Subject-specific Model Estimation of Cardiac Output and Blood Volume During Hemorrhage

Abstract

We have developed a novel method for estimating subject-specific hemodynamics during hemorrhage. First, a mathematical model representing a closed-loop circulation and baroreceptor feedback system was parameterized to match the baseline physiology of individual experimental subjects by fitting model results to 1 min of pre-injury data. This automated parameterization process matched pre-injury measurements within 1.4  ±  1.3% SD. Tuned parameters were then used in similar open-loop models to simulate dynamics post-injury. Cardiac output (CO) estimates were obtained continuously using post-injury measurements of arterial blood pressure (ABP) and heart rate (HR) as inputs to the first open-loop model. Secondarily, total blood volume (TBV) estimates were obtained by summing the blood volumes in all the circulatory segments of a second open-loop model that used measured CO as an additional input. We validated the estimation method by comparing model CO results to flowprobe measurements in 14 pigs. Overall, CO estimates had a Bland-Altman bias of  −0.30 l/min with upper and lower limits of agreement 0.80 and  −1.40 l/min. The negative bias is likely due to overestimation of the peripheral resistance response to hemorrhage. There was no reference measurement of TBV; however, the estimates appeared reasonable and clearly predicted survival versus death during the post-hemorrhage period. Both open-loop models ran in real time on a computer with a 2.4 GHz processor, and their clinical applicability in emergency care scenarios is discussed.

Appendix

Equations of the Open-loop Cardiac Output Estimation Model

Lower-case subscript key:

• 0: unstressed volume

• a: left or right atrium

• d: diastolic

• i: left atrium, left ventricle, right atrium or right ventricle

• j: circulatory segment immediately downstream of the i segment

• s: systolic

• v: left or right ventricle

Four-chamber Heart

Heart chamber activation function

$$ y_{\rm i}= \left( \begin{array}{ll} \frac{1.0 - \cos\left(\frac{\pi \times t_{\rm i, REL}}{Ts_{\rm i}}\right)}{2.0}&\hbox{for}\,(0.0\le t_{\rm i, REL} < T_{S_{\rm i}})\\ \frac{1.0+\cos\left(2.0\times \pi \times {\frac{t_{\rm i, REL} - Ts_{\rm i}}{{Ts_{\rm i}}}}\right)}{2.0}&\hbox{for}\,(Ts_{\rm i}\le t_{\rm i, REL} < 1.5 \times Ts_{\rm i})\\ 0&\hbox{for}\,(t_{\rm i, REL}\ge 1.5 \times Ts_{\rm i})\\ \end{array} \right.$$

(A.1)

$$ t_{\rm a,REL}=t - t_{\rm PWAVE} $$

(A.2)

$$ t_{\rm v,REL}=t - t_{\rm RWAVE}$$

(A.3)

Discrete functions to set heart beat start time and heart period

In the model the x domain was created as the set of all positive integers, and used to access R wave event times and heart periods indexed by heartbeat number in an external data file created from empirical ECG data. The functions m and n increased by 1 after each heart cycle completed, and t _HB was set to the R wave event time for each heartbeat. The heart period HP was also set discretely for each indexed heartbeat.

$$ m=m(x) $$

(A.4)

$$ n=n(x) $$

(A.5)

$$ t_{\rm HB}=t_{\rm HB}(x) $$

(A.6)

$$ {HP}={HP}(x) $$

(A.7)

for $(t\ge t_{\rm HB}(n+1) - \hbox{PR}_{\rm INT} - offv)\{$

$$ \hbox{HR}_{\rm a}=\frac{1}{{HP}(n+1)} $$

(A.8)

$$ Ts_{\rm a}=Ts1_{\rm a}\times \sqrt{Ts2/\hbox{HR}_{\rm a}} $$

(A.9)

$$ t_{\rm PWAVE}=t_{\rm HB}(n+1) - \hbox{PR}_{\rm INT} - offv $$

(A.10)

$$ n=n+1 \} $$

(A.11)

for $(t\ge t_{\rm HB}(m+1) - offv)\{$

$$ \hbox{HR}_{\rm v}=\frac{1}{HP(m+1)} $$

(A.12)

