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Towards optimum chest compression performance during constant peak displacement cardiopulmonary resuscitation

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Abstract

The aim of this study is to determine the conditions necessary to achieve optimum chest compression (CC) performance during constant peak displacement cardiopulmonary resuscitation (CPR). This was accomplished by first performing a sensitivity analysis on a theoretical constant peak displacement CPR CC model to identify the parameters with the highest sensitivity. Next, the most sensitive parameters were then optimized for net sternum-to-spine compression depth, using a two-variable non-linear least squares method. The theoretical CC model was found to be most sensitive to: thoracic stiffness, maximum sternal displacement, CC rate, and back support stiffness. Based on a two-variable, non-linear least squares analysis to optimize the model for the net sternum-to-spine compression depth during constant peak displacement CPR, it was found that the optimum ranges for the CC rate and back support stiffness are between 40–120 cpm and 241.0–1198.5 Ncm−1, respectively. Clinically, this suggests that current ERC guidelines for the CC rate during peak displacement CPR are appropriate; however, practitioners should be aware that the stiffness of the back support surfaces found in many hospitals may be sub-optimal and should consider using a backboard or a concrete floor to enhance CPR effectiveness.

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Abbreviations

m 1 :

Sternal mass (g)

m 2 :

Thoracic mass (g)

μ 1 :

Sternal damping coefficient (Ncm−1 s−1)

μ 2 :

Back support damping coefficient (Ncm−1 s−1)

k 1 :

Sternal spring constant (Ncm−1)

k 2 :

Back support spring constant (Ncm−1)

r :

Residual (cm)

t :

Time (s)

x :

Displacement (cm)

x 1 :

Absolute sternal displacement (cm)

x 1,max :

Maximum sternal displacement (cm)

x 2 :

Back support displacement (cm)

ω :

Angular frequency of compression (rads−1)

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Author information

Authors and Affiliations

  1. Department of Mechanical and Mechatronic Engineering, University of Stellenbosch, Private Bag X1 Matieland, Stellenbosch, 7602, Western Cape, South Africa

    Kiran H. J. Dellimore, Garth Cloete & Cornie Scheffer

Authors
  1. Kiran H. J. Dellimore
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  2. Garth Cloete
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  3. Cornie Scheffer
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Corresponding author

Correspondence to Cornie Scheffer.

Appendix

Appendix

The equations governing the chest compression (CC) depth can be expressed in terms of the two most sensitive variables, back support stiffness (k 2) and CC rate, as follows:

$$ x_{1} = \frac{{x_{1,\max } }}{2}\left[ {1 - \cos \left( {\frac{{2\pi \;{\text{CCrate}} \cdot t}}{60}} \right)} \right] $$
(10)
$$ x_{2} = a\left( {{\text{CCrate}},k_{2} } \right)\sin \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right) + b\left( {{\text{CCrate}},k_{2} } \right)\cos \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right) + {F \mathord{\left/ {\vphantom {F {\left( {k_{1} + k_{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {k_{1} + k_{2} } \right)}} $$
(11)

In Eqs. 10 and 11, a(CCrate, k 2) implies a = f(CCrate, k 2) and b(CCrate, k 2 ) implies b = f(CCrate, k 2) where f indicates ‘function of’, such that a(CCrate, k 2) and b(CCrate, k 2) are defined as follows:

$$ a\left( {{\text{CCrate}},k_{2} } \right) = \frac{{\left( {\frac{{\mu_{1} x_{1,\max } }}{2}} \right)\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)\left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right] + BE\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)}}{{B^{2} \left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} + \left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right]^{2} }} $$
(12)
$$ b\left( {{\text{CCrate}},k_{2} } \right) = \frac{{E\left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right] - B\left( {\frac{{\mu_{1} x_{1,\max } }}{2}} \right)\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} }}{{B^{2} \left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} + \left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right]^{2} }} $$
(13)

Subtracting Eqs. 11 from 10 yields an expression for the net sternum-to-spine compression depth, g = x 1 − x 2:

$$ \begin{gathered} g = x_{1} - x_{2} = \frac{{x_{1,\max } }}{2}\left[ {1 - \cos \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right)} \right] - a\left( {{\text{CCrate}},k_{2} } \right)\sin \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right) \hfill \\ - b\left( {{\text{CCrate}},k_{2} } \right)\cos \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right) - {F \mathord{\left/ {\vphantom {F {\left( {k_{1} + k_{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {k_{1} + k_{2} } \right)}} \hfill \\ \end{gathered} $$
(14)

