## The Ca Model and Ca Buffer Pool

The model is again depicted in figure 2 where another compartment, termed a Ca 'buffer pool', is shown. This buffer pool is defined as a source of very rapid Ca mobilization (M+Ca) into extracellular fluid (ECW) or sequestration (M-Ca) beyond ECW during dialysis when there is a concentration gradient between plasma and dialysate. As shown quantitatively below, such a Ca pool rapidly buffering changes in plasma concentration was found to be required mathematically to close mass balance of clinical data with the model. It must be emphasized that only the intradialytic portion of the dialysis cycle is considered in the following and no attempt has been made here to close mass balance over the complete cycle which will include accumulation or depletion during the interdialytic phase of the cycle. Good estimates of Ca2+ balance between dialyses will be required to optimize prescription writing.

Mass balance of Ca2+ during a dialysis treatment is defined as change in Ca content of the single well-mixed Ca distribution volume, A(CpCa2+)*(VCa2+) + MCa during dialysis as shown in figure 2. Thus we can write:

DCa2+((1 - Qf/Qe) + Qf)*CpiCa2+ + MCa dVCa2+/dt = - Qf (7)

where Qf is the ultrafiltration rate during dialysis. Solution of equations 6 and 7 over one dialysis for the end dialysis concentration results in:

CptCa2+ = CdiCa2+ - (CdiCa2+ - CpoCa2+)*((VtCa2+/VoCa2+)ADCa2+ (1/Qf-1/Qe))+MCa/DCa2+(1-Qf/Qe)*(1-((VtCa2+/VoCa2+)ADCa2+

where Cpi Ca = end dialysis plasma Ca, Cdi Ca = dialysate inlet Ca, CpoCa = predialysis Ca, VtCa and VoCa are VCa post- and pre-dialysis, Qf is the ultrafiltration rate and Qe is the effective blood flow rate equal to plasma volume flow rate; and the term MCa represents sequestration (M-Ca) or mobilization of Ca2+ (M+Ca) into VCa2+ from the CaBP induced during dialysis as discussed above.

A direct analytic solution for MCa can be obtained from simple rearrangement of equation 8 to give:

Cpt, - Cdi + (Cdi - Cpo) ((Vt/Vo) (1/Qf - 1/Qe))(DCa (l ~Qf/Qe)) 1-(Vt/Vo)A(a (1/Qf-1/Qe))

If serial values are available for Cpi Ca, Cdi Ca, VCa and Q d, mass balance can be evaluated from calculated serial values for change in content of Ca 2+ in its distribution volume plus mobilization of Ca, i.e., A(CpiCa*VtCa) + (MCa*t) and compared to serial measurements of dialysate content, (CdiCa - CdoCa)*QdT. In this way the inter nal consistency of plasma and dialysate concentrations and the validity of the kinetic model can be evaluated from the mass balance: if we are accounting for all Ca2+ removed from the body it should equal that recovered in dialysate, i.e.:

A(CpiCa*VCa) - MCa*t = (CdiCa - Cdc>Ca)*QdT (10)

recalling that the MCa is negative if Ca2+ is sequestered in and positive if it is removed from the CaBP so that mass balance in the body is change in body content minus total MCa.

If only serial plasma concentrations are available, total Ca removal from only its distribution volume can be calculated from:

over the total dialysis.

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