$$ Ts_{\rm v}=Ts1_{\rm v}\times\sqrt{Ts2/\hbox{HR}_{\rm v}} $$

(A.13)

$$ t_{\rm RWAVE}=t_{\rm HB}(m+1) - \,offv $$

(A.14)

$$ m=n $$

(A.15)

$$ Vvar_{\rm vs0}=\left( \begin{array}{ll} V_{\rm vd0}&\hbox{for}\,(V_{\rm v} < V_{\rm vd0})\\ V_{\rm vs0}&\hbox{for}\,(V_{\rm v} > \hbox{EDV}_{\rm v})\\ \frac{(V_{vs0} - V_{\rm vd0})\times (V_v - V_{\rm vd0})}{(\hbox{EDV}_v - V_{\rm vd0})}+V_{\rm vd0}&\hbox{for}\,(V_{\rm vd0}\le V_{\rm v}\le \hbox{EDV}_{\rm v})\\ \end{array}\right. $$

(A.16)

$$ af_{\rm CON2}=af_{\rm CON} \} $$

(A.17)

$$ \hbox{HR}=\hbox{HR}_{\rm v} $$

(A.18)

Heart chamber pressure-volume relationships

$$ P_{\rm v}=E_{\rm v}\times (V_{\rm v} - V_{\rm v0})\times af_{\rm CON2} - \Uppsi $$

(A.19)

$$ P_{\rm a}=E_{\rm a}\times (V_{\rm a} - V_{\rm a0}) - \Uppsi $$

(A.20)

$$ E_{\rm i}=y_{\rm i}\times (E_{\rm MAX,i} - E_{\rm MIN,i})+E_{\rm MIN,i} $$

(A.21)

$$ E_{\rm MAX,v}=\hbox{KE}_{\rm v}\times E_{\rm MAX,v1} $$

(A.22)

$$ V_{\rm v0}=(1 - y_{\rm v})\times (V_{\rm vd0} - Vvar_{\rm vs0})+Vvar_{\rm vs0} $$

(A.23)

$$ V_{\rm a0}=(1 - y_{\rm a})\times(V_{\rm ad0} - V_{\rm as0})+V_{\rm as0} $$

(A.24)

Heart flows from the ith to jth chamber

$$ F_{\rm i}= \begin{array}{ll} \frac{P_{\rm i} - P_{\rm j}}{R_{\rm i}}&\hbox{for}\,(P_{\rm i} > P_{\rm j})\\ 0&\hbox{for}\,(P_{\rm i}\le P_{\rm j})\\ \end{array} $$

(A.25)

$$ \frac{{\rm d}V_{\rm i}}{{\rm d}t}=F\hbox{in}_{\rm i} - F\hbox{out}_{\rm i} $$

(A.26)

Systemic Circulation

Aortic flow estimate

$$ \frac{{\rm d}\hbox{AOF}_{\rm MOD}}{{\rm d}t}=(\hbox{MAP}_{\rm MEAS} - \hbox{MAP}_{\rm MOD})\times K_{\rm CO,MAP} $$

(A.27)

Smoothed pulmonary valve flow

$$ \frac{d{F_{RV,SM}}}{dt}=\frac{F_{\rm RV} - F_{\rm RV,SM}}{\tau_{\rm CO}} $$

(A.28)

Cardiac output estimate

$$ \hbox{CO}_{\rm MOD}=F_{\rm RV,SM} $$

(A.29)

Stroke volume

$$ \hbox{SV}=\hbox{CO}_{\rm MOD}\times \hbox{HR} $$

(A.30)

Time-shifted, measured arterial blood pressure

$$ \hbox{ABP}_{\rm SHIFT}=\hbox{ABP}_{\rm MEAS}(t+offv) $$

(A.31)

Approximate “derivative” of arterial blood pressure, produced using a lag operator with a small time constant

$$ \frac{{\rm d}\hbox{ABP}_{\rm FOL}}{{\rm d}t}=\frac{\hbox{ABP}_{\rm SHIFT} - \hbox{ABP}_{\rm FOL}}{\tau_{\rm ABP}} $$

