Bulletin of Mathematical Biology

, Volume 77, Issue 12, pp 2264–2293 | Cite as

Mathematical Model of Cardiovascular and Metabolic Responses to Umbilical Cord Occlusions in Fetal Sheep

  • Qiming Wang
  • Nathan Gold
  • Martin G. Frasch
  • Huaxiong HuangEmail author
  • Marc Thiriet
  • Xiaogang Wang
Original Article


Fetal acidemia during labor is associated with an increased risk of brain injury and lasting neurological deficits. This is in part due to the repetitive occlusions of the umbilical cord (UCO) induced by uterine contractions. Whereas fetal heart rate (FHR) monitoring is widely used clinically, it fails to detect fetal acidemia. Hence, new approaches are needed for early detection of fetal acidemia during labor. We built a mathematical model of the UCO effects on FHR, mean arterial blood pressure (MABP), oxygenation and metabolism. Mimicking fetal experiments, our in silico model reproduces salient features of experimentally observed fetal cardiovascular and metabolic behavior including FHR overshoot, gradual MABP decrease and mixed metabolic and respiratory acidemia during UCO. Combined with statistical analysis, our model provides valuable insight into the labor-like fetal distress and guidance for refining FHR monitoring algorithms to improve detection of fetal acidemia and cardiovascular decompensation.


Mathematical modeling Fetal blood circulation Fetal acidemia Hemodynamics Labor Neural control Sensitivity analysis 



Fetal heart rate


Mean arterial blood pressure


Umbilical cord occlusions


Base deficit


Root-mean-square of the successive differences

List of Symbols


Coefficient related to ventricular elastance during relaxation (mmHg/ml\(^2\))

\(\alpha _b\)

Maximum binding capacity of hemoglobin (ml \(\mathrm {O_2}\)/g Hb)


Parameter related to ventricular volume for zero diastolic pressure (ml)

\(\beta _d\)

Scaling factor on dissolved oxygen (1/mmHg)

\(\beta _h(H)\)

Time for the onset of ventricular relaxation (s)

\(\eta \)

Parameter controlling the steepness of the relation for heart pressure

\(\theta \)

Median value of time for peak pressure in heart model (1/s)

\(\phi \)

Median value of heart pressure in heart model (1/s)

\(\psi \)

Scaling factor in mean arterial pressure (1/s)

\(\mathrm {[CO_2]}_i\)

Carbon dioxide concentration in compartment \(i=\mathrm{nc,c,um}\) (ml \(\mathrm {CO_2}\)/ml blood)

\(\mathrm {[CO_2]}_{a}\)

Feeding Carbon dioxide concentration in fetus (ml \(\mathrm {CO_2}\)/ml blood)

\(c_\mathrm{f}, c_\mathrm{m}\)

Heart contractility for fetus/mother (mmHg/ml)

\(c_{1,\mathrm m}, c_{1,\mathrm f}\)

Scaling constant in the relation between Oxygen content and its partial pressure for mother and fetus (\(\hbox {mmHg}^3\))

\(c_{2,\mathrm{m}}, c_{2,\mathrm{f}}\)

Scaling constant in the relation between Oxygen content and its partial pressure for mother and fetus (\(\hbox {mmHg}^2\))


Compliance in compartment \(i=\mathrm{nc,c,um,ut}\) (ml/mmHg)

\(d_\mathrm{f}, d_\mathrm{m}\)

Parameter related to the volume-dependent and volume-independent components of the developed pressure (mmHg)

\(\delta G_{T}\)

Scaling factor for sympathetic gain (\(\hbox {s}^2\))


Mass transfer coefficient in Oxygen model (\(\hbox {ml }\mathrm {O_2}\hbox {/s/mmHg}\))


Mass transfer coefficient in Carbon dioxide model (\(\hbox {ml }\mathrm {CO_2} \hbox {/s/mmHg}\))

\(\Delta C_v\)

Effector in venous compliance (ml/mmHg)

\(\Delta c\)

Effector in heart contractility (mmHg/ml)

\(f_{\mathrm{br}}, f_{\mathrm{cr}}\)

Afferent firing rate via stimulation of baro/chemoreceptor (1/s)


Offset values that create a threshold for vagal or sympathetic response \(i=\mathrm{va},s\beta ,s\alpha \) (1/s)

\(f_{i,\mathrm{min}}, f_{i,\mathrm{max}}\)

Control constants in sigmoid functions \(i=s,\hbox {va},\hbox {hy},\hbox {br},\hbox {cr}\) (1/s)


