In silico ordinary differential equation/partial differential equation hemodialysis model estimates methadone removal during dialysis

Figures & data

Table 1 Patient demographics

Patient	Age (years)	BMI	Sex	Cause of pain
1	70	20.2	F	Colon cancer
2	66	21.1	M	Chronic low back pain
3	84	19.5	F	Osteoarthritis
4	81	24.6	M	Chronic low back pain
5	84	25.3	M	Chronic low back pain
6	60	32.6	M	Peripheral vascular disease
7	70	29	M	Polyneuropathy
8	81	24.6	M	Chronic low back pain
9	41	46.8	F	SLE myopathy/neuropathy
10	71	26.6	F	Rhabdomyosarcoma
11	84	25.3	M	Chronic low back pain

Abbreviations: BMI, body mass index; F, female; M, male; SLE, systemic lupus erythematosus.

Figure 1 ODE/PDE model of one-dimensional (1-D) counter current hemodialyzer with distance z of the spatial boundary independent variable along the dialyzer.

Notes: Time t is the temporal initial value independent variable. The two PDE dependent variables, u₁(z, t) and u₂(z, t), represent normalized methadone concentrations in the blood and dialysate, respectively, Z_M stands for the length of the dialyzer membrane excluding the headers. In this model, blood enters the left end of the dialyzer at a normalized methadone blood concentration of u_1L(t). V_1L is the header flow volume at the left end. The exiting normalized methadone blood concentration at the right end is u_1R(t). V_1R is the header flow volume at the right end. The entering and exiting dialysate concentrations are u₂(z = z_M, t) and u₂(z = 0, t). The header is the space enclosed by the end cap of the dialyzer and the polyurethane potting material that holds the hollow fibers together (inset).
Abbreviations: ODE, ordinary differential equation; PDE, partial differential equation.

Table 2 Glossary of symbols and units

Symbolism	Interpretation	Unitsa
u1	Plasma methadone concentrations	ng/mL
u2	Dialysate methadone concentrations	ng/mL
E	Extraction	%
T	Throughput	%
VlL, V1R	Left and right header flow volumes	cm^3
t	Time	min
z	Axial spatial position along dialyzer	cm
ZM	Dialyzer membrane length excluding headers	cm
A	Dialyzer cross-sectional area (transverse to z)	cm^2
ξ	Blood flow fraction of A	–
v1, v2	Blood and dialysate linear flow velocities, respectively	cm^3/min
AM	Area for mass transfer per unit length in z	cm^2/cm
km	Mass transfer coefficient	cm/s
D	Dialyzer diameter	cm
D	Methadone’s diffusivity	cm^2/s
jME	Methadone’s intradialytic removal rate	mg/kg/h
q1, q2	Blood and dialysate volumetric flow rates, respectively	cm^3/min
u1L	Normalized inflow blood methadone concentration	Dimensionless
u1R	Normalized outflow blood methadone concentration	Dimensionless
v1L, v2R	Header volumes	cm
zM	Dialyzer length excluding headers	cm
ξ	Fraction of A available for blood flow (0 ≤ ξ ≤ 1)	Dimensionless
VlL	Volume on the left	cm^3
V1R	Volume on the right	cm^3

Notes:

a units: 1 mcg = 1,000 ng and 1 L = 1,000 mL; converting mL to L gives 1 mcg/L/1 L = 1,000 ng/1,000 mL, so that 1 mcg/L = 1 ng/mL; 1 mm = 0.1 cm; 1 h = 3,600 s; 1 mcg = 0.001 mg.

Abbreviations: h, hours; s, seconds; min, minutes.

Table 3 Experimentally measured plasma methadone concentrations versus ODE/PDE model predictionsa,b

	Measured	Model	Error (%)
	N=11	N=11
Inflow	46.14±14.07	46.13±14.07	0.36±0.15
Outflow	36.36±10.79	36.33±10.78	1.85±4.43
E	19.84±3.01	20.52±2.92	0.61±0.49
T	80.16±3.01	79.58±2.89	0.68±0.60

Notes:

a Values are mean ± SE; inflow and outflow are ng/mL (equal to μg/L); E is extraction and T is throughput in percent.

b Measured versus model all P>0.05 by dependent 2-group Wilcoxon signed rank test.

Abbreviations: ODE, ordinary differential equation; PDE, partial differential equation; SE, standard error.

Figure 2 One-dimensional (1-D) ODE/PDE model’s perfect reference standard response with zero mass transfer of methadone into dialysate.

Abbreviations: ODE, ordinary differential equation; PDE, partial differential equation.

Figure 3 One-dimensional (1-D) ODE/PDE model’s response with estimated mass transfers of methadone into dialysate.

Note: Although both solutions have the same normalized inflow values, their normalized outflow values are significantly different due to methadone’s extraction.
Abbreviations: ODE, ordinary differential equation; PDE, partial differential equation.

Figure 4 One-dimensional (1-D) ODE/PDE model’s normalized methadone’s response to its removal via the dialysate at low dialysate flow rates of 250 mL/min and high dialysate flow rates of 800 mL/min.

Note: Significant increase in methadone’s mass transferred across the dialyzer membrane with the high dialysate flow rate.
Abbreviations: ODE, ordinary differential equation; PDE, partial differential equation.