Hemodialysis Hollow-Fiber Capsule Volume Maximization (Biomedical Engineering)

In biomedical engineering and artificial organ design, hemodialysis dialyzers (artificial kidneys) purify blood for patients with end-stage renal disease. A dialyzer consists of a cylindrical polycarbonate housing packed with thousands of semipermeable hollow fibers. Blood flows inside the fibers while cleansing dialysate fluid circulates through the surrounding shell space.

To prevent acute clinical hypotension and reduce patient circulatory stress, clinical protocols place a strict ceiling on the total external surface area of the dialyzer capsule, limiting polymer shell contact and priming surface to exactly $S_0 = 150\pi\text{ cm}^2$.

                        Blood Inlet Header Plate
                           +-----------------+
                          /                   \
                         +---------------------+
                         |                     |
                         |  Hollow Membrane    |
                         |  Fibers & Dialysate |  Length h
                         |                     |
                         +---------------------+
                          \                   /
                           +-----------------+
                        Blood Outlet Header Plate
                                Radius r

The dialyzer housing is modeled as a closed right circular cylinder with radius $r$ and barrel length $h$, including its two circular end-manifold caps.

1. State the objective function for total dialyzer internal volume $V(r, h)$ and the surface area constraint equation.
2. Use the Applied Optimization Modeling Framework to eliminate $h$ and express the internal dialysate volume as a single-variable function $V(r)$.
3. Determine the Context-Driven Domain for radius $r$, noting both the non-negativity of radius and the physical constraint on length $h \ge 0$.
4. Apply the Closed Interval Method algorithm (guaranteed by the Extreme Value Theorem) to determine the optimal radius $r$ and length $h$ that maximize internal filtration volume.
5. Compute the maximum internal volume and verify whether the optimal geometry satisfies the general $h = 2r$ aspect ratio theorem from Optimal Aspect Ratios in Geometric Containers.