A normal value for corresponding afferent or efferent firing rate \(i=\hbox {br}, \hbox {cr}, \hbox {va},s\beta ,s\alpha \) (1/s)

\(f_\mathrm{s,0}, f_{\mathrm{s},\infty }\)

Constants in sympathetic firing rate function (1/s)

\(f_{\mathrm{s}\beta }, f_{\mathrm{s}\alpha }\)

Efferent \(\alpha \)- and \(\beta \)- sympathetic firing rate (1/s)


Control parameter in sympathetic firing rate function (1/s)


Efferent vagal firing rate (1/s)


Gain constants \(i=T_\mathrm{va}, T_s\) (\(\hbox { s}^2\))

\(G_c, G_\mathrm{va}\)

Constants in effectors (mmHg s/ml)

\(\mathrm {[GL]}_a\)

Arterial glucose concentration (mM)

\(\mathrm {[GL]}_i\)

Glucose concentration in compartment \(i=\hbox {nc}, \hbox {c}, \hbox {um}\) (mM)


Activation function in heart model


Heart rate (1/s)

\(\mathrm {[H^+]}_a\)

Arterial proton concentration (nM)

\(\mathrm {[H^+]}_i\)

Proton concentration in compartment \(i=\hbox {nc}, \hbox {c}, \hbox {um}\) (nM)

\(Hb_\mathrm{f}, Hb_\mathrm{m}\)

Hemoglobin concentration in fetus or mother (g Hb/ml blood)


Constant in Oxygen metabolic model (ml blood/s)


Kinetic constant (mmol/s)


Kinetic constant \(i=2,3,4,5,6\) (ml/s)


Kinetic constant (\(10^{-6}\,\hbox {ml/ s}\))

\(K_{8}, K_9\)

Kinetic constant (nmol/s)


Kinetic constant (\(10^{6}\,\hbox {ml/s}\))

\(K_{\mathrm {CO_2}}\)

Scaling constants (ml \(\mathrm {CO_2}\)/ml blood/mmHg)

\(k_{\mathrm {CO_2}}\)

Scaling constants ml (\(\mathrm {CO_2}/\hbox {ml blood}\))


Control constant

\(k_\mathrm{hy}, k_{\mathrm{pr},s}\)

Control constant (mmHg)


Constant \(i=R_\mathrm{nc},\hbox {br}, \hbox {cr}, \hbox {s}, \hbox {va}\) (1/s)

\(\mathrm {[LA]}_{a}\)

Arterial lactate concentration (mM)

\(\mathrm {[LA]}_{i}\)

Lactate concentration in compartment \(i=\hbox {nc}, \hbox {c}, \hbox {um}\) (mM)


Production rate of \(\mathrm {[CO_2]_i}\) with \(i=\hbox {nc}, \hbox {c}, \hbox {um}\) (ml/s)


Production rate due to \(\mathrm {[H^+]}\) accumulation (ml/s)


Normalization factor (s)

\(\nu \)

Parameter related to steepness of the time for peak pressure in heart model


Parameters related to the onset time of ventricular relaxation in heart model

\(\mathrm {[O_{2}]}_\mathrm{th}\)

Threshold value to control Oxygen metabolic uptake (\(\hbox {ml }\mathrm {O_2}/ \hbox {ml blood}\))

\(\mathrm {[O_{2}]}_{a}, \mathrm {[O_{2}]}_{m,a}\)

Feeding Oxygen concentration in fetus and mother (\(\hbox {ml }\mathrm {O_2}/\hbox {ml blood}\))

\(\mathrm {[O_2]}_{a,n}\)

A normal value to control the sigmoid function in systemic resistance (\(\hbox {ml }\mathrm {O_2}/\hbox {ml blood}\))

\(\mathrm {[O_2]_i}\)

Oxygen concentration in compartment \(i=\hbox {nc}, \hbox {c}, \hbox {um}, \hbox {ivs}\) (ml \(\mathrm {O_2}/\hbox {ml blood}\))

\(\mathrm {O_{\mathrm{met},i}}\)

Metabolic Oxygen uptake in compartment \(i=\hbox {nc, c}\) (ml \(\mathrm {O_2}/\hbox {s}\))


Control constants in heart model (mmHg)


Blood pressure in compartment \(i=\hbox { nc, c, um, h, ut, ivs}\) (mmHg)

\(\bar{p}_a\) (MABP)

Mean arterial blood pressure (mmHg)