(A.32)

Tuned scaling factor for systemic veins PV relationship

$$ K_{\rm V}=K_{\rm V1}\times K_{\rm SV} $$

(A.33)

Aortic afterload set by time-shifted arterial blood pressure data

$$ P_{\rm AOD}=\hbox{ABP}_{\rm SHIFT} $$

(A.34)

Systemic circulation pressures

$$ \frac{{\rm d}P_{\rm AOP}}{{\rm d}t}=\frac{F_{\rm LV} - \frac{{\rm d}V_{\rm AOP}}{{\rm d}t} - F_{\rm AOP} - F_{\rm COREPI}}{C_{\rm COREPI}} $$

(A.35)

$$ \hbox{MAP}_{\rm MOD}= \frac{R_{\rm CRB}\times \left[(R_{\rm TAOD}\times\hbox{AOF}_{\rm MOD}) - (F_{\rm AOD}\times R_{\rm TAOD})+\frac{V_{\rm AOD} - V_{\rm AOD,0}}{C_{\rm AOD}} - \Uppsi\right]+(P_{\rm VC}\times R_{\rm TAOD})}{R_{\rm CRB}+R_{\rm TAOD}} $$

(A.36)

$$ P_{\rm SAP}=(V_{\rm SAP} - V_{\rm SAP,0})/C_{\rm SAP} - \Uppsi $$

(A.37)

Ψ is given by $\Uppsi=K_{\rm XP}/(e^{v/k_{XV}} - 1)$ where V is volume and K _XP and K _XV are curve-shaping constants (see Table 3).

$$ P_{\rm SA,A}=Kc\times\log\left[\frac{V_{\rm SA} - V_{\rm SA,0}}{D_0}+1\right] $$

(A.38)

$$ P_{\rm SA,P}=Kp1\times e^{[\tau_{\rm P}\times (V_{\rm SA} - V_{\rm SA,0})]}+Kp2\times (V_{\rm SA} - V_{\rm SA,0})^2 $$

(A.39)

$$ P_{\rm SA}=(f_{\rm VASO}\times P_{\rm SA,A})+[(1 - f_{\rm VASO})\times P_{\rm SA,P}]$$

(A.40)

$$ P_{\rm SC}=(V_{\rm SC} - V_{\rm SC,0})/C_{\rm SC} - \Uppsi $$

(A,41)

$$ P_{\rm SV}= - K_{\rm V}\times \log \left[\frac{V_{\rm MAX,SV}}{V_{\rm SV}} - 0.99\right] $$

(A.42)

$$ P_{\rm VC}= \begin{array}{ll} K1\times (V_{\rm VC} - V_{\rm{VC},0}) - \Uppsi& \hbox{for}\,(V_{\rm VC} > V_{\rm{VC},0})\\ D2+\left[ K2\times e^{\frac{V_{\rm VC}}{V_{\rm MIN,VC}}}\right] - \Uppsi& \hbox{for}\,(V_{\rm VC}\le V_{\rm{VC},0})\\ \end{array} $$

(A.43)

Systemic circulation forward flow

$$ \frac{{\rm d}F_{\rm AOP}}{{\rm d}t}= \frac{P_{\rm AOP} - (F_{\rm AOP}\times R_{\rm AOP}) - \hbox{ABP}_{\rm MEAS}}{L_{\rm AOP}} $$

(A.44)

$$ \frac{{\rm d}F_{\rm AOD}}{{\rm d}t}= \frac{\hbox{MAP}_{\rm MOD} - (F_{\rm AOD}\times R_{\rm AOD}) - P_{\rm SAP}}{L_{\rm AOD}} $$

