Hi med, I think what you want to do is compare the linear trends in pulse rate over time between the two drugs. I think doing this will be most straightforward in a repeated measures ANOVA framework. To do this, first compute two linear contrasts for each person, lin_A being the linear trend for Drug A and lin_B being the linear trend for Drug B, as:
lin_A = -4.5*t1_A - 3.5*t2_A - 2.5*t3_A - 1.5*t4_A - 0.5*t5_A + 0.5*t6_A + 1.5*t7_A + 2.5*t8_A + 3.5*t9_A + 4.5*t10_A
lin_B = -4.5*t1_B - 3.5*t2_B - 2.5*t3_B - 1.5*t4_B - 0.5*t5_B + 0.5*t6_B + 1.5*t7_B + 2.5*t8_B + 3.5*t9_B + 4.5*t10_B
Where ti_X is the ith pulse measurement in time under drug X (A or B). Given your balanced design, these lin scores will be equivalent (up to a multiplicative constant) to the regression coefficients you would get by running separate within-participant regressions on the centered time interval variable for both drugs for each person.
Second, compute a difference score between these two lin scores for each person, e.g., lin_diff = lin_A - lin_B.
Finally, the actual analysis consists simply of a one-sample t-test examining whether these lin_diff scores are significantly different from 0 on average. The procedure I have outlined is the piecemeal version of a repeated-measures ANOVA.