\(\mathrm {P}_{\mathrm {O_2},i}\)

Oxygen partial pressure in compartment \(i=\hbox {nc, c, um, ivs}\) (mmHg)

\(\mathrm {P}_{\mathrm {CO_2},i}\)

Carbon dioxide partial pressure in compartment \(i=\hbox {nc, c, um}\) (mmHg)

\(\mathrm {P}_{\mathrm {CO_2},\hbox {nc, n}}\)

A normal value for Carbon dioxide partial pressure in systemic compartment (mmHg)

\(\mathrm {P}_{\mathrm {O_{2}},c,0}\)

Threshold value for cerebral Oxygen partial pressure to control the sigmoid function in firing rate function \(f_\mathrm{hy}\) (mmHg)

\(\mathrm {[PY]_a}\)

Arterial pyruvate concentration (mM)

\(\mathrm {[PY]_i}\)

Pyruvate concentration in compartment \(i=\hbox {nc, c, um}\) (mM)


Arterial flow rate into compartment \(i=\hbox {nc, c, um}\) (ml/s)


Venous flow rate out of compartment \(i=\hbox {nc, c, um}\) (ml/s)

\(q_\mathrm{in}, q_\mathrm{out}\)

Flow rate in/out of corresponding compartment by summing over flow rate from local branches (ml/s)


Resistance in compartment \(i=\mathrm{nc, c, um, ut, ivs, m}\) (mmHg s/ml)

\(R_{c,a,0}, R_{cn,0}\)

Baseline values for cerebral and systemic resistance (mmHg s/ml)

\(R_{c, \mathrm{min}}, R_{c, \mathrm{max}}\)

Constants in cerebral autoregulation model (mmHg s/ml)

\(R_{\mathrm{nc, min}}, R_{\mathrm{nc, max}}\)

Constants in systemic resistance model (mmHg s/ml)

\(\Delta R_\mathrm{nc}\)

Effector in systemic resistance (mmHg s/ml)


Relative sensitivity matrix

\(S_{\mathrm {[O_2]},d}\)

Oxygen diffusion (ml \(\mathrm {O_2}/\hbox {s}\))

\(S_{\mathrm {P[O_2]}}\)

Scaling function to relate Oxygen content and its partial pressure

\(s_\mathrm{mv}, s_\mathrm{av}\)

Indicator function of mitral/aortic valve


Peak time (s)


Control constants in heart model (s)


Base line value for heart period (s)

\(T_{v,0}, T_{\mathrm{s},0}\)

Contributions from vagal and sympathetic activities on the base heart period (s)

\(\tau _i\)

Control constant \(i=\hbox {gs},\hbox {gv}\) (1/s)

\(\tau _0\)

Offset constant in heart period (s)

\(\tau _\mathrm{pr,br},\tau _\mathrm{pr,cr}\)

Control constant (mmHg)

\(\tau _\mathrm{br}, \tau _\mathrm{cr}, \tau _s, \tau _{Rc}\)

Time constant (s)

\(\tau _{T,s}, \tau _{T,v}\)

Time constant (s)

\(\Delta T_s,\Delta T_\mathrm{va}\)

Contribution to heart period from sympathetic/vagal activity (s)


Volume in compartment \(i=\hbox {nc, c, um, h, ut}\) (ml)


Total volume of all compartments in the circulation system (ml)

\(W_{\mathrm{br},i}, W_{\mathrm{cr},i}\)

Weight factor \(i=s\alpha , s\beta \)



The authors are grateful to Dr. Josh Chang for some useful discussions. This work is partially supported by MITACS, NeuroDevNet, NSERC, CIHR and the Fields Institute.


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Copyright information

© Society for Mathematical Biology 2015

Authors and Affiliations

  • Qiming Wang
    • 1
  • Nathan Gold
    • 1
  • Martin G. Frasch
    • 2
    • 3
    • 4
  • Huaxiong Huang
    • 1
  • Marc Thiriet
    • 5
  • Xiaogang Wang
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada
  2. 2.Department of Obstetrics and Gynecology, Faculty of MedicineCHU Sainte-Justine Research CenterMontréalCanada
  3. 3.Department of Neurosciences, Faculty of MedicineCHU Sainte-Justine Research CenterMontréalCanada
  4. 4.Centre de Recherche en Reproduction Animale (CRRA)Université de MontréalQCCanada
  5. 5.UPMC, Laboratoire Jacques-Louis Lions, CNRS, UMR 7598, INRIA, EPI REOSorbonne UniversityParisFrance

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