Biopharma

% One-compartment model, 500 mg every 8 hours (sum of Bateman curves)
ka = 1.2; ke = 0.25;                     % absorption and elimination rates (1/h)
D = 500; V = 40; F = 0.8;                % dose (mg), volume (L), bioavailability
tt = linspace(0, 72, 1500);
C = zeros(size(tt));
for td = 0:8:64
  d = tt - td;
  C = C + (d > 0) .* (F*D*ka / (V*(ka - ke))) .* (exp(-ke*d) - exp(-ka*d));
end
fill([0 72 72 0], [4 4 10 10], [0.55 0.8 0.55], 'FaceAlpha', 0.2)
hold on
plot(tt, C, 'LineWidth', 2)
yline(4, 'k:', 'minimum effective')
yline(10, 'r:', 'toxic')
hold off
xlabel('time (h)'), ylabel('plasma concentration (mg/L)')
title('Repeated dosing settles into the therapeutic window')

c = logspace(-3, 2, 300);                 % concentration (µM)
EC50 = 0.5; Emax = 100;
for nH = [0.5 1 2 4]
  E = Emax * c.^nH ./ (EC50^nH + c.^nH);
  semilogx(c, E, 'LineWidth', 2, 'DisplayName', sprintf('Hill coefficient %g', nH))
  hold on
end
xline(EC50, 'k--', 'EC50')
yline(50, 'k:')
hold off
legend show
xlabel('concentration (µM)'), ylabel('response (% of maximum)')
title('Emax model: steeper curves for higher cooperativity')

rng(4)
Vmax = 2.4; Km = 1.8;                     % true values
S = [0.25 0.5 1 2 4 8 16];               % substrate (mM)
v = Vmax*S ./ (Km + S) .* (1 + 0.04*randn(size(S)));

p = polyfit(1./S, 1./v, 1);               % Lineweaver–Burk: 1/v = (Km/Vmax)(1/S) + 1/Vmax
VmaxFit = 1/p(2); KmFit = p(1)*VmaxFit;

subplot(1, 2, 1)
scatter(S, v, 50, 'filled'), hold on
s = linspace(0, 16, 200);
plot(s, VmaxFit*s ./ (KmFit + s), 'LineWidth', 2), hold off
xlabel('[S] (mM)'), ylabel('v (µmol/min)')
title(sprintf('Vmax = %.2f,  Km = %.2f', VmaxFit, KmFit))

subplot(1, 2, 2)
scatter(1./S, 1./v, 50, 'filled'), hold on
x = linspace(0, 4.2, 50);
plot(x, polyval(p, x), 'LineWidth', 2), hold off
xlabel('1/[S]'), ylabel('1/v')
title('Lineweaver–Burk plot')