the instructions comprising:

(a) a first set of instructions for defining a first parameter as a change in pressure within the vessel, said change in pressure being due to a contraction of smooth muscles of a wall of the vessel; and

(b) a second set of instructions for defining a second parameter as an active compliance for the vessel, said active compliance varying with internal pressure, external pressure, and said first parameter.

36. A computer readable medium according to claim 35, further comprising a third set of instructions for relating said first parameter and said second parameter via a differential equation.

37. A computer readable medium according to claim 36, wherein said third set of instructions comprises a fourth set of instructions for relating said first parameter and said second parameter according to the following equation:

{dot over (V)}=C·

({dot over (P)}−

({dot over (φ

)}+{dot over (P)}*))

wherein {dot over (V)} is a time derivative of a volume of said vessel, {dot over (P)} is a time derivative of the pressure inside said vessel, {dot over (φ

)} is a time derivative of said first parameter, {dot over (P)}* is a time derivative of the pressure outside of said vessel, and C is said second parameter.

38. A computer readable medium according to claim 36, wherein said second set of instructions comprises a fifth set of instructions for defining said second parameter according to the following equation:

$\mathrm{??}=\frac{r<br><br>\text{\hspace{1em}<br><br>}<br><br>V^{\mathrm{Max}}<br><br>{\mathrm{e<br><br>}}^{-r<br><br>\left(P-\left(\mathrm{\varphi <br><br>}+P^{*}\right)\right)}}{{\left(1+{\mathrm{e<br><br>}}^{-r<br><br>\left(P-\left(\mathrm{\varphi <br><br>}+P^{*}\right)\right)}\right)}^2}$

wherein C is said second parameter, r>

0, V^Max is a maximum volume of said vessel, P is the pressure inside said vessel, P* is the pressure outside said vessel, and φ

is said first parameter.

39. A computer readable medium according to claim 35, further comprising a sixth set of instructions for using said first parameter and said second parameter in modeling a circulatory system.