(A.45)

$$ F_{\rm CRB}=(\hbox{MAP}_{\rm MOD} - P_{\rm VC})/R_{\rm CRB} $$

(A.46)

$$ F_{\rm SAP}=(P_{\rm SAP} - P_{\rm SA})/R_{\rm SAP} $$

(A.47)

$$ F_{\rm SA}=(P_{\rm SA} - P_{\rm SC})/R_{\rm SA} $$

(A.48)

$$ F_{\rm SC}=(P_{\rm SC} - P_{\rm SV})/R_{\rm SC} $$

(A.49)

$$ F_{\rm SV}=(P_{\rm SV} - P_{\rm VC})/R_{\rm SV} $$

(A.50)

$$ F_{\rm VC}=(P_{\rm VC} - P_{\rm RA})/R_{\rm VC} $$

(A.51)

Systemic circulation radial flow

$$ \frac{{\rm d}V_{\rm AOP}}{{\rm d}t}=\frac{P_{\rm AOP} - \frac{V_{\rm AOP} - V_{\rm AOP,0}}{c_{\rm AOP}}}{R_{\rm TAOP}} $$

(A.52)

$$ \frac{{\rm d}V_{\rm AOP}}{{\rm d}t}=\hbox{AOF}_{\rm MOD} - F_{\rm AOD} - F_{\rm CRB} $$

(A.53)

$$ \frac{{\rm d}V_{\rm SA}}{{\rm d}t}=F_{\rm SAP} - F_{\rm SA} $$

(A.54)

$$ \frac{{\rm d}V_{\rm SAP}}{{\rm d}t}=F_{\rm AOD} - F_{\rm SAP} $$

(A.55)

$$ \frac{{\rm d}V_{\rm SC}}{{\rm d}t}=F_{\rm SA} - F_{\rm SC} $$

(A.56)

$$ \frac{{\rm d}V_{\rm SV}}{{\rm d}t}=F_{\rm SC} - F_{\rm SV} $$

(A.57)

$$ \frac{{\rm d}V_{\rm VC}}{{\rm d}t}=F_{\rm SV} - F_{\rm CRB} - F_{\rm VC} $$

(A.58)

Nonlinear systemic resistances

$$ R_{\rm SA}=[Kr\times e^{(4\times f_{\rm VASO})}]+\left[ Kr+\left({\frac{V_{\rm SA,MAX}}{V_{\rm SA}}}^2\right)\right]+R_{\rm SA0} $$

(A.59)

$$ R_{\rm VC}=\left[ \hbox{KR}\times\left( {\frac{V_{\rm MAX,VC}}{V_{\rm VC}}}\right)^2\right]+R_0 $$

(A.60)

Pulmonary Circulation

Pulmonary circulation pressures

$$ P_{\rm PAP}= \left(\begin{array}{ll}P_{\rm PAP1} =\frac{(R_{\rm TPAP}\times P_{\rm RV}) - (R_{\rm RV}\times F_{\rm PAP}\times R_{\rm TPAP})+\left[R_{\rm RV}\times\frac{V_{\rm PAP} - V_{\rm PAP,0}}{C_{\rm PAP}} - \Uppsi\right]}{R_{\rm TPAP}+R_{\rm RV}}& \hbox{for}\,(P_{\rm RV} > P_{\rm PAP1})\\P_{\rm PAP2}= \frac{\left[R_{\rm RV}\times\frac{V_{\rm PAP} - V_{\rm PAP,0}}{C_{\rm PAP}} - \Uppsi\right](R_{\rm RV}\times F_{\rm PAP}\times R_{\rm TPAP})}{R_{\rm RV}}& \hbox{for}\,(P_{\rm RV}\le P_{\rm PAP1})\\ \end{array} \right.$$

(A.61)

$$ P_{\rm PAD}=(F_{\rm PAP}\times R_{\rm TPAD}) - (F_{\rm PAD}\times R_{\rm TPAD})+\frac{V_{\rm PAD} - V_{\rm PAD,0}}{C_{\rm PAD}} - \Uppsi $$

(A.62)

$$ P_{\rm PA}=(V_{\rm PA} - V_{\rm PA,0})/C_{\rm PA} - \Uppsi $$

(A.63)

$$ P_{\rm PC}=(V_{\rm PC} - V_{\rm PC,0})/C_{\rm PC} - \Uppsi $$

(A.64)

$$ P_{\rm PV}=(V_{\rm PV} - V_{\rm PV,0})/C_{\rm PV} - \Uppsi $$

(A.65)