Next, differentiating f in Eq. 14 with respect to each variable gives the Jacobian matrix:

$$ A = \left| {{{ \, \partial g\left( {{\text{CCrate}},k_{2} } \right)} \mathord{\left/ {\vphantom {{ \, \partial g\left( {{\text{CCrate}},k_{2} } \right)} {\partial {\text{CCrate}}}}} \right. \kern-\nulldelimiterspace} {\partial {\text{CCrate}}}}\,{{\partial g\left( {{\text{CCrate}},k_{2} } \right)} \mathord{\left/ {\vphantom {{\partial g\left( {{\text{CCrate}},k_{2} } \right)} {\partial k_{2} }}} \right. \kern-\nulldelimiterspace} {\partial k_{2} }}} \right| $$
(15)

where \(\partial g({\text{CCrate}},k_{2} )/\partial {\text{CCrate and}}\;\partial g({\text{CCrate}},k_{2} )/\partial k_{2}\) respectively, are given by:

$$ \begin{gathered} {{ \, \partial g\left( {{\text{CCrate}},k_{2} } \right)} \mathord{\left/ {\vphantom {{ \, \partial g\left( {{\text{CCrate}},k_{2} } \right)} {\partial {\text{CCrate}} = \frac{{x_{1,\max } }}{2} \cdot }}} \right. \kern-\nulldelimiterspace} {\partial {\text{CCrate}} = \frac{{x_{1,\max } }}{2} \cdot }}\sin \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right)\left( {\frac{2\pi \cdot t}{60}} \right) \hfill \\ \quad - a\left( {{\text{CCrate}},k_{2} } \right)\cos \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right)\left( {\frac{2\pi \cdot t}{60}} \right) -{{\partial a\left( {{\text{CCrate}},k_{2} } \right)} \mathord{\left/ {\vphantom {{\partial a\left( {{\text{CCrate}},k_{2} } \right)} \partial }} \right. \kern-\nulldelimiterspace} \partial }{\text{CCrate}} \cdot \sin \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right) \hfill \\ \quad + b\left( {{\text{CCrate}},k_{2} } \right)\sin \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right)\left( {\frac{2\pi \cdot t}{60}} \right) - {{\partial b\left( {{\text{CCrate}},k_{2} } \right)} \mathord{\left/ {\vphantom {{\partial b\left( {{\text{CCrate}},k_{2} } \right)} \partial }} \right. \kern-\nulldelimiterspace} \partial }{\text{CCrate}} \cdot \cos \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right) \hfill \\ \end{gathered} $$
(16)
$$ \begin{gathered} {{ \, \partial g\left( {{\text{CCrate}},k_{2} } \right)} \mathord{\left/ {\vphantom {{ \, \partial g\left( {{\text{CCrate}},k_{2} } \right)} {\partial k_{2} = {F \mathord{\left/ {\vphantom {F {\left( {k_{1} + k_{2} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {k_{1} + k_{2} } \right)^{2} }} - \sin \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right) \cdot }}} \right. \kern-\nulldelimiterspace} {\partial k_{2} = {F \mathord{\left/ {\vphantom {F {\left( {k_{1} + k_{2} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {k_{1} + k_{2} } \right)^{2} }} - \sin \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right) \cdot }}{{\partial a} \mathord{\left/ {\vphantom {{\partial a} {\partial k_{2} }}} \right. \kern-\nulldelimiterspace} {\partial k_{2} }}\left( {{\text{CCrate}},k_{2} } \right) \hfill \\ \quad - \cos \left( {\frac{{2\pi {\text{CCrate}} \cdot t}}{60}} \right) \cdot {{\partial b} \mathord{\left/ {\vphantom {{\partial b} {\partial k_{2} }}} \right. \kern-\nulldelimiterspace} {\partial k_{2} }}\left( {{\text{CCrate}},k_{2} } \right) \hfill \\ \end{gathered} $$
(17)

In Eqs. 16 and 17 \( {{\partial a} \mathord{\left/ {\vphantom {{\partial a} {\partial k_{2} }}} \right. \kern-\nulldelimiterspace} {\partial k_{2} }},\,{{\partial b} \mathord{\left/ {\vphantom {{\partial b} {\partial k_{2} }}} \right. \kern-\nulldelimiterspace} {\partial k_{2} }},\,{{\partial a} \mathord{\left/ {\vphantom {{\partial a} {\partial {\text{CCrate}}}}} \right. \kern-\nulldelimiterspace} {\partial {\text{CCrate}}}}\,{\text{and}}\,{{\partial b} \mathord{\left/ {\vphantom {{\partial b} {\partial {\text{CCrate}}}}} \right. \kern-\nulldelimiterspace} {\partial {\text{CCrate}}}} \) are defined as:

$$ \begin{gathered} {{ \, \partial a({\text{CCrate}},k_{2} )} \mathord{\left/ {\vphantom {{ \, \partial a({\text{CCrate}},k_{2} )} {\partial k_{2} = }}} \right. \kern-\nulldelimiterspace} {\partial k_{2} = }}\frac{{\left( {\frac{{\mu_{1} x_{1,\max } }}{2}} \right)\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)}}{{B^{2} \left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} + \left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right]^{2} }} \hfill \\ \quad - \left[ {2k_{1} + 2k_{2} - 2A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right] \hfill \\ \cdot \left[ {\frac{{\left( {\frac{{\mu_{1} x_{1,\max } }}{2}} \right)\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)\left( {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right) + BE\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)}}{{\left( {B^{2} \left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} + \left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right]^{2} } \right)^{2} }}} \right] \hfill \\ \end{gathered} $$
(18)
$$ \begin{gathered} {{ \, \partial b\left( {{\text{CCrate}},k_{2} } \right)} \mathord{\left/ {\vphantom {{ \, \partial b\left( {{\text{CCrate}},k_{2} } \right)} {\partial k_{2} = }}} \right. \kern-\nulldelimiterspace} {\partial k_{2} = }}\frac{E}{{B^{2} \left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} + \left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right]^{2} }} \hfill \\ \quad - \left[ {2k_{1} + 2k_{2} - 2A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right] \hfill \\ \cdot \left[ {\frac{{\left( {E\left( {k_{1} + k_{2} } \right) - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right) - B\left( {\frac{{\mu_{1} x_{1,\max } }}{2}} \right)\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} }}{{\left( {B^{2} \left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} + \left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right]^{2} } \right)^{2} }}} \right] \hfill \\ \end{gathered} $$
(19)
$$ \begin{gathered} {{ \, \partial a\left( {{\text{CCrate}},k_{2} } \right)} \mathord{\left/ {\vphantom {{ \, \partial a\left( {{\text{CCrate}},k_{2} } \right)} {\partial {\text{CCrate}} = }}} \right. \kern-\nulldelimiterspace} {\partial {\text{CCrate}} = }}\frac{{\left( {\frac{{\mu_{1} x_{1,\max } }}{2}} \right)\left( {\frac{2\pi }{60}} \right)\left( {k_{1} + k_{2} } \right) - 3A\left( {\frac{{\mu_{1} x_{1,\max } }}{2}} \right)\left( {\frac{2\pi }{60}} \right)^{3} {\text{CCrate}}^{2} + BE\left( {\frac{2\pi }{60}} \right)}}{{B^{2} \left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} + \left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right]^{2} }} \hfill \\ \quad - \left[ {2B^{2} \left( {\frac{2\pi }{60}} \right)^{2} {\text{CCrate}} + 2\left( {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right) \cdot \left( { - 2A} \right)\left( {\frac{2\pi }{60}} \right)^{2} {\text{CCrate}}} \right] \hfill \\ \cdot \left[ {\frac{{\left( {\frac{{\mu_{1} x_{1,\max } }}{2}} \right)\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)\left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right] + BE\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)}}{{\left( {B^{2} \left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} + \left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right]^{2} } \right)^{2} }}} \right] \hfill \\ \end{gathered} $$
(20)
$$ \begin{gathered} {{ \, \partial b\left( {{\text{CCrate}},k_{2} } \right)} \mathord{\left/ {\vphantom {{ \, \partial b\left( {{\text{CCrate}},k_{2} } \right)} {\partial {\text{CCrate}} = }}} \right. \kern-\nulldelimiterspace} {\partial {\text{CCrate}} = }}\frac{{ - EA \cdot 2 \cdot \left( {\frac{2\pi }{60}} \right)^{2} {\text{CCrate}} - 2 \cdot B\left( {\frac{{\mu_{1} x_{1,\max } }}{2}} \right)\left( {\frac{2\pi }{60}} \right)^{2} {\text{CCrate}}}}{{B^{2} \left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} + \left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right]^{2} }} \hfill \\ - \left[ {2B^{2} \left( {\frac{2\pi }{60}} \right)^{2} {\text{CCrate}} + 2\left( {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right) \cdot \left( { - 2A} \right)\left( {\frac{2\pi }{60}} \right)^{2} {\text{CCrate}}} \right] \hfill \\ \cdot \frac{{E\left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right] - B\left( {\frac{{\mu_{1} x_{1,\max } }}{2}} \right)\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} }}{{\left( {B^{2} \left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} + \left[ {k_{1} + k_{2} - A\left( {\frac{{2\pi {\text{CCrate}}}}{60}} \right)^{2} } \right]^{2} } \right)^{2} }} \hfill \\ \end{gathered} $$
(21)

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Dellimore, K.H.J., Cloete, G. & Scheffer, C. Towards optimum chest compression performance during constant peak displacement cardiopulmonary resuscitation. Med Biol Eng Comput 49, 1057–1065 (2011). https://doi.org/10.1007/s11517-011-0812-5

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