// MODEL NUMBER: 0435
//
// TITLE: Transdermal ethanol transport, Ethanol diffusion through the skin
// By:  Joseph C. Anderson
//
// Describes the diffusion of ethanol from the skin capillaries, through the epidermis 
// and stratum corneum and into the supradermal space of the detector.  Ethanol enters
// the capillary bed at a rate Qdot (ml/s) and blood alcohol concentration (BAC(t)).
//

JSim v1.1  
import nsrunit;
unit dL=0.1 L;
unit conversion on;  

math Skin_Diff  {

realDomain t hour; t.min=0; t.max=10.0; t.delta=0.005;


// -------------  PARAMETERS  ----------------------------

real BAC = 0.1 g/dL,		// Maximum blood alcohol concentration
     Tpeak = 2 hr,		// Time to peak ethanol BAC
     BOR = 0.018 g/(dL*hr);	// Elimination rate of ethanol 
 
real Bbld = 232 mmHg^(-1),	// Ethanol solubility in blood
     Bepi = 232	mmHg^(-1),	// Ethanol solubility in epidermis
     Bstc = 211	mmHg^(-1);	// Ethanol solubility in stratum corneum

real Depi = 5E-6 cm^2/s, 	// Diffusivity through epidermis
     Dstc = 5E-10 cm^2/s; 	// Diffusivity through epidermis

real Lepi = 0.02 cm,		// Thickness of epidermis layer
     Lstc = 0.0015 cm,		// Thickness of stratum corneum layer
     Lcap = 7E-4 cm,		// Thickness of capillary vessel
     Lgas = 0.5 cm;		// Thickness of gas layer

real Acap = 7.5E-2 cm^2,	// Capillary surface area
     Area = 1 cm^2;		// Surface area

real Qdot = 4E-4 cm^3/s,	// Capillary blood flow
     Vdot = 5E-5 cm^3/s;	// Convective gas flow

real Pin = 0 mmHg;		// EtOH partial pressure entering gas comp.


// -------------  FIXED PARAMETERS  -----------------------

private real 
    Bgas = 0.132 mmHg^(-1),	// Ethanol solubility in gas phase
    Rgc=62.363 L*mmHg/(mol*K), // Universal gas constant
    Tbody = 37+273.15 K,	// Body temperature
    MW = 46.07 g/mol;		// Molecular weight of ethanol


// -------------  VARIABLES  ------------------------------

real kes cm/(s*mmHg);		// Epi:SC Mass transfer coeffcient
real Pmax mmHg;			// Partial pressure equivalent of maximum BAC
real Tfinal hr;			// Time when Part=0

real Part(t) mmHg,		// EtOH partial pressure in artery
     Pcap(t) mmHg,		// EtOH partial pressure in capillary
     Pepi(t) mmHg,		// EtOH partial pressure in epithelium
     Pstc(t) mmHg,		// EtOH partial pressure in stratum corneum
     Pgas(t) mmHg;		// EtOH partial pressure in gas compartment
     
real Cart(t) g/dL,		// Blood alcohol concentration profile
     Cgas(t) g/dL;		// Equivalent BAC in gas phase

// ----------  STATE EQUATIONS & BAC PROFILE  ---------------

 Pmax=(Rgc*Tbody/MW)*(Bgas/Bbld)*BAC;
 kes=1/(Lepi/(2*Depi*Bepi)+Lstc/(2*Dstc*Bstc));
 Tfinal = Tpeak+BAC/BOR;

 Part = if (t<=Tpeak) Pmax*t/(Tpeak) else
	(if (t>Tpeak and t<Tfinal)
	  Pmax*(Tfinal-t)/(Tfinal-Tpeak) else 0);

 Cart = (Bbld/Bgas)*MW/(Tbody*Rgc)*Part;
 Cgas = (Bbld/Bgas)*MW/(Tbody*Rgc)*Pgas;

// --------------  INITIAL CONDITIONS ---------------------

 when (t=t.min) { Pcap = 0; Pepi=0; Pstc=0;Pgas=0;}

// --------------  MODEL EQUATIONS (ODE) ------------------

 Pcap:t=Qdot/(Acap*Lcap)*(Part-Pcap)-(Depi*Bepi)/(0.5*Lepi*Bbld*Lcap)*(Pcap-Pepi);
 Pepi:t=(Depi*Acap)/(0.5*Lepi^2*Area)*(Pcap-Pepi)-kes/(Bepi*Lepi)*(Pepi-Pstc);
 Pstc:t=kes/(Bstc*Lstc)*(Pepi-Pstc)-Dstc/(0.5*Lstc^2)*(Pstc-Pgas);
 Pgas:t=Vdot/(Area*Lgas)*(Pin-Pgas)+(Dstc*Bstc)/(0.5*Lstc*Bgas*Lgas)*(Pstc-Pgas);


}

/*


 DETAILED DESCRIPTION:
  The kinetics of ethanol transport from the blood to the skin surface are
incompletely understood. We present a mathematical model to predict the
transient exchange of ethanol across the skin while it is being absorbed
from the gut and eliminated from the body. The model simulates the
behavior of a commercial device that is used to estimate the blood alcohol
concentration (BAC). During the elimination phase, the stratum corneum
of the skin has a higher ethanol concentration than the blood. We studied
the effect of varying the maximum BAC and the absorption rate from the
gut on the relationship between BAC and equivalent concentration in the
gas phase above the skin. The results showed that the ethanol concentration 
in the gas compartment always took longer to reach its maximum,
had a lower maximum, and had a slower apparent elimination rate than
the BAC. These effects increased as the maximum BAC increased. Our
model’s predictions are consistent with experimental data from the
literature. We performed a sensitivity analysis (using Latin hypercube
sampling) to identify and rank the importance of parameters. The analysis
showed that outputs were sensitive to solubility and diffusivity within the
stratum corneum, to stratum corneum thickness, and to the volume of gas
in the sampling chamber above the skin. We conclude that ethanol
transport through the skin is primarily governed by the washin and
washout of ethanol through the stratum corneum. The dynamics can be
highly variable from subject to subject because of variability in the
physical properties of the stratum corneum.

 SHORTCOMINGS/GENERAL COMMENTS:
	- Specific inadequacies or next level steps
 
 KEY WORDS: diffusion, mathematical modeling, blood alcohol concentration, 
  skin alcohol, fuel cell.  

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 REVISION HISTORY:
	Original Author : Joe Anderson, clarkja@u.washington.edu  Date: 01/July/24
	Revised by: BEJ Date:10july24 : Add metadata( model description, refs,etc)
	

 COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:   
  Copyright (C) 1999-2022 University of Washington. From the National Simulation Resource,  
  Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061. 
  Academic use is unrestricted. Software may be copied so long as this copyright notice is included.

  When citing JSim please use this reference: Butterworth E, Jardine BE, Raymond GM, Neal ML, Bassingthwaighte JB. 
  JSim, an open-source modeling system for data analysis [v3; ref status: indexed, http://f1000r.es/3n0] 
  F1000Research 2014, 2:288 (doi: 10.12688/f1000research.2-288.v3)  

  JSim software was developed with support from NIH grants HL088516 and HL073598, NIBIB grant BE08417,
  the Cardiac Energy Grid HL122199 (PI: J.B. Bassingthwaighte), and the Virtual Physiological Rat program 
  GM094503 (PI: D.A.Beard). Please cite these grants in any publication for which this software is used and 
  send an email with the citation and, if possible, a PDF file of the paper to: staff@physiome.org. 

*/