Pulmonary circulation forward flows

$$ F_{\rm PS}=(P_{\rm PA} - P_{\rm PV})/R_{\rm PS} $$

(A.66)

$$ F_{\rm PA}=(P_{\rm PA} - P_{\rm PC})/R_{\rm PA} $$

(A.67)

$$ F_{\rm PC}=(P_{\rm PC} - P_{\rm PV})/R_{\rm PC} $$

(A.68)

$$ F_{\rm PV}=(P_{\rm PV} - P_{\rm LA})/R_{\rm PV} $$

(A.69)

Pulmonary circulation radial flows

$$ \frac{{\rm d}V_{\rm PAD}}{{\rm d}t}=F_{\rm PAP} - F_{\rm PAD} $$

(A.70)

$$ \frac{{\rm d}V_{\rm PAP}}{{\rm d}t}=F_{\rm RV} - F_{\rm PAP} $$

(A.71)

$$ \frac{{\rm d}V_{\rm PA}}{{\rm d}t}=F_{\rm PAD} - F_{\rm PS} - F_{\rm PA} $$

(A.72)

$$ \frac{{\rm d}V_{\rm PC}}{{\rm d}t}=F_{\rm PA} - F_{\rm PC} $$

(A.73)

$$ \frac{{\rm d}V_{\rm PV}}{{\rm d}t}=F_{\rm PC}+F_{\rm PS} - F_{\rm PV} $$

(A.74)

$$ \frac{{\rm d}F_{\rm PAP}}{{\rm d}t}=\frac{P_{\rm PAP} - P_{\rm PAD} - (F_{\rm PAP}\times R_{\rm PAP})}{L_{\rm PAP}} $$

(A.75)

$$ \frac{{\rm d}F_{\rm PAD}}{{\rm d}t}=\frac{P_{\rm PAD} - P_{\rm PA} - (F_{\rm PAD}\times R_{\rm PAD})}{L_{\rm PAD}} $$

(A.76)

Coronary Circulation

Coronary circulation pressures

$$ P_{\rm COREPI}=P_{\rm AOP} $$

(A.77)

$$ P_{\rm CORINTRA}=(V_{\rm CORINTRA} - V_{\rm CORINTRA,0})/C_{\rm CORINTRA} - \Uppsi $$

(A.78)

$$ P_{\rm CORCAP}=(V_{\rm CORCAP} - V_{\rm CORCAP,0})/C_{\rm CORCAP} - \Uppsi $$

(A.79)

$$ P_{\rm CORVN}=(V_{\rm CORVN} - V_{\rm CORVN,0})/C_{\rm CORVN} - \Uppsi $$

(A.80)

$$ P_{\rm CORINTRA,C}=P_{\rm CORINTRA}+P_{\rm IM} $$

(A.81)

$$ P_{\rm CORCAP,C}=P_{\rm CORCAP}+P_{\rm IM} $$

(A.82)

$$ P_{\rm CORVN,C}=P_{\rm CORVN} $$

(A.83)

$$ P_{\rm IM}=|P_{\rm LV}/2| $$

(A.84)

Coronary circulation flows

$$ F_{\rm COREPI}=(P_{\rm COREPI} - P_{\rm CORINTRA,C})/R_{\rm COREPI} $$

(A.85)

$$ F_{\rm CORINTRA}=(P_{\rm CORINTRA,C} - P_{\rm CORCAP,C})/R_{\rm CORINTRA} $$

(A.86)

$$ F_{\rm CORCAP}=(P_{\rm CORCAP,C} - P_{\rm CORVN,C})/R_{\rm CORCAP} $$

(A.87)

$$ F_{\rm CORVN}=(P_{\rm CORVN,C} - P_{\rm RA})/R_{\rm CORVN} $$

(A.88)

$$ \frac{{\rm d}V_{\rm COREPI}}{{\rm d}t}=F_{\rm LV} - \frac{{\rm d}V_{\rm AOP}}{{\rm d}t} - F_{\rm AOP} - F_{\rm COREPI} $$

(A.89)

$$ \frac{{\rm d}V_{\rm CORINTRA}}{{\rm d}t}=F_{\rm COREPI} - F_{\rm CORINTRA} $$

(A.90)

$$ \frac{{\rm d}V_{\rm CORCAP}}{{\rm d}t}=F_{\rm CORINTRA} - F_{\rm CORCAP} $$

(A.91)

$$ \frac{{\rm d}V_{\rm CORVN}}{{\rm d}t}=F_{\rm CORCAP} - F_{\rm CORVN} $$

(A.92)

Baroreceptor

Transfer function for carotid sinus firing frequency

$$ \begin{array}{l} a2\times a \times \frac{{\rm d}^2}{{\rm d}t}\hbox{Nbr}+\left[(a2 + a) \times \frac{{\rm d}}{{\rm d}t}\hbox{Nbr}\right]+\hbox{Nbr} \\ =(K\times \hbox{ABP}_{\rm MEAS})+\left(a1\times K \times \frac{{\rm d}\hbox{ABP}_{\rm FOL}}{{\rm d}t}\right)\\ \end{array} $$

(A.93)

Equations of efferent pathways

$$ \frac{{\rm d}N_{\rm CON}}{{\rm d}t}= \begin{array}{ll} \frac{- N_{\rm CON}[K_{\rm CON}\times \hbox{Nbr}(t - l_{\rm CON})]}{T_{\rm CON}}& \hbox{for}\,(t - t_{\rm MIN} > l_{\rm CON})\\ 0&\hbox{for}\,(t - t_{\rm MIN}\le l_{\rm CON})\\ \end{array} $$

(A.94)

$$ \frac{dN_{\rm VASO}}{dt}= \begin{array}{ll} \frac{- N_{\rm VASO}[K_{\rm VASO}\times \hbox{Nbr}(t - l_{\rm VASO})]}{T_{\rm VASO}}& \hbox{for}\,(t - t_{\rm MIN} > l_{\rm VASO})\\ 0&\hbox{for}\,(t - t_{\rm MIN}\le l_{\rm VASO})\\ \end{array} $$

(A.95)

$$ f_{\rm CON}=a_{\rm CON}+\frac{b_{\rm CON}}{e^{\tau_{\rm CON}\times(N_{\rm CON} - No_{\rm CON})}+1} $$

(A.96)

$$ f_{\rm VASO}=a_{\rm VASO}+\frac{b_{\rm VASO}}{e^{\tau_{\rm VASO}\times(N_{\rm VASO} - No_{\rm VASO})}+1} $$

(A.97)

$$ b_{\rm VASO}=1 - a_{\rm VASO} $$

(A.98)

$$ af_{\rm CON}= a_{\rm MIN}+(Ka \times f_{\rm CON}) $$

(A.99)

Blood Volumes

Total blood volume

$$ \hbox{TBV}=V_{\rm HEART}+V_{\rm SYSART}+V_{\rm SC}+V_{\rm SYSVEN}+V_{\rm PULART}+V_{\rm PC}+V_{\rm PV} $$

(A.100)

Blood volume in heart

$$ V_{\rm HEART}=V_{\rm RA}+V_{\rm RV}+V_{\rm LA}+V_{\rm LV}+V_{\rm CORCIRC} $$

(A.101)

Blood volume in coronary circulation

$$ V_{\rm CORCIRC}=V_{\rm COREPI}+V_{\rm CORINTRA}+V_{\rm CORCAP}+V_{\rm CORVN} $$

(A.102)

Blood volume in systemic arterial system

$$ V_{\rm SYSART}=V_{\rm AOP}+V_{\rm AOD}+V_{\rm SAP}+V_{\rm SA} $$

(A.103)

Blood volume in systemic venous system

$$ V_{\rm SYSVEN}=V_{\rm SV}+V_{\rm VC} $$

(A.104)

Blood volume in pulmonary arterial system

$$ V_{\rm PULART}=V_{\rm PAP}+V_{\rm PAD}+V_{\rm PA} $$

